A new formulation of Hermite–Hadamard–Fejer inequalities connected with an extended fractional operator

Abstract

In this work, we classify the extensions of Hermite–Hadamard(H–H)–Fejer-type inequalities for the fractional operators involving nonlinear kernel. By utilizing these inequalities, we develop many kinds of fractional integral (FI) inequalities. By considering the limiting cases of our main results, we attain the inequalities that already exist in the literature. In our work, we calculate the bounds of well-known fractional problems involving extended fractional operators. As implementations of the proved results, we calculate the midpoint-type inequalities. In the last section as the application of our defined operator, we present a generalized Abel integral equation and compute its solution. Also, we define the nonlinear form of a weakly singular Volterra-type integral equation and investigate its solution. These results might be useful in the investigation of the uniqueness of mathematical models and applied problems.

Introduction and preliminaries

The study of fractional calculus and its widespread applications has recently been paid more and more attention. If we outline the history of this theory, Abel was the first to apply fractional calculus by solving the Tautocrone problem [1]. For further advancements in fractional calculus, researchers published their own books [2, 3]. These books play an important role in the applications and achievements of fractional calculus in mathematical modeling and applied analysis. In [4], Caputo et al. developed the well-known Caputo sense derivative without a nonsingular kernel. However, researchers have still found many gaps in the theory. To overcome these gaps, many researchers have developed their own fractional operators with nonsingular kernel [5–8]. Furthermore, in [9], Wu et al. proved the nonlinear and nonsingular extensions of fractional operators and their symmetric properties. In [10], Samraiz et al. developed the fractional derivative with nonlinear and nonsingular kernel and discussed its applications in applied analysis. Continuing in this way, a fractional operator involving different kinds of Mittag–Leffler function as a kernel developed in [11, 12]. For more applications and developments on fractional calculus, we refer the readers to [13–18]. Roman et al. described the fractional derivatives and their applications in random walks, time series with long-term memory in [19]. Moreles et al. presented mathematical modeling of fractional-order circuit components and applications of bioimpedance in [20]. Melo et al. gave the application of fractional calculus to the modeling of dielectric relaxation phenomena in polymeric materials in [21]. A fractional integral with application in shares with wave theory are presented in [22]. Convex functions (Cf) and their extensions s-Cf are very close to the theory of inequalities in [23, 24]. Chen et al. presented H–H and H–H–Fejer-type inequalities for generalized FI in [25]. Khan et al. generalized the conformable fractional operators and a new class of fractional operators in [26, 27]. When discussing recent advancements in fractional inequalities, several notable contributions stand out. In [28], the authors investigate parameterized inequalities for s-Cf in the fourth sense using Katugampola fractional integrals, offering novel insights into fractional-calculus applications for convex-function analysis. In [29], they present fractional Ostrowski-type inequalities for $(s,m)$-Cf, accompanied by practical applications that deepen the understanding of fractional inequalities in mathematical analysis. Moving to discrete settings, [30] introduces discrete versions of H–H inequalities for $(s,m)$-Cf, enriching the study of fractional calculus in discrete frameworks. Additionally, in [31], the authors derive H–H–Fejer inequalities via Katugampola fractional integrals for s-Cf in the second sense, providing new analytical tools for fractional equation analysis. Hakiki et al. described H–H–Fejer inequalities involving second kind of s-Cf via Riemann–Liouville (RL) FI and novel inequalities of H–H–Fejer kind for Cf via FI in [32, 33].

Definition 1.1

[23] A function $\mathrm{\Upsilon }:[a,b]\mapsto \mathbb{R}$ satisfying the following expression known as Cf: $\mathrm{\Upsilon }({\lambda }_{1}x+(1-{\lambda }_1)y)\le {\lambda }_1\mathrm{\Upsilon }(x)+(1-{\lambda }_1)\mathrm{\Upsilon }(y),$$\mathrm{\forall }x,y\in [a,b]$ and ${\lambda }_1\in (0,1]$.

In [24], Brechner gives the definition of s-Cf, stated by the following definition.

Definition 1.2

A function $\mathrm{\Upsilon }:(0,\mathrm{\infty })\mapsto \mathbb{R}$ satisfying the below expression is called second sense s-Cf for some fixed $sϵ(0,1)$: $\mathrm{\Upsilon }({\lambda }_{1}x+{\lambda }_{2}y)\le {\lambda }_1^s\mathrm{\Upsilon }(x)+{\lambda }_2^s\mathrm{\Upsilon }(y),$$\mathrm{\forall }x,y\in (0,\mathrm{\infty }),{\lambda }_1,{\lambda }_2>0$, where ${\lambda }_1+{\lambda }_2=1$. This class of function is denoted as $K_s^2$.

Definition 1.3

[34] The incomplete beta function $B_x({\lambda }_1,{\lambda }_2)$ is defined by $B_x({\lambda }_1,{\lambda }_2)={\int }_0^x{y}^{{\lambda }_1-1}{(1-y)}^{{\lambda }_2-1}dy,$ where $0<x<1,{\lambda }_1,{\lambda }_2>0$.

Definition 1.4

[35] Let $p,q\in R$ with $p,q>1$ and $\frac{1}{p}+\frac{1}{q}=1$, then the integral form of the Hölder inequality is presented as $\underset{a}{\overset{e}{\int }}|f(ȷ)g(ȷ)|dȷ\le {(\underset{a}{\overset{e}{\int }}|f(ȷ){|}^{p}dȷ)}^{\frac{1}{p}}{(\underset{a}{\overset{e}{\int }}|g(ȷ){|}^{q}dȷ)}^{\frac{1}{q}},$ with $f,g\in {\mathbb{C}}^1[a,e]$.

Definition 1.5

The left- and right-sided RLFI $I_{{\lambda }_1^{+}}^{\varrho }(\mathrm{\Upsilon })$ and $I_{{\lambda }_2^{-}}^{\varrho }(\mathrm{\Upsilon })$, of order $\varrho >0$ on $[{\lambda }_1,{\lambda }_2]$ are defined as $I_{{\lambda }_1^{+}}^{\varrho }\mathrm{\Upsilon }(p_1)=\frac{1}{\mathrm{\Gamma }(\varrho )}{\int }_{{\lambda }_1}^{p_1}{(p_1-\nu )}^{\varrho -1}\mathrm{\Upsilon }(\nu )d\nu ,p_1>{\lambda }_1$ and $I_{{\lambda }_2^{-}}^{\varrho }\mathrm{\Upsilon }(p_1)=\frac{1}{\mathrm{\Gamma }(\varrho )}{\int }_{p_1}^{{\lambda }_2}{(\nu -p_1)}^{\varrho -1}\mathrm{\Upsilon }(\nu )d\nu ,p_1<{\lambda }_2,$ respectively. Here, Γ represents the Gamma function and its integral representation is given below: $\mathrm{\Gamma }(p_1)={\int }_0^{\mathrm{\infty }}{\nu }^{p_1-1}{e}^{-\nu }d\nu ,R(p_1)>0.$

Now, we present the definition of the RLFI, which is defined in [36].

Definition 1.6

If a function f is continuous on $[a,b]$, then the $(k,s)$-RLF integral for order $\delta >0$ can be defined as

1.1

${}_k{}^s\mathfrak{J}_{a^{+}}^{\delta }f(t_1)=\frac{{(s+1)}^{1-\frac{\delta }{k}}}{k{\mathrm{\Gamma }}_k(\delta )}{\int }_{a^{+}}^{t_1}{(t_1^{s+1}-{\nu }^{s+1})}^{\frac{\delta }{k}-1}{\nu }^{s}f(\nu )d\nu .$

Now, we are going to present the generalized form of fractional operator (1.1).

Definition 1.7

If a function f is continuous on $[a,b]$, $s\in R/\{-1\}$, $k>0$ and Ω is an increasing function then the generalized $(k,s)$-RLF integrals for order $\delta >0$ can be defined as ${}_k{}^s\mathfrak{J}_{a^{+}}^{\delta }f(t_1)=\frac{{(s+1)}^{1-\frac{\delta }{k}}}{k{\mathrm{\Gamma }}_k(\delta )}{\int }_{a^{+}}^{t_1}{({\mathrm{\Omega }}^{s+1}(t_1)-{\mathrm{\Omega }}^{s+1}(\nu ))}^{\frac{\delta }{k}-1}{\mathrm{\Omega }}^s(\nu ){\mathrm{\Omega }}^{\prime }(\nu )f(\nu )d\nu .$

Definition 1.8

If a function f is continuous on $[a,b]$, $s\in R/\{-1\}$, $k>0$ and Ω is an increasing function then the left- and right-sided generalized $(k,s)$-RLF integrals for order $\delta >0$ can be defined as ${}_k{}^s\mathfrak{J}_{a^{+}}^{\delta }f(t_1)=\frac{{(s+1)}^{1-\frac{\delta }{k}}}{k{\mathrm{\Gamma }}_k(\delta )}{\int }_{a^{+}}^{t_1}{({\mathrm{\Omega }}^{s+1}(t_1)-{\mathrm{\Omega }}^{s+1}(\nu ))}^{\frac{\delta }{k}-1}{\mathrm{\Omega }}^s(\nu ){\mathrm{\Omega }}^{\prime }(\nu )f(\nu )d\nu ,t_1>a^{+}$ and ${}_k{}^s\mathfrak{J}_{{\zeta }^{-}}^{\delta }f(t_1)=\frac{{(s+1)}^{1-\frac{\delta }{k}}}{k{\mathrm{\Gamma }}_k(\delta )}{\int }_{t_1}^{\zeta }{({\mathrm{\Omega }}^{s+1}(\nu )-{\mathrm{\Omega }}^{s+1}(t_1))}^{\frac{\delta }{k}-1}{\mathrm{\Omega }}^s(\nu ){\mathrm{\Omega }}^{\prime }(\nu )f(\nu )d\nu ,t_1<\zeta .$

Remark 1.9

The above fractional operator generalizes the following operators.

1. If we take $\mathrm{\Omega }(x)=x$, then we obtain Definition 1.6.

2. If we take $k=1$, $s=0$ then we obtain a fractional integral concerning another function defined in [37].

3. If we take $k=1$, $s=0$, and $\mathrm{\Omega }(x)=\frac{x^v}{v}$, $v>0$, then we obtain the definition of the Katugumpola fractional integral given in [25].

4. If we take $k=1$, $s=0$, and $\mathrm{\Omega }(x)=\frac{x^{v+p}}{v+p}$, then we obtain the definition of the generalized conformable fractional integral given in [26].

5. If we take $k=1$, $s=0$, and $\mathrm{\Omega }(x)=x$, then we obtain Definition 1.5.

Theorem 1.10

If a functionfis continuous on$[a,b]$, $s\in R/\{-1\}$, $k\ge 0$and Ω is an increasing function then the left- and right-sided generalized$(k,s)$-RLF integrals for order$\delta >0$if we choose$f(t_1)={({\mathrm{\Omega }}^{s+1}(t_1)-{\mathrm{\Omega }}^{s+1}(\nu ))}^{\frac{\beta }{k}}$, then

1.2

${}_k{}^s\mathfrak{J}_{a^{+}}^{\delta }f(t_1)=\frac{{(s+1)}^{-\frac{\delta }{k}}}{{\mathrm{\Gamma }}_k(\delta )}\frac{{({\mathrm{\Omega }}^{s+1}(t_1)-{\mathrm{\Omega }}^{s+1}(a^{+}))}^{\frac{\delta +\beta }{k}}}{\delta +\beta }$and if we choose$f(t_1)={({\mathrm{\Omega }}^{s+1}(\nu )-{\mathrm{\Omega }}^{s+1}(t_1))}^{\frac{\beta }{k}}$, then

1.3

${}_k{}^s\mathfrak{J}_{{\zeta }^{-}}^{\delta }f(t_1)=-\frac{{(s+1)}^{-\frac{\delta }{k}}}{{\mathrm{\Gamma }}_k(\delta )}\frac{{({\mathrm{\Omega }}^{s+1}(\zeta )-{\mathrm{\Omega }}^{s+1}(t_1))}^{\frac{\delta +\beta }{k}}}{\delta +\beta }.$

Proof

By utilizing Definition 1.8, we have $\begin{array}{rl}& {}_k{}^s\mathfrak{J}_{a^{+}}^{\delta }f(t_1)=\frac{{(s+1)}^{1-\frac{\delta }{k}}}{k{\mathrm{\Gamma }}_k(\delta )}{\int }_{a^{+}}^{t_1}{({\mathrm{\Omega }}^{s+1}(t_1)-{\mathrm{\Omega }}^{s+1}(\nu ))}^{\frac{\delta }{k}-1}\\ & \times \left({({\mathrm{\Omega }}^{s+1}(t_1)-{\mathrm{\Omega }}^{s+1}(\nu ))}^{\frac{\beta }{k}}\right){\mathrm{\Omega }}^s(\nu ){\mathrm{\Omega }}^{\prime }(\nu )d\nu \\ & =\frac{{(s+1)}^{1-\frac{\delta }{k}}}{k{\mathrm{\Gamma }}_k(\delta )}{\int }_{a^{+}}^{t_1}{({\mathrm{\Omega }}^{s+1}(t_1)-{\mathrm{\Omega }}^{s+1}(\nu ))}^{\frac{\delta +\beta }{k}-1}{\mathrm{\Omega }}^s(\nu ){\mathrm{\Omega }}^{\prime }(\nu )d\nu \\ & =\frac{{(s+1)}^{-\frac{\delta }{k}}}{{\mathrm{\Gamma }}_k(\delta )}\frac{{({\mathrm{\Omega }}^{s+1}(t_1)-{\mathrm{\Omega }}^{s+1}(a^{+}))}^{\frac{\delta +\beta }{k}}}{\delta +\beta }\end{array}$ and $\begin{array}{rl}& {}_k{}^s\mathfrak{J}_{{\zeta }^{-}}^{\delta }f(t_1)=\frac{{(s+1)}^{1-\frac{\delta }{k}}}{k{\mathrm{\Gamma }}_k(\delta )}{\int }_{t_1}^{\zeta }{({\mathrm{\Omega }}^{s+1}(\nu )-{\mathrm{\Omega }}^{s+1}(t_1))}^{\frac{\delta }{k}-1}\\ & \times \left({({\mathrm{\Omega }}^{s+1}(\nu )-{\mathrm{\Omega }}^{s+1}(t_1))}^{\frac{\beta }{k}}\right){\mathrm{\Omega }}^s(\nu ){\mathrm{\Omega }}^{\prime }(\nu )d\nu \\ & =\frac{{(s+1)}^{1-\frac{\delta }{k}}}{k{\mathrm{\Gamma }}_k(\delta )}{\int }_{t_1}^{\zeta }{({\mathrm{\Omega }}^{s+1}(\nu )-{\mathrm{\Omega }}^{s+1}(t_1))}^{\frac{\delta +\beta }{k}-1}{\mathrm{\Omega }}^s(\nu ){\mathrm{\Omega }}^{\prime }(\nu )d\nu \\ & =-\frac{{(s+1)}^{-\frac{\delta }{k}}}{{\mathrm{\Gamma }}_k(\delta )}\frac{{({\mathrm{\Omega }}^{s+1}(\zeta )-{\mathrm{\Omega }}^{s+1}(t_1))}^{\frac{\delta +\beta }{k}}}{\delta +\beta }.\end{array}$ □

Example 1.11

If we choose $s=2$, $k=1$, $\beta =3$, $a^{+}=0$, and $\mathrm{\Omega }(x)=x$ and $\delta =0.2,0.5,0.8$ in (1.2) then we have

1.4

${}_k{}^s\mathfrak{J}_{a^{+}}^{\delta }f(t_1)=\frac{1}{3^{\delta }\mathrm{\Gamma }(\delta )}\frac{t_1^{3(\delta +3)}}{\delta +3}.$

Example 1.12

If we choose $s=2$, $k=1$, $\beta =3$, $a^{+}=0$, and $\mathrm{\Omega }(x)=ln(x+1)$ and $\delta =0.2,0.5,0.8$ in (1.2) then we have

1.5

${}_k{}^s\mathfrak{J}_{a^{+}}^{\delta }f(t_1)=\frac{1}{3^{\delta }\mathrm{\Gamma }(\delta )}\frac{{(ln(t_1+1))}^{3(\delta +3)}}{\delta +3}.$

Example 1.13

If we choose $s=2$, $k=1$, $\beta =3$, ${\zeta }^{-}=1$, and $\mathrm{\Omega }(x)=x$ and $\delta =0.2,0.5,0.8$ in (1.3) then we have

1.6

${}_k{}^s\mathfrak{J}_{{\zeta }^{-}}^{\delta }f(t_1)=-\frac{1}{3^{\delta }\mathrm{\Gamma }(\delta )}\frac{{(1-t_1)}^{3(\delta +3)}}{\delta +3}.$

Example 1.14

If we choose $s=2$, $k=1$, $\beta =3$, ${\zeta }^{-}=1$, and $\mathrm{\Omega }(x)=ln(x+1)$ and $\delta =0.2,0.5,0.8$ in (1.3) then we have

1.7

${}_k{}^s\mathfrak{J}_{{\zeta }^{-}}^{\delta }f(t_1)=-\frac{1}{3^{\delta }\mathrm{\Gamma }(\delta )}\frac{{(ln(\frac{2}{t_1+1}))}^{3(\delta +3)}}{\delta +3}.$

To explore the versatility of fractional operators, we investigated the behavior of both left and right sided fractional operators across various fractional orders and kernel types. Through the derivation of multiple examples and by fixing specific parameters, we analyzed the operators behavior using graphical representations. Figure 1 illustrates the behavior of the left-sided fractional operator for different fractional orders with a linear kernel. Figure 2 depicts the behavior of the left-sided fractional operator for varying fractional orders using a logarithmic kernel. Figure 3 shows the behavior of the right-sided fractional operator for different fractional orders with a linear kernel. Finally, Fig. 4 presents the behavior of the right-sided fractional operator for varying fractional orders with a logarithmic kernel.

[See PDF for image]

Figure 1

This presents the graphical representation of (1.4) corresponding to the choice $0\le \upsilon \le 1$

[See PDF for image]

Figure 2

This presents the graphical representation of (1.5) corresponding to the choice $0\le \upsilon \le 1$

[See PDF for image]

Figure 3

This represents the graphical representation of (1.6) corresponding to the choice $0\le \upsilon \le 1$

[See PDF for image]

Figure 4

This represents the graphical representation of (1.7) corresponding to the choice $0\le \upsilon \le 1$

Fejer-type inequalities involving $(k,s)$-fractional operators

In this section we develop H–H–Fejer-type inequalities for generalized Riemann–Liouville fractional operators. We will use the following notation ${}_k{}^m\mathfrak{J}_a^{\delta }f(t_1)$ instead of ${}_k{}^s\mathfrak{J}_a^{\delta }f(t_1)$ in the following results.

Lemma 2.1

Let$m\in R\setminus \{-1\}$, $k\ge 0$, and ϒ be a differentiable on$({\lambda }_1,{\lambda }_2)$with${\lambda }_1<{\lambda }_2$and$\chi :[{\lambda }_1,{\lambda }_2]\mapsto \mathbb{R}$be bounded. If${\mathrm{\Upsilon }}^{\prime },\chi \in L[{\lambda }_1,{\lambda }_2]$, then the following expression holds:

2.1

$\begin{array}{rl}& \mathrm{\Upsilon }\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi ({\lambda }_2)+{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi ({\lambda }_1)]\\ & -[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi \mathrm{\Upsilon }({\lambda }_2)+{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi \mathrm{\Upsilon }({\lambda }_1)]\\ & =\frac{{(m+1)}^{1-\frac{\varrho }{k}}}{k{\mathrm{\Gamma }}_k(\varrho )}{\int }_{{\lambda }_1}^{{\lambda }_2}\kappa (\nu ){\mathrm{\Upsilon }}^{\prime }(\nu )d\nu ,\end{array}$where$\kappa (t_1)=\{\begin{array}{c}{\int }_{{\lambda }_1}^{t_1}\frac{{\mathrm{\Omega }}^m(u){\mathrm{\Omega }}^{\prime }(u)\chi (u)du}{{({\mathrm{\Omega }}^{m+1}(u)-{\mathrm{\Omega }}^{m+1}({\lambda }_1))}^{1-\frac{\varrho }{k}}},t_1\in [{\lambda }_1,\frac{{\lambda }_1+{\lambda }_2}{2}],\\ {\int }_{{\lambda }_2}^{t_1}\frac{{\mathrm{\Omega }}^m(u){\mathrm{\Omega }}^{\prime }(u)\chi (u)du}{{({\mathrm{\Omega }}^{m+1}({\lambda }_2)-{\mathrm{\Omega }}^{m+1}(u))}^{1-\frac{\varrho }{k}}},t_1\in [\frac{{\lambda }_1+{\lambda }_2}{2},{\lambda }_2].\end{array}$

Proof

Consider $\begin{array}{rl}& I={\int }_{{\lambda }_1}^{{\lambda }_2}\kappa (\nu ){\mathrm{\Upsilon }}^{\prime }(\nu )d\nu \\ & ={\int }_{{\lambda }_1}^{\frac{{\lambda }_1+{\lambda }_2}{2}}\kappa (\nu ){\mathrm{\Upsilon }}^{\prime }(\nu )d\nu +{\int }_{\frac{{\lambda }_1+{\lambda }_2}{2}}^{{\lambda }_2}\kappa (\nu ){\mathrm{\Upsilon }}^{\prime }(\nu )d\nu \\ & =I_1+I_2.\end{array}$ Now, taking $I_1$ and after employing integration by parts

2.2

$\begin{array}{rl}& I_1={\int }_{{\lambda }_1}^{\frac{{\lambda }_1+{\lambda }_2}{2}}\kappa (\nu ){\mathrm{\Upsilon }}^{\prime }(\nu )d\nu \\ & =\kappa \left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)\mathrm{\Upsilon }\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)-{\int }_{{\lambda }_1}^{\frac{{\lambda }_1+{\lambda }_2}{2}}\frac{{\mathrm{\Omega }}^m(\nu ){\mathrm{\Omega }}^{\prime }(\nu )}{{[{\mathrm{\Omega }}^{m+1}(\nu )-{\mathrm{\Omega }}^{m+1}({\lambda }_1)]}^{1-\frac{\varrho }{k}}}\chi (\nu )\mathrm{\Upsilon }(\nu )d(\nu )\\ & =\mathrm{\Upsilon }\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right){\int }_{{\lambda }_1}^{\frac{{\lambda }_1+{\lambda }_2}{2}}\frac{{\mathrm{\Omega }}^m(u){\mathrm{\Omega }}^{\prime }(u)}{{[{\mathrm{\Omega }}^{m+1}(u)-{\mathrm{\Omega }}^{m+1}({\lambda }_1)]}^{1-\frac{\varrho }{k}}}\chi (u)d(u)\\ & -{\int }_{{\lambda }_1}^{\frac{{\lambda }_1+{\lambda }_2}{2}}\frac{{\mathrm{\Omega }}^m(\nu ){\mathrm{\Omega }}^{\prime }(\nu )}{{[{\mathrm{\Omega }}^{m+1}(\nu )-{\mathrm{\Omega }}^{m+1}({\lambda }_1)]}^{1-\frac{\varrho }{k}}}\chi (\nu )\mathrm{\Upsilon }(\nu )d(\nu )\\ & =\frac{k{\mathrm{\Gamma }}_k(\varrho )}{{(m+1)}^{1-\frac{\varrho }{k}}}[\mathrm{\Upsilon }{\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)}_k^m{\mathfrak{J}}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi ({\lambda }_1)-{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi \mathrm{\Upsilon }({\lambda }_1)].\end{array}$ By similar arguments, it can be verified that

2.3

$I_2=\frac{k{\mathrm{\Gamma }}_k(\varrho )}{{(m+1)}^{1-\frac{\varrho }{k}}}[\mathrm{\Upsilon }{\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)}_k^m{\mathfrak{J}}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi ({\lambda }_2)-{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi \mathrm{\Upsilon }({\lambda }_2)].$ Hence, by adding (2.2) and (2.3), we obtain $\begin{array}{rl}& \frac{{(m+1)}^{1-\frac{\varrho }{k}}}{k{\mathrm{\Gamma }}_k(\varrho )}{\int }_{{\lambda }_1}^{{\lambda }_2}\kappa (\nu ){\mathrm{\Upsilon }}^{\prime }(\nu )d\nu \\ & =\mathrm{\Upsilon }\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi ({\lambda }_2)+{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi ({\lambda }_1)]\\ & -[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi \mathrm{\Upsilon }({\lambda }_2)+{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi \mathrm{\Upsilon }({\lambda }_1)].\end{array}$ The required result is proved. □

Corollary 2.2

If we replace$\mathrm{\Omega }(u)=u$, $m=0$, and$k=1$in Lemma2.1then we have [[32], Lemma 4], $\begin{array}{rl}& \mathrm{\Upsilon }\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi ({\lambda }_2)+{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi ({\lambda }_1)]\\ & -[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi \mathrm{\Upsilon }({\lambda }_2)+{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi \mathrm{\Upsilon }({\lambda }_1)]\\ & =\frac{1}{\mathrm{\Gamma }(\varrho )}{\int }_{{\lambda }_1}^{{\lambda }_2}\kappa (\nu ){\mathrm{\Upsilon }}^{\prime }(\nu )d\nu ,\end{array}$where$\kappa (t_1)=\{\begin{array}{c}{\int }_{{\lambda }_1}^{t_1}\frac{\chi (u)du}{{(u-{\lambda }_1)}^{1-\varrho }},t_1\in [{\lambda }_1,\frac{{\lambda }_1+{\lambda }_2}{2}],\\ {\int }_{{\lambda }_2}^{t_1}\frac{\chi (u)du}{{({\lambda }_2-u)}^{1-\varrho }},t_1\in [\frac{{\lambda }_1+{\lambda }_2}{2},{\lambda }_2].\end{array}$

Theorem 2.3

Let ϒ be a Cf on$({\lambda }_1,{\lambda }_2)$with${\lambda }_1<{\lambda }_2$and$\chi :[{\lambda }_1,{\lambda }_2]\mapsto \mathbb{R}$be a nonnegative integrable on$[{\lambda }_1,{\lambda }_2]$, then the H–H–Fejer kind of inequality for the generalized RLFI holds:

2.4

$\begin{array}{rl}& Ϝ\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi ({\lambda }_2)+{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi ({\lambda }_1)]\\ & \le \mathrm{\Upsilon }(\frac{{\lambda }_1+{\lambda }_2}{2})[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi Ϝ({\lambda }_2)+{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi Ϝ({\lambda }_1)]\\ & \le \frac{Ϝ({\lambda }_1)+Ϝ({\lambda }_2)}{2}[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi ({\lambda }_2)+{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi ({\lambda }_1)],\end{array}$where$Ϝ(\nu )=\mathrm{\Upsilon }(\nu )+\tilde{\mathrm{\Upsilon }}(\nu )$, and$\tilde{\mathrm{\Upsilon }}(\nu )=\mathrm{\Upsilon }({\lambda }_1+{\lambda }_2-\nu )$.

Proof

As we know that ϒ is Cf for all $x_1,y_1\in [{\lambda }_1,{\lambda }_2]$, that is $\mathrm{\Upsilon }(\frac{x_1+y_1}{2})\le \frac{\mathrm{\Upsilon }(x_1)+\mathrm{\Upsilon }(y_1)}{2},$ this implies, for $\nu \in [0,1]$ and substituting $x_1=\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2,y_1=\frac{\nu }{2}{\lambda }_2+\frac{2-\nu }{2}{\lambda }_1$$2\mathrm{\Upsilon }(\frac{{\lambda }_1+{\lambda }_2}{2})\le \mathrm{\Upsilon }(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2)+\mathrm{\Upsilon }(\frac{\nu }{2}{\lambda }_2+\frac{2-\nu }{2}{\lambda }_1).$ By multiplying $\frac{{(m+1)}^{1-\frac{\varrho }{k}}({\lambda }_2-{\lambda }_1)}{2k{\mathrm{\Gamma }}_k(\varrho )}\frac{{\mathrm{\Omega }}^m(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2){\mathrm{\Omega }}^{\prime }(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2)\chi (\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2)}{{({\mathrm{\Omega }}^{m+1}({\lambda }_2)-{\mathrm{\Omega }}^{m+1}(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2))}^{1-\frac{\rho }{k}}}$ and after simplifying and taking integration on $[0,1]$ with respect to ν, we can have $\begin{array}{rl}& \frac{{(m+1)}^{1-\frac{\varrho }{k}}({\lambda }_2-{\lambda }_1)}{2k{\mathrm{\Gamma }}_k(\varrho )}\\ & \times Ϝ(\frac{{\lambda }_1+{\lambda }_2}{2}){\int }_0^1\frac{{\mathrm{\Omega }}^m(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2){\mathrm{\Omega }}^{\prime }(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2)\chi (\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2)}{{({\mathrm{\Omega }}^{m+1}({\lambda }_2)-{\mathrm{\Omega }}^{m+1}(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2))}^{1-\frac{\rho }{k}}}d\nu \\ & \le \frac{{(m+1)}^{1-\frac{\varrho }{k}}({\lambda }_2-{\lambda }_1)}{2k{\mathrm{\Gamma }}_k(\varrho )}{\int }_0^1\frac{{\mathrm{\Omega }}^m(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2){\mathrm{\Omega }}^{\prime }(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2)}{{({\mathrm{\Omega }}^{m+1}({\lambda }_2)-{\mathrm{\Omega }}^{m+1}(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2))}^{1-\frac{\rho }{k}}}\\ & \times \chi (\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2)\mathrm{\Upsilon }(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2)d\nu \\ & +\frac{{(m+1)}^{1-\frac{\varrho }{k}}({\lambda }_2-{\lambda }_1)}{2k{\mathrm{\Gamma }}_k(\varrho )}{\int }_0^1\frac{{\mathrm{\Omega }}^m(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2){\mathrm{\Omega }}^{\prime }(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2)}{{({\mathrm{\Omega }}^{m+1}({\lambda }_2)-{\mathrm{\Omega }}^{m+1}(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2))}^{1-\frac{\rho }{k}}}\\ & \times \chi (\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2)\mathrm{\Upsilon }(\frac{\nu }{2}{\lambda }_2+\frac{2-\nu }{2}{\lambda }_1)d\nu .\end{array}$ By replacing $u=\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2$, we arrive at $\begin{array}{rl}& Ϝ\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi ({\lambda }_2)\le \mathrm{\Upsilon }(\frac{{\lambda }_1+{\lambda }_2}{2}\left)\right[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi \mathrm{\Upsilon }({\lambda }_2)\\ & +{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi \tilde{\mathrm{\Upsilon }}({\lambda }_2)],\end{array}$ that is

2.5

$Ϝ\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi ({\lambda }_2)]\le \mathrm{\Upsilon }\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi Ϝ({\lambda }_2)].$ Continuing in the same way, if we multiply the following term on both sides $\frac{{(m+1)}^{1-\frac{\varrho }{k}}({\lambda }_2-{\lambda }_1)}{2k{\mathrm{\Gamma }}_k(\varrho )}\frac{{\mathrm{\Omega }}^m(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2){\mathrm{\Omega }}^{\prime }(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2)\chi (\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2)}{{{\mathrm{\Omega }}^{m+1}(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2)-{\mathrm{\Omega }}^{m+1}({\lambda }_1))}^{1-\frac{\rho }{k}}},$ then after simplifying and performing integration on $[0,1]$ with respect to ν, we arrive at

2.6

$Ϝ\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi ({\lambda }_1)]\le \mathrm{\Upsilon }\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi Ϝ({\lambda }_1)].$ Adding (2.5) and (2.6), we arrive at

2.7

$\begin{array}{rl}& Ϝ\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi ({\lambda }_2)+{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi ({\lambda }_1)]\\ & \le \mathrm{\Upsilon }\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi Ϝ({\lambda }_2)+{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi Ϝ({\lambda }_1)].\end{array}$ This shows the first half of the required inequality. Now, we are going to prove the second half of the inequality. Since we know that ϒ is Cf on $[{\lambda }_1,{\lambda }_2]$, after implementation, we arrive at $\mathrm{\Upsilon }(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2)+\mathrm{\Upsilon }(\frac{2-\nu }{2}{\lambda }_1+\frac{\nu }{2}{\lambda }_2)\le \mathrm{\Upsilon }({\lambda }_1)+\mathrm{\Upsilon }({\lambda }_2).$ Continuing in the same way, if we multiply the following term on both sides as in the first part of the inequality $\frac{{(m+1)}^{1-\frac{\varrho }{k}}({\lambda }_2-{\lambda }_1)}{2k{\mathrm{\Gamma }}_k(\varrho )}\frac{{\mathrm{\Omega }}^m(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2){\mathrm{\Omega }}^{\prime }(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2)\chi (\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2)}{{({\mathrm{\Omega }}^{m+1}({\lambda }_2)-{\mathrm{\Omega }}^{m+1}(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2))}^{1-\frac{\rho }{k}}},$ then after simplifying and integrating over $[0,1]$ with respect to ν, we arrive at $\begin{array}{rl}& \frac{{(m+1)}^{1-\frac{\varrho }{k}}({\lambda }_2-{\lambda }_1)}{2k{\mathrm{\Gamma }}_k(\varrho )}{\int }_0^1\frac{{\mathrm{\Omega }}^m(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2){\mathrm{\Omega }}^{\prime }(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2)}{{({\mathrm{\Omega }}^{m+1}({\lambda }_2)-{\mathrm{\Omega }}^{m+1}(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2))}^{1-\frac{\rho }{k}}}\\ & \times \chi (\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2)\mathrm{\Upsilon }(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2)d\nu \\ & +\frac{{(m+1)}^{1-\frac{\varrho }{k}}({\lambda }_2-{\lambda }_1)}{2k{\mathrm{\Gamma }}_k(\varrho )}{\int }_0^1\frac{{\mathrm{\Omega }}^m(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2){\mathrm{\Omega }}^{\prime }(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2)}{{({\mathrm{\Omega }}^{m+1}({\lambda }_2)-{\mathrm{\Omega }}^{m+1}(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2))}^{1-\frac{\rho }{k}}}\\ & \times \chi (\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2)\mathrm{\Upsilon }(\frac{2-\nu }{2}{\lambda }_1+\frac{\nu }{2}{\lambda }_2)d\nu \\ & \le [\mathrm{\Upsilon }({\lambda }_1)+\mathrm{\Upsilon }({\lambda }_2)]\frac{{(m+1)}^{1-\frac{\varrho }{k}}({\lambda }_2-{\lambda }_1)}{2k{\mathrm{\Gamma }}_k(\varrho )}\\ & \times {\int }_0^1\frac{{\mathrm{\Omega }}^m(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2){\mathrm{\Omega }}^{\prime }(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2)\chi (\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2)}{{({\mathrm{\Omega }}^{m+1}({\lambda }_2)-{\mathrm{\Omega }}^{m+1}(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2))}^{1-\frac{\rho }{k}}}d\nu .\end{array}$ By replacing $u=\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2$, we obtain $\begin{array}{rl}& \mathrm{\Upsilon }\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi \mathrm{\Upsilon }({\lambda }_2)+{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi \tilde{\mathrm{\Upsilon }}({\lambda }_2)]\\ & \le [\mathrm{\Upsilon }({\lambda }_1)+\mathrm{\Upsilon }({\lambda }_2)]\mathrm{\Upsilon }\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi ({\lambda }_2).\end{array}$ This implies

2.8

$\begin{array}{rl}& \mathrm{\Upsilon }\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi Ϝ({\lambda }_2)\le \frac{Ϝ({\lambda }_1)+Ϝ({\lambda }_2)}{2}\\ & \mathrm{\Upsilon }\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi ({\lambda }_2).\end{array}$ Now, by multiplying on both sides of the first half of the inequality $\frac{{(m+1)}^{1-\frac{\varrho }{k}}({\lambda }_2-{\lambda }_1)}{2k{\mathrm{\Gamma }}_k(\varrho )}\frac{{\mathrm{\Omega }}^m(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2){\mathrm{\Omega }}^{\prime }(\frac{\nu }{2}{\lambda }_1+\frac{2-\nu }{2}{\lambda }_2)\chi (\frac{2-\nu }{2}{\lambda }_1+\frac{\nu }{2}{\lambda }_2)}{{({\mathrm{\Omega }}^{m+1}({\lambda }_2)-{\mathrm{\Omega }}^{m+1}(\frac{2-\nu }{2}{\lambda }_1+\frac{\nu }{2}{\lambda }_2))}^{1-\frac{\rho }{k}}},$ then after simplifying and integrating over $[0,1]$ with respect to ν, we obtain

2.9

$\begin{array}{rl}& \mathrm{\Upsilon }\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi Ϝ({\lambda }_1)\\ & \le \frac{Ϝ({\lambda }_1)+Ϝ({\lambda }_2)}{2}\mathrm{\Upsilon }\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi ({\lambda }_1).\end{array}$ Adding (2.8) and (2.9), we obtain

2.10

$\begin{array}{rl}& \mathrm{\Upsilon }\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi Ϝ({\lambda }_2)+\mathrm{\Upsilon }(\frac{{\lambda }_1+{\lambda }_2}{2}\left)\right[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi Ϝ({\lambda }_1)\\ & \le \frac{Ϝ({\lambda }_1)+Ϝ({\lambda }_2)}{2}[\mathrm{\Upsilon }(\frac{{\lambda }_1+{\lambda }_2}{2}\left)\right[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi ({\lambda }_2)\\ & +\mathrm{\Upsilon }\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi ({\lambda }_1].\end{array}$ Hence, by adding (2.7) and (2.10), we obtain our desired inequality. □

Corollary 2.4

If we replace$m=0$and$k=1$in Theorem2.3, chooseχto be symmetric about$\frac{{\lambda }_1+{\lambda }_2}{2}$, then we have$\begin{array}{rl}& Ϝ\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)[{\mathfrak{J}}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi ({\lambda }_2)+{\mathfrak{J}}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi ({\lambda }_1)]\\ & \le \mathrm{\Upsilon }\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)[{\mathfrak{J}}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi Ϝ({\lambda }_2)+{\mathfrak{J}}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi Ϝ({\lambda }_1)]\\ & \le \frac{Ϝ({\lambda }_1)+Ϝ({\lambda }_2)}{2}[{\mathfrak{J}}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi ({\lambda }_2)+{\mathfrak{J}}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi ({\lambda }_1)].\end{array}$

Theorem 2.5

Let ϒ be a differentiable on$({\lambda }_1,{\lambda }_2)$and$\chi :[{\lambda }_1,{\lambda }_2]\mapsto \mathbb{R}$be a continuous function. If$|{\mathrm{\Upsilon }}^{\prime }|\in K_s^2$on$[{\lambda }_1,{\lambda }_2]$for any fixed$s\in (0,1]$, then the H–H–Fejer kind of inequality for the generalized RLFI holds: $\begin{array}{rl}& \left|Ϝ\right(\frac{{\lambda }_1+{\lambda }_2}{2}\left)\right[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi ({\lambda }_2)+{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi ({\lambda }_1)\left]\right|\\ & -\left|\right[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi Ϝ({\lambda }_2)+{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi Ϝ({\lambda }_1)\left]\right|\\ & \le \frac{{(m+1)}^{1-\frac{\varrho }{k}}}{{({\lambda }_2-{\lambda }_1)}^p)k{\mathrm{\Gamma }}_k(\varrho +1)}||\chi |{|}_{\mathrm{\infty }},[{\lambda }_1,{\lambda }_2](|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1)|+|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2)|)\\ & \times \left[\right({\int }_{{\lambda }_1}^{\frac{{\lambda }_1+{\lambda }_2}{2}}{({\mathrm{\Omega }}^{m+1}(\nu )-{\mathrm{\Omega }}^{m+1}({\lambda }_1))}^{\frac{\varrho }{k}}\\ & +{\int }_{\frac{{\lambda }_1+{\lambda }_2}{2}}^{{\lambda }_2}{({\mathrm{\Omega }}^{m+1}({\lambda }_2)-{\mathrm{\Omega }}^{m+1}(\nu ))}^{\frac{\varrho }{k}}\left)\right({({\lambda }_2-\nu )}^s+{(\nu -{\lambda }_1)}^s\left)d\nu \right],\end{array}$whereϜfulfils the condition as in Theorem2.3.

Proof

By using the defined s-convexity of the second kind, we can write $\begin{array}{rl}& |{Ϝ}^{\prime }(\nu )|=|{\mathrm{\Upsilon }}^{\prime }(\nu )-{\mathrm{\Upsilon }}^{\prime }({\lambda }_1+{\lambda }_2-\nu )|\le |{\mathrm{\Upsilon }}^{\prime }(\nu )|+|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1+{\lambda }_2-\nu )|\\ & =\left|{\mathrm{\Upsilon }}^{\prime }\right(\frac{{\lambda }_2-\nu }{{\lambda }_2-{\lambda }_1}{\lambda }_1+\frac{\nu -{\lambda }_1}{{\lambda }_2-{\lambda }_1}{\lambda }_2\left)\right|\\ & +\left|{\mathrm{\Upsilon }}^{\prime }\right(\frac{\nu -{\lambda }_1}{{\lambda }_2-{\lambda }_1}{\lambda }_1+\frac{{\lambda }_2-\nu }{{\lambda }_2-{\lambda }_1}{\lambda }_2\left)\right|\\ & \le {\left(\frac{{\lambda }_2-\nu }{{\lambda }_2-{\lambda }_1}\right)}^s|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1)|\\ & +{\left(\frac{\nu -{\lambda }_1}{{\lambda }_2-{\lambda }_1}\right)}^s|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2)|+{\left(\frac{\nu -{\lambda }_1}{{\lambda }_2-{\lambda }_1}\right)}^s|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1)|+{\left(\frac{{\lambda }_2-\nu }{{\lambda }_2-{\lambda }_1}\right)}^s|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2)|\\ & =({\left(\frac{{\lambda }_2-\nu }{{\lambda }_2-{\lambda }_1}\right)}^s+{\left(\frac{\nu -{\lambda }_1}{{\lambda }_2-{\lambda }_1}\right)}^s)(|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1)|+|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2)|).\end{array}$ Using Theorem 2.3, we reach $\begin{array}{rl}& \left|Ϝ\right(\frac{{\lambda }_1+{\lambda }_2}{2}\left)\right[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi ({\lambda }_2)+{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi ({\lambda }_1)\left]\right|\\ & -\left|\right[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi Ϝ({\lambda }_2)+{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi Ϝ({\lambda }_1)\left]\right|\\ & \le \frac{{(m+1)}^{1-\frac{\varrho }{k}}}{k{\mathrm{\Gamma }}_k(\varrho )}{\int }_{{\lambda }_1}^{{\lambda }_2}|\kappa (\nu )||{Ϝ}^{\prime }(\nu )|d\nu \\ & \le \frac{{(m+1)}^{1-\frac{\varrho }{k}}}{k{\mathrm{\Gamma }}_k(\varrho )}\left({\int }_{{\lambda }_1}^{\frac{{\lambda }_1+{\lambda }_2}{2}}\right|{\int }_{{\lambda }_1}^{\nu }\frac{{\mathrm{\Omega }}^m(u){\mathrm{\Omega }}^{\prime }(u)\chi (u)du}{{({\mathrm{\Omega }}^{m+1}(u)-{\mathrm{\Omega }}^{m+1}({\lambda }_1))}^{1-\frac{\varrho }{k}}}\\ & \times \left|{\int }_{\frac{{\lambda }_1+{\lambda }_2}{2}}^{{\lambda }_2}\right|{\int }_{{\lambda }_2}^{\nu }\frac{{\mathrm{\Omega }}^m(u){\mathrm{\Omega }}^{\prime }(u)\chi (u)du}{{({\mathrm{\Omega }}^{m+1}({\lambda }_2)-{\mathrm{\Omega }}^{m+1}(u))}^{1-\frac{\varrho }{k}}}\left|\right){Ϝ}^{\prime }(\nu )d\nu \\ & \le \frac{{(m+1)}^{1-\frac{\varrho }{k}}}{k{\mathrm{\Gamma }}_k(\varrho )}||\chi |{|}_{\mathrm{\infty }}({\int }_{{\lambda }_1}^{\frac{{\lambda }_1+{\lambda }_2}{2}}\left|{\int }_{{\lambda }_2}^{\nu }\frac{{\mathrm{\Omega }}^m(u){\mathrm{\Omega }}^{\prime }(u)du}{{({\mathrm{\Omega }}^{m+1}({\lambda }_2)-{\mathrm{\Omega }}^{m+1}(u))}^{1-\frac{\varrho }{k}}}\right|\\ & +{\int }_{\frac{{\lambda }_1+{\lambda }_2}{2}}^{{\lambda }_2}\left|{\int }_{{\lambda }_1}^{\nu }\frac{{\mathrm{\Omega }}^m(u){\mathrm{\Omega }}^{\prime }(u)du}{{({\mathrm{\Omega }}^{m+1}(u)-{\mathrm{\Omega }}^{m+1}({\lambda }_1))}^{1-\frac{\varrho }{k}}}\right|){Ϝ}^{\prime }(\nu )d\nu \\ & \le \frac{{(m+1)}^{-\frac{\varrho }{k}}}{k{\mathrm{\Gamma }}_k(\varrho )}||\chi |{|}_{\mathrm{\infty }}[({\int }_{{\lambda }_1}^{\frac{{\lambda }_1+{\lambda }_2}{2}}\frac{{[{\mathrm{\Omega }}^{m+1}(\nu )-{\mathrm{\Omega }}^{m+1}({\lambda }_1)]}^{\frac{\varrho }{k}}}{\frac{\varrho }{k}}\\ & +{\int }_{\frac{{\lambda }_1+{\lambda }_2}{2}}^{{\lambda }_2}\frac{{[{\mathrm{\Omega }}^{m+1}({\lambda }_2)-{\nu }^{s+}]}^{\frac{\varrho }{k}}}{\frac{\varrho }{k}})\\ & \times ({\left(\frac{{\lambda }_2-\nu }{{\lambda }_2-{\lambda }_1}\right)}^p+{\left(\frac{\nu -{\lambda }_1}{{\lambda }_2-{\lambda }_1}\right)}^p)(|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1)|+|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2)|d\nu )]\\ & \le \frac{{(m+1)}^{-\frac{\varrho }{k}}}{{({\lambda }_2-{\lambda }_1)}^p{\mathrm{\Gamma }}_k(\varrho +k)}||\chi |{|}_{\mathrm{\infty }}(|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1)|+|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2)|)\end{array}$$\begin{array}{rl}& \times \left[\right({\int }_{{\lambda }_1}^{\frac{{\lambda }_1+{\lambda }_2}{2}}{({\mathrm{\Omega }}^{m+1}(\nu )-{\mathrm{\Omega }}^{m+1}({\lambda }_1))}^{\frac{\varrho }{k}}+{\int }_{\frac{{\lambda }_1+{\lambda }_2}{2}}^{{\lambda }_2}{({\mathrm{\Omega }}^{m+1}({\lambda }_2)-{\mathrm{\Omega }}^{m+1}(\nu ))}^{\frac{\varrho }{k}})\\ & \times ({({\lambda }_2-\nu )}^p+{(\nu -{\lambda }_1)}^p)d\nu ].\end{array}$ We reach our desired outcome. □

Corollary 2.6

If we fix$p=k=1$and$m=0$in Theorem2.5andχis symmetric about$\frac{{\lambda }_1+{\lambda }_2}{2}$, then we have$\begin{array}{rl}& \left|Ϝ\right(\frac{{\lambda }_1+{\lambda }_2}{2}\left)\right[{\mathfrak{J}}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi ({\lambda }_2)+{\mathfrak{J}}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi ({\lambda }_1)\left]\right|\\ & -\left|\right[{\mathfrak{J}}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi Ϝ({\lambda }_2)+{\mathfrak{J}}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi Ϝ({\lambda }_1)\left]\right|\\ & \le \frac{{(m+1)}^{1-\varrho }({\lambda }_2-{\lambda }_1)}{{\lambda }_2-{\lambda }_1)\mathrm{\Gamma }(\varrho +1)}||\chi |{|}_{\mathrm{\infty }}(|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1)|+|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2)|\left)\right[({\int }_{{\lambda }_1}^{\frac{{\lambda }_1+{\lambda }_2}{2}}{(\nu -{\lambda }_1)}^{\varrho }d\nu \\ & +{\int }_{\frac{{\lambda }_1+{\lambda }_2}{2}}^{{\lambda }_2}{({\lambda }_2-\nu )}^{\varrho }\left)d\nu \right]=\frac{{(m+1)}^{1-\varrho }{({\lambda }_2-{\lambda }_1)}^{\varrho +1}}{(\varrho +1)\mathrm{\Gamma }(\varrho +1)2^{\varrho +1}}||\chi |{|}_{\mathrm{\infty }}(|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1)|+|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2)|).\end{array}$

Proposition 2.7

Furthermore, if we take$\chi =1$and$\varrho =1$in Corollary2.6, we obtain the following midpoint inequality: $|({\lambda }_2-{\lambda }_1)\mathrm{\Upsilon }(\frac{{\lambda }_1+{\lambda }_2}{2})-{\int }_{{\lambda }_1}^{{\lambda }_2}\mathrm{\Upsilon }(\nu )d\nu |\le \frac{{({\lambda }_2-{\lambda }_1)}^2}{8}(|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1)|+|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2)|).$

Theorem 2.8

Let ϒ be a Cf on$({\lambda }_1,{\lambda }_2)$with${\lambda }_1<{\lambda }_2$and$\chi :[{\lambda }_1,{\lambda }_2]\mapsto \mathbb{R}$be a nonnegative integrable on$[{\lambda }_1,{\lambda }_2]$, then the H–H–Fejer-type inequality for generalized RLFI holds: $\begin{array}{rl}& \left|Ϝ\right(\frac{{\lambda }_1+{\lambda }_2}{2}\left)\right[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi ({\lambda }_2)+{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi ({\lambda }_1)\left]\right|\\ & -\left|\right[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi Ϝ({\lambda }_2)+{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi Ϝ({\lambda }_1)\left]\right|\\ & \le \frac{{(m+1)}^{-\frac{\varrho }{k}}||\chi |{|}_{\mathrm{\infty }}}{{\mathrm{\Gamma }}_k(\varrho +k)}\\ & \times \left({\int }_{{\lambda }_1}^{\frac{{\lambda }_1+{\lambda }_2}{2}}{\left(\right|{({\mathrm{\Omega }}^{m+1}(\nu )-{\mathrm{\Omega }}^{m+1}({\lambda }_1))}^{\frac{\varrho }{k}}\left|d\nu \right)}^{1-\frac{1}{q}}{\int }_{{\lambda }_1}^{\frac{{\lambda }_1+{\lambda }_2}{2}}\right|{({\mathrm{\Omega }}^{m+1}(\nu )-{\mathrm{\Omega }}^{m+1}({\lambda }_1))}^{\frac{\varrho }{k}}|\\ & {\times ({\left(\frac{{\lambda }_2-\nu }{{\lambda }_2-{\lambda }_1}\right)}^s(|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1){|}^q+{\left(\frac{\nu -{\lambda }_1}{{\lambda }_2-{\lambda }_1}\right)}^s)|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2){|}^{q}d\nu )}^{\frac{1}{q}})\\ & +\frac{{(m+1)}^{-\frac{\varrho }{k}}|\chi |{|}_{\mathrm{\infty }},[{\lambda }_1,{\lambda }_2]}{{\mathrm{\Gamma }}_k(\varrho +k)}\\ & \times \left({\int }_{\frac{{\lambda }_1+{\lambda }_2}{2}}^{{\lambda }_2}{\left(\right|{({\mathrm{\Omega }}^{m+1}({\lambda }_2)-{\mathrm{\Omega }}^{m+1}(\nu ))}^{\frac{\varrho }{k}}\left|d\nu \right)}^{1-\frac{1}{q}}{\int }_{\frac{{\lambda }_1+{\lambda }_2}{2}}^{{\lambda }_2}\right|{({\mathrm{\Omega }}^{m+1}({\lambda }_2)-{\mathrm{\Omega }}^{m+1}(\nu ))}^{\frac{\varrho }{k}}|\\ & {\times ({\left(\frac{{\lambda }_2-\nu }{{\lambda }_2-{\lambda }_1}\right)}^s(|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1){|}^q+{\left(\frac{\nu -{\lambda }_1}{{\lambda }_2-{\lambda }_1}\right)}^s)|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2){|}^{q}d\nu )}^{\frac{1}{q}}).\end{array}$

Proof

By using Theorem 2.3, we can write $\begin{array}{rl}& \left|Ϝ\right(\frac{{\lambda }_1+{\lambda }_2}{2}\left)\right[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi ({\lambda }_2)+{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi ({\lambda }_1)\left]\right|\\ & -\left|\right[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi Ϝ({\lambda }_2)+{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi Ϝ({\lambda }_1)\left]\right|\\ & \le \frac{{(m+1)}^{1-\frac{\varrho }{k}}||\chi |{|}_{\mathrm{\infty }}}{k{\mathrm{\Gamma }}_k(\varrho )}{\int }_{{\lambda }_1}^{{\lambda }_2}|\kappa (\nu )||{Ϝ}^{\prime }(\nu )|d\nu \\ & \le \frac{{(m+1)}^{1-\frac{\varrho }{k}}}{k{\mathrm{\Gamma }}_k(\varrho )}{({\int }_{{\lambda }_1}^{{\lambda }_2}|\kappa (\nu )|d\nu )}^{1-\frac{1}{q}}{({\int }_{{\lambda }_1}^{{\lambda }_2}|\kappa (\nu )||{Ϝ}^{\prime }(\nu ){|}^{q}d\nu )}^{\frac{1}{q}}\\ & \le \frac{{(m+1)}^{1-\frac{\varrho }{k}}||\chi |{|}_{\mathrm{\infty }}}{k{\mathrm{\Gamma }}_k(\varrho )}\{{\int }_{{\lambda }_1}^{\frac{{\lambda }_1+{\lambda }_2}{2}}{\left(\right|{\int }_{{\lambda }_1}^{\nu }\frac{{\mathrm{\Omega }}^m(u){\mathrm{\Omega }}^{\prime }(u)\chi (u)du}{{({\mathrm{\Omega }}^{m+1}(u)-{\mathrm{\Omega }}^{m+1}({\lambda }_1))}^{1-\frac{\varrho }{k}}}\left|d\nu \right)}^{1-\frac{1}{q}}\\ & \times {\int }_{{\lambda }_1}^{\frac{{\lambda }_1+{\lambda }_2}{2}}\left|{\int }_{{\lambda }_1}^{\nu }\frac{{\mathrm{\Omega }}^m(u){\mathrm{\Omega }}^{\prime }(u)\chi (u)du}{{({\mathrm{\Omega }}^{m+1}(u)-{\mathrm{\Omega }}^{m+1}({\lambda }_1))}^{1-\frac{\varrho }{k}}}\right|\\ & {\times ({\left(\frac{{\lambda }_2-\nu }{{\lambda }_2-{\lambda }_1}\right)}^s(|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1){|}^q+{\left(\frac{\nu -{\lambda }_1}{{\lambda }_2-{\lambda }_1}\right)}^s)|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2){|}^{q}d\nu )}^{\frac{1}{q}}\}\\ & +\frac{{(m+1)}^{1-\frac{\varrho }{k}}||\chi |{|}_{\mathrm{\infty }}}{k{\mathrm{\Gamma }}_k(\varrho )}\{{\left({\int }_{\frac{{\lambda }_1+{\lambda }_2}{2}}^{{\lambda }_2}\right|{\int }_{{\lambda }_2}^{\nu }\frac{{\mathrm{\Omega }}^m(u){\mathrm{\Omega }}^{\prime }(u)\chi (u)du}{{({\mathrm{\Omega }}^{m+1}({\lambda }_2)-{\mathrm{\Omega }}^{m+1}(u))}^{1-\frac{\varrho }{k}}}\left|d\nu \right)}^{1-\frac{1}{q}}\\ & \times \left({\int }_{\frac{{\lambda }_1+{\lambda }_2}{2}}^{{\lambda }_2}\right|{\int }_{{\lambda }_2}^{\nu }\frac{{\mathrm{\Omega }}^m(u){\mathrm{\Omega }}^{\prime }(u)\chi (u)du}{{({\mathrm{\Omega }}^{m+1}({\lambda }_2)-{\mathrm{\Omega }}^{m+1}(u))}^{1-\frac{\varrho }{k}}}|\\ & {\times {\left(\frac{{\lambda }_2-\nu }{{\lambda }_2-{\lambda }_1}\right)}^s(|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1){|}^q+{\left(\frac{\nu -{\lambda }_1}{{\lambda }_2-{\lambda }_1}\right)}^s)|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2){|}^{q}d\nu )}^{\frac{1}{q}}\}\\ & =\frac{{(m+1)}^{-\frac{\varrho }{k}}||\chi |{|}_{\mathrm{\infty }}}{{\mathrm{\Gamma }}_k(\varrho +k)}\{{\int }_{{\lambda }_1}^{\frac{{\lambda }_1+{\lambda }_2}{2}}{\left(\right|{({\mathrm{\Omega }}^{m+1}(\nu )-{\mathrm{\Omega }}^{m+1}({\lambda }_1))}^{\frac{\varrho }{k}}\left|d\nu \right)}^{1-\frac{1}{q}}\\ & \times {\int }_{{\lambda }_1}^{\frac{{\lambda }_1+{\lambda }_2}{2}}\left|{({\mathrm{\Omega }}^{m+1}(\nu )-{\mathrm{\Omega }}^{m+1}({\lambda }_1))}^{\frac{\varrho }{k}}\right|\\ & {\times ({\left(\frac{{\lambda }_2-\nu }{{\lambda }_2-{\lambda }_1}\right)}^s(|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1){|}^q+{\left(\frac{\nu -{\lambda }_1}{{\lambda }_2-{\lambda }_1}\right)}^s)|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2){|}^{q}d\nu )}^{\frac{1}{q}}\}\\ & +\frac{{(m+1)}^{-\frac{\varrho }{k}}||\chi |{|}_{\mathrm{\infty }}}{{\mathrm{\Gamma }}_k(\varrho +k)}\{{\int }_{\frac{{\lambda }_1+{\lambda }_2}{2}}^{{\lambda }_2}{\left(\right|{({\mathrm{\Omega }}^{m+1}({\lambda }_2)-{\mathrm{\Omega }}^{m+1}(\nu ))}^{\frac{\varrho }{k}}\left|d\nu \right)}^{1-\frac{1}{q}}\\ & \times {\int }_{\frac{{\lambda }_1+{\lambda }_2}{2}}^{{\lambda }_2}\left|{({\mathrm{\Omega }}^{m+1}({\lambda }_2)-{\mathrm{\Omega }}^{m+1}(\nu ))}^{\frac{\varrho }{k}}\right|\\ & {\times ({\left(\frac{{\lambda }_2-\nu }{{\lambda }_2-{\lambda }_1}\right)}^s(|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1){|}^q+{\left(\frac{\nu -{\lambda }_1}{{\lambda }_2-{\lambda }_1}\right)}^s)|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2){|}^{q}d\nu )}^{\frac{1}{q}}\}.\end{array}$ This is our desired result. □

Remark 2.9

For the choice of $s=k=1$ and $m=0$ in Theorem 2.8, then we obtain [[32], Theorem 7].

Remark 2.10

For the choice of $k=1$ and $m=0$ in Theorem 2.8, we obtain [[33], Theorem 5].

Corollary 2.11

For the choice of$k=s=1$, $\varrho =1$, and$m=0$in Theorem2.8, we arrive at the following midpoint inequality: $\begin{array}{rl}& |\mathrm{\Upsilon }(\frac{{\lambda }_1+{\lambda }_2}{2}){\int }_{{\lambda }_1}^{{\lambda }_2}\chi (\nu )d\nu -{\int }_{{\lambda }_1}^{{\lambda }_2}(\mathrm{\Upsilon }\chi )(\nu )d\nu |\\ & \le \frac{||\chi ||{({\lambda }_2-{\lambda }_1)}^2}{{(3)}^{\frac{1}{q}}8}[{(2|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1){|}^q+|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2){|}^q)}^{\frac{1}{q}}+{(|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1){|}^q+2|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2){|}^q)}^{\frac{1}{q}}].\end{array}$

Theorem 2.12

Let ϒ be a Cf on$({\lambda }_1,{\lambda }_2)$with${\lambda }_1<{\lambda }_2$and$\chi :[{\lambda }_1,{\lambda }_2]\mapsto \mathbb{R}$be a nonnegative integrable on$[{\lambda }_1,{\lambda }_2]$, then the following expression holds: $\begin{array}{rl}& \left|Ϝ\right(\frac{{\lambda }_1+{\lambda }_2}{2}\left)\right[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi ({\lambda }_2)+{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi ({\lambda }_1)\left]\right|\\ & -\left|\right[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi Ϝ({\lambda }_2)+{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi Ϝ({\lambda }_1)\left]\right|\\ & \le \frac{{(m+1)}^{-\frac{\varrho }{k}}||\chi |{|}_{\mathrm{\infty }}{({\lambda }_2-{\lambda }_1)}^{\frac{1}{q}}}{2^{\frac{s}{q}+\frac{1}{q}}{(s+1)}^{\frac{1}{q}}{\mathrm{\Gamma }}_k(\varrho +k)}\{{\int }_{{\lambda }_1}^{\frac{{\lambda }_1+{\lambda }_2}{2}}{\left(\right|{({\mathrm{\Omega }}^{m+1}(\nu )-{\mathrm{\Omega }}^{m+1}({\lambda }_1))}^{\frac{p\varrho }{k}}\left|d\nu \right)}^{\frac{1}{p}}\\ & \times {((2^{s+1}-1)(|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1)){|}^q+|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2){|}^q)}^{\frac{1}{q}}+{\int }_{\frac{{\lambda }_1+{\lambda }_2}{2}}^{{\lambda }_2}{\left(\right|{({\mathrm{\Omega }}^{m+1}({\lambda }_2)-{\mathrm{\Omega }}^{m+1}(\nu ))}^{\frac{p\varrho }{k}}\left|d\nu \right)}^{\frac{1}{p}}\\ & \times {(|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1)|)}^q+{((2^{s+1}-1)|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2){|}^q)}^{\frac{1}{q}}\}.\end{array}$

Proof

Using Hölder’s inequality, we have $\begin{array}{rl}& \left|Ϝ\right(\frac{{\lambda }_1+{\lambda }_2}{2}\left)\right[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi ({\lambda }_2)+{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi ({\lambda }_1)\left]\right|\\ & -\left|\right[{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi Ϝ({\lambda }_2)+{}_k{}^m\mathfrak{J}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi Ϝ({\lambda }_1)\left]\right|\\ & \le \frac{{(m+1)}^{1-\frac{\varrho }{k}}||\chi |{|}_{\mathrm{\infty }}}{k{\mathrm{\Gamma }}_k(\varrho )}{\int }_{{\lambda }_1}^{{\lambda }_2}|{Ϝ}^{\prime }(\nu )|d\nu \\ & \le \frac{{(m+1)}^{1-\frac{\varrho }{k}}}{k{\mathrm{\Gamma }}_k(\varrho )}{({\int }_{{\lambda }_1}^{{\lambda }_2}|\kappa (\nu ){|}^{p}d\nu )}^{\frac{1}{p}}{({\int }_{{\lambda }_1}^{{\lambda }_2}|{Ϝ}^{\prime }(\nu ){|}^{q}d\nu )}^{\frac{1}{q}}\\ & \le \frac{{(m+1)}^{1-\frac{\varrho }{k}}||\chi |{|}_{\mathrm{\infty }}}{k{\mathrm{\Gamma }}_k(\varrho )}({\int }_{{\lambda }_1}^{\frac{{\lambda }_1+{\lambda }_2}{2}}{\left(\right|{\int }_{{\lambda }_1}^{\nu }\frac{{\mathrm{\Omega }}^m(u){\mathrm{\Omega }}^{\prime }(u)\chi (u)du}{{({\mathrm{\Omega }}^{m+1}(u)-{\mathrm{\Omega }}^{m+1}({\lambda }_1))}^{1-\frac{\varrho }{k}}}{|}^{p}d\nu )}^{\frac{1}{p}}\\ & {\times ({\int }_{{\lambda }_1}^{\nu }{\left(\frac{{\lambda }_2-\nu }{{\lambda }_2-{\lambda }_1}\right)}^s(|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1){|}^q+{\left(\frac{\nu -{\lambda }_1}{{\lambda }_2-{\lambda }_1}\right)}^s)|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2){|}^{q}d\nu )}^{\frac{1}{q}}\\ & +{\left({\int }_{\frac{{\lambda }_1+{\lambda }_2}{2}}^{{\lambda }_2}\right|{\int }_{{\lambda }_2}^{\nu }\frac{{\mathrm{\Omega }}^m(u){\mathrm{\Omega }}^{\prime }(u)\chi (u)du}{{({\mathrm{\Omega }}^{m+1}({\lambda }_2)-{\mathrm{\Omega }}^{m+1}(u))}^{1-\frac{\varrho }{k}}}{|}^{p}d\nu )}^{\frac{1}{p}}\end{array}$$\begin{array}{rl}& {\times ({\int }_{\nu }^{{\lambda }_2}{\left(\frac{{\lambda }_2-\nu }{{\lambda }_2-{\lambda }_1}\right)}^s(|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1){|}^q+{\left(\frac{\nu -{\lambda }_1}{{\lambda }_2-{\lambda }_1}\right)}^s)|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2){|}^{q}d\nu )}^{\frac{1}{q}})\\ & =\frac{{(m+1)}^{-\frac{\varrho }{k}}||\chi |{|}_{\mathrm{\infty }}}{{\mathrm{\Gamma }}_k(\varrho +k)}({\int }_{{\lambda }_1}^{\frac{{\lambda }_1+{\lambda }_2}{2}}{\left(\right|{({\mathrm{\Omega }}^{m+1}(\nu )-{\mathrm{\Omega }}^{m+1}({\lambda }_1))}^{\frac{\varrho }{k}}{|}^{p}d\nu )}^{\frac{1}{p}}\\ & {\times ({\int }_{{\lambda }_1}^{\nu }{\left(\frac{{\lambda }_2-\nu }{{\lambda }_2-{\lambda }_1}\right)}^s(|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1){|}^q+{\left(\frac{\nu -{\lambda }_1}{{\lambda }_2-{\lambda }_1}\right)}^s)|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2){|}^{q}d\nu )}^{\frac{1}{q}}\\ & +{\int }_{\frac{{\lambda }_1+{\lambda }_2}{2}}^{{\lambda }_2}{\left(\right|{({\mathrm{\Omega }}^{m+1}({\lambda }_2)-{\mathrm{\Omega }}^{m+1}(\nu ))}^{\frac{\varrho }{k}}{|}^{p}d\nu )}^{\frac{1}{p}}\\ & {\times ({\int }_{\nu }^{{\lambda }_2}{\left(\frac{{\lambda }_2-\nu }{{\lambda }_2-{\lambda }_1}\right)}^s(|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1){|}^q+{\left(\frac{\nu -{\lambda }_1}{{\lambda }_2-{\lambda }_1}\right)}^s)|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2){|}^{q}d\nu )}^{\frac{1}{q}})\\ & =\frac{{(m+1)}^{-\frac{\varrho }{k}}||\chi |{|}_{\mathrm{\infty }}}{{({\lambda }_2-{\lambda }_1)}^{\frac{s}{q}}{\mathrm{\Gamma }}_k(\varrho +k)}\{{\int }_{{\lambda }_1}^{\frac{{\lambda }_1+{\lambda }_2}{2}}{\left(\right|{({\mathrm{\Omega }}^{m+1}(\nu )-{\mathrm{\Omega }}^{m+1}({\lambda }_1))}^{\frac{\varrho }{k}}{|}^{p}d\nu )}^{\frac{1}{p}}\\ & \times {(\frac{{({\lambda }_2-{\lambda }_1)}^{s+1}(2^{s+1}-1)}{2^{s+1}(s+1)}(|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1)){|}^q+\frac{{({\lambda }_2-{\lambda }_1)}^{s+1}}{2^{s+1}(s+1)}|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2){|}^q)}^{\frac{1}{q}}\\ & +{\int }_{\frac{{\lambda }_1+{\lambda }_2}{2}}^{{\lambda }_2}{\left(\right|{({\mathrm{\Omega }}^{m+1}({\lambda }_2)-{\mathrm{\Omega }}^{m+1}(\nu ))}^{\frac{\varrho }{k}}{|}^{p}d\nu )}^{\frac{1}{p}}\\ & \times \frac{{({\lambda }_2-{\lambda }_1)}^{s+1}}{2^{s+1}(s+1)}{(|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1)|)}^q+{(\frac{{({\lambda }_2-{\lambda }_1)}^{s+1}(2^{s+1}-1)}{2^{s+1}(s+1)}|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2){|}^q)}^{\frac{1}{q}}\}\\ & \le \frac{{(m+1)}^{-\frac{\varrho }{k}}||\chi |{|}_{\mathrm{\infty }}{({\lambda }_2-{\lambda }_1)}^{\frac{1}{q}}}{2^{\frac{s}{q}+\frac{1}{q}}{(s+1)}^{\frac{1}{q}}{\mathrm{\Gamma }}_k(\varrho +k)}\{{\int }_{{\lambda }_1}^{\frac{{\lambda }_1+{\lambda }_2}{2}}{\left(\right|{({\mathrm{\Omega }}^{m+1}(\nu )-{\mathrm{\Omega }}^{m+1}({\lambda }_1))}^{\frac{p\varrho }{k}}\left|d\nu \right)}^{\frac{1}{p}}\\ & \times {((2^{s+1}-1)(|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1)){|}^q+|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2){|}^q)}^{\frac{1}{q}}+{\int }_{\frac{{\lambda }_1+{\lambda }_2}{2}}^{{\lambda }_2}{\left(\right|{({\mathrm{\Omega }}^{m+1}({\lambda }_2)-{\mathrm{\Omega }}^{m+1}(\nu ))}^{\frac{p\varrho }{k}}\left|d\nu \right)}^{\frac{1}{p}}\\ & \times {(|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1)|)}^q+{((2^{s+1}-1)|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2){|}^q)}^{\frac{1}{q}}\}\end{array}$ This is our desired result. □

Corollary 2.13

If we replace$m=0$and$s=k=1$, then we obtain [[32], Theorem 8]: $\begin{array}{rl}& \left|Ϝ\right(\frac{{\lambda }_1+{\lambda }_2}{2}\left)\right[{\mathfrak{J}}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi ({\lambda }_2)+{\mathfrak{J}}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi ({\lambda }_1)\left]\right|\\ & -\left|\right[{\mathfrak{J}}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{+}}^{\varrho }\chi Ϝ({\lambda }_2)+{\mathfrak{J}}_{{(\frac{{\lambda }_1+{\lambda }_2}{2})}^{-}}^{\varrho }\chi Ϝ({\lambda }_1)\left]\right|\\ & \le \frac{||\chi |{|}_{\mathrm{\infty }}{({\lambda }_2-{\lambda }_1)}^{\frac{1}{q}}}{2^{\frac{3}{q}}{\mathrm{\Gamma }}_{(}\varrho +1)}({\int }_{{\lambda }_1}^{\frac{{\lambda }_1+{\lambda }_2}{2}}{\left(\right|{(\nu -{\lambda }_1)}^{p\varrho }\left|d\nu \right)}^{\frac{1}{p}}\\ & \times {(3(|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1)){|}^q+|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2){|}^q)}^{\frac{1}{q}}+{\int }_{\frac{{\lambda }_1+{\lambda }_2}{2}}^{{\lambda }_2}{\left(\right|{({\lambda }_2-\nu )}^{p\varrho }\left|d\nu \right)}^{\frac{1}{p}}{(|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1)|)}^q\\ & +{(3|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2){|}^q)}^{\frac{1}{q}}).\end{array}$

Proposition 2.14

For the choice of$s=k=1$, $\varrho =1$, and$m=0$, we obtain the following: $\begin{array}{rl}& |\mathrm{\Upsilon }(\frac{{\lambda }_1+{\lambda }_2}{2}){\int }_{{\lambda }_1}^{{\lambda }_2}\chi (\nu )d\nu -{\int }_{{\lambda }_1}^{{\lambda }_2}(\mathrm{\Upsilon }\chi )(\nu )d\nu |\\ & \le \frac{||\chi ||{({\lambda }_2-{\lambda }_1)}^2}{{(m+1)}^{\frac{1}{p}}{2}^{\frac{4p-2}{p}}}[{(3|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1){|}^q+|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2){|}^q)}^{\frac{1}{q}}+{(|{\mathrm{\Upsilon }}^{\prime }({\lambda }_1){|}^q+3|{\mathrm{\Upsilon }}^{\prime }({\lambda }_2){|}^q)}^{\frac{1}{q}}].\end{array}$

Applications

It is useful to introduce a further generalization of Abel’s Integral equation by considering a generalized singular kernel instead of $K(\gamma ,t)=\frac{1}{\sqrt{\gamma -t}}$, the generalized kernel will be of the form $K(x,t)=\frac{1}{{[g^{m+1}(\gamma )-g^{m+1}(t)]}^{\frac{\delta }{k}}},0<\delta <1,$ where $g(x)$ is an increasing function with $g(x)\ne 0$ on the interval $(0,b)$ and $\gamma \in (0,b)$. The main generalized Abel’s integral equation is given by

3.1

$f(\gamma )=\frac{(m+1)}{k}{\int }_0^{\gamma }\frac{g^m(t)g^{\prime }(t)}{{[g^{m+1}(\gamma )-g^{m+1}(t)]}^{\frac{\delta }{k}}}u(t)dt,0<\delta <1.$

Theorem 3.1

The solution of Abel’s integral equation (3.1) is$u(x)=\frac{k}{{\mathrm{\Gamma }}_k(1-\delta ){(m+1)}^{1-\frac{\delta }{k}}}\frac{d}{dx}({}_k{}^m\mathfrak{J}_0^{\delta }f(x)).$

Proof

Consider the integral ${\int }_0^x\frac{g^m(y)g^{\prime }(y)}{{[g^{m+1}(x)-g^{m+1}(y)]}^{1-\frac{\delta }{k}}}f(y)dy.$ By substituting the values of $f(t)$, we have $\frac{(m+1)}{k}{\int }_0^x{\int }_0^y\frac{g^m(y)g^{\prime }(y)}{{[g^{m+1}(x)-g^{m+1}(y)]}^{1-\frac{\delta }{k}}{[g^{m+1}(y)-g^{m+1}(t)]}^{\frac{\delta }{k}}}u(t)dtdy.$ By changing the order of integration we have $\begin{array}{rl}& \frac{(m+1)}{k}{\int }_0^{x}u(t)dt{\int }_0^y\frac{g^m(y)g^{\prime }(y)}{{[g^{m+1}(x)-g^{m+1}(y)]}^{1-\frac{\delta }{k}}{[g^{m+1}(y)-g^{m+1}(t)]}^{\frac{\delta }{k}}}dy\\ & =B_k(\delta ,1-\delta ){\int }_0^{x}u(t)dt.\end{array}$ Hence, we have ${\int }_0^x\frac{g^m(y)g^{\prime }(y)}{{[g^{m+1}(x)-g^{m+1}(y)]}^{1-\frac{\delta }{k}}}f(y)dy=B_k(\delta ,1-\delta ){\int }_0^{x}u(t)dt.$ Differentiating both sides gives $u(x)=\frac{1}{B_k(\delta ,1-\delta )}\frac{d}{dx}{\int }_0^x\frac{g^m(y)g^{\prime }(y)}{{[g^{m+1}(x)-g^{m+1}(y)]}^{1-\frac{\delta }{k}}}f(y)dy.$ This is the required result. □

Now, we define the application of our generalized kernel by presenting the $(k,m)$ form of weakly singular Volterra-type integral equations of the second kind as follows:

3.2

$u(\gamma )=g(\gamma )+{\int }_0^{\gamma }\frac{\beta u(t)dt}{{(g^{m+1}(\gamma )-g^{m+1}(t))}^{\frac{\delta }{k}}},$ where g is a strictly monotonically increasing function on the interval $(0,b)$, $g(\gamma )\ne 0$ for all values of γ in $(0,b)$ and β is constant. If we put $m=0$ and $k=1$ in (3.2) then we obtain the generalized weakly singular Volterra integral equation defined in [38].

Theorem 3.2

Let$g(\gamma )=\theta {(g^{m+1}(\gamma )-g^{m+1}(0))}^{\frac{\mu }{k}}$be a strictly monotonically increasing function on the interval$(0,b)$, $g(\gamma )\ne 0$for all values ofγin$(0,b)$andβis a constant. Then, the solution of the$(k,m)$form of the weakly singular Volterra-type integral equation of the second kind (3.2) is$=\theta {\mathrm{\Gamma }}_k(1+\mu ){(g^{m+1}(\gamma )-g^{m+1}(0))}^{\frac{\mu }{k}}{\mathcal{E}}_{1-\delta ,1+\mu }^k(\beta ({\mathrm{\Gamma }}_k(1-\delta )){(g^{m+1}(\gamma )-g^{m+1}(0))}^{(1-\frac{\delta }{k})}).$

Proof

We use the Adomian decomposition method and substitute the following decomposition series:

3.3

$u(\gamma )=\sum _{n=0}^{\mathrm{\infty }}{u}_n(\gamma ),$ into equation (3.2), then we have $\sum _{n=0}^{\mathrm{\infty }}{u}_n(\gamma )=g(\gamma )+{\int }_0^{\gamma }\frac{\beta {\sum }_{n=0}^{\mathrm{\infty }}{u}_n(t)dt}{{(g^{m+1}(\gamma )-g^{m+1}(t))}^{\frac{\delta }{k}}}.$ By comparing, we have $\begin{array}{rl}& u_0(\gamma )=g(\gamma )=\theta {(g^{m+1}(\gamma )-g^{m+1}(0))}^{\frac{\mu }{k}}\\ & u_1(\gamma )=\beta \theta {\int }_0^{\gamma }\frac{{(g^{m+1}(\gamma )-g^{m+1}(0))}^{\frac{\mu }{k}}}{{(g^{m+1}(\gamma )-g^{m+1}(t))}^{\frac{\delta }{k}}}\\ & \phantom{u_1(\gamma )}=\theta \beta \frac{{\mathrm{\Gamma }}_k(1-\delta ){\mathrm{\Gamma }}_k(1+\mu )}{{\mathrm{\Gamma }}_k(2-\delta +\mu )}{(g^{m+1}(\gamma )-g^{m+1}(0))}^{1-\frac{\delta }{k}+\frac{\mu }{k}}\\ & u_2(\gamma )=\beta {\int }_0^{\gamma }\frac{u_1(t)}{{(g^{m+1}(\gamma )-g^{m+1}(t))}^{\frac{\delta }{k}}}\\ & \phantom{u_2(\gamma )}=\theta {\beta }^2\frac{{({\mathrm{\Gamma }}_k(1-\delta ))}^2{\mathrm{\Gamma }}_k(1+\mu )}{{\mathrm{\Gamma }}_k(3-2\delta +\mu )}{(g^{m+1}(\gamma )-g^{m+1}(0))}^{2(1-\frac{\delta }{k})+\frac{\mu }{k}}\\ & u_3(\gamma )=\beta {\int }_0^{\gamma }\frac{u_2(t)}{{(g^{m+1}(\gamma )-g^{m+1}(t))}^{\frac{\delta }{k}}}\\ & \phantom{u_3(\gamma )}=\theta {\beta }^3\frac{{({\mathrm{\Gamma }}_k(1-\delta ))}^3{\mathrm{\Gamma }}_k(1+\mu )}{{\mathrm{\Gamma }}_k(4-3\delta +\mu )}{(g^{m+1}(\gamma )-g^{m+1}(0))}^{3(1-\frac{\delta }{k})+\frac{\mu }{k}}\\ ⋮\end{array}$ By substituting these values into (3.3), we have $\begin{array}{rl}& u(\gamma )=\theta ({(g^{m+1}(\gamma )-g^{m+1}(0))}^{\frac{\mu }{k}}+\beta \frac{{\mathrm{\Gamma }}_k(1-\delta ){\mathrm{\Gamma }}_k(1+\mu )}{{\mathrm{\Gamma }}_k(2-\delta +\mu )}{(g^{m+1}(\gamma )-g^{m+1}(0))}^{1-\frac{\delta }{k}+\frac{\mu }{k}}\\ & +{\beta }^2\frac{{({\mathrm{\Gamma }}_k(1-\delta ))}^2{\mathrm{\Gamma }}_k(1+\mu )}{{\mathrm{\Gamma }}_k(3-2\delta +\mu )}{(g^{m+1}(\gamma )-g^{m+1}(0))}^{2(1-\frac{\delta }{k})+\frac{\mu }{k}}+\cdots )\\ & =\theta {\mathrm{\Gamma }}_k(1-\mu ){(g^{m+1}(\gamma )-g^{m+1}(0))}^{\frac{\mu }{k}}\sum _{n=0}^{\mathrm{\infty }}\frac{{(\beta ({\mathrm{\Gamma }}_k(1-\delta )){(g^{m+1}(\gamma )-g^{m+1}(0))}^{(1-\frac{\delta }{k})})}^n}{{\mathrm{\Gamma }}_k((1-\delta )n+\mu +1)}\\ & =\theta {\mathrm{\Gamma }}_k(1+\mu ){(g^{m+1}(\gamma )-g^{m+1}(0))}^{\frac{\mu }{k}}{\mathcal{E}}_{1-\delta ,1+\mu }^k(\beta ({\mathrm{\Gamma }}_k(1-\delta )){(g^{m+1}(\gamma )-g^{m+1}(0))}^{(1-\frac{\delta }{k})}).\end{array}$ □

Conclusion

Applications of mathematical fractional inequalities have been found in human-body modeling, heat conduction, viscoelasticity, time-series analysis, circuits, material sciences, shear waves, etc. Inequalities themselves may not directly address educational access, quality research and advancement in mathematics, which includes topics likes inequalities, contribute to the broader goal of enhancing educational by promoting knowledge dissemination, fostering critical thinking skills and supporting the development of mathematical literacy among researchers. In the present article, we establish novel H–H–Fejer-type inequalities connected with generalized RLFI. From our study, we attain the bounds for well-known fractional problems, which are discussed in detail. As an application of our work by making useful substitutions, we obtain midpoint inequalities and different kinds of inequalities as a special case of these inequalities by utilizing different convexities. Researchers can use our defined operator to generalize Chebyshev-type inequalities, Opial-type inequalities, inequalities involving composite Cf, etc. These inequalities play vital roles to compute the special means. Finally, we compute the solution of a generalized Abel integral equation as a function of our defined operator. Weakly Volterra-type fractional integral equations arise while solving many mathematical and applied problems. By using the Adomian decomposition method, we investigate the solutions of generalized weakly Volterra-type fractional integral equations. The outcomes will be useful to find solutions of nonlinear fractional integral equations.

Author contributions

Conceptualization, Ahsan Mehmood and Muhammad Younis; methodology, Muhammad Samraiz and Zhi-Guo Liu; software, Muhammad Samraiz, Muhammad Younis and Zhi-Guo Liu; validation, Ahsan Mehmood and Muhammad Younis; formal analysis, Muhammad Samraiz and Zhi-Guo Liu; investigation, Muhammad Younis and Ahsan Mehmood; writing -original draft preparation, Muhammad Younis and Ahsan Mehmood; writing-review and editing, Muhammad Younis, Muhammad Samraiz and Zhi-Guo Liu; visualization, Ahsan Mehmood; supervision, Zhi-Guo Liu.

Funding information

There is no funding for this research article.

Data availability

No datasets were generated or analysed during the current study.

Declarations

Competing interests

The authors declare no competing interests.

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

References

1. Abel, N.: Solution de quelques problemes l’aide d’integrales definies. Mag. Naturvidensk. (1823)

2. Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; 2006; Amsterdam, Elsevier:

3. Oldham, K.B.; Spanier, J. The Fractional Calculus; 1974; New York, Academic Press:

4. Caputo, M.; Fabrizio, M. A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl.; 2015; 2, pp. 73-85.

5. Losada, J.; Nieto, J.J. Properties of a new fractional derivative without singular kernel. Prog. Fract. Differ. Appl.; 2015; 1, pp. 87-92.

6. Yang, X.J.; Srivastava, H.M.; Machado, J.A.T. A new fractional derivative without a singular kernel: application to the modeling of steady heat flow. Therm. Sci.; 2016; 20, pp. 753-756.

7. Fernandez, A.; Ozarslan, M.A.; Baleanu, D. On fractional calculus with general analytic kernels. Appl. Math. Comput.; 2019; 354, pp. 248-265.3921011

8. Teodoro, G.S.; Machado, J.A.T.; de Oliveira, E.C. A review of definitions of fractional derivatives and other operators. J. Comp. Physiol.; 2019; 388, pp. 195-208.3933276

9. Wu, S.; Samraiz, M.; Mehmood, A.; Jarad, F.; Naheed, S. Some symmetric properties and applications of weighted fractional integral operator. Fractals; 2023; 31, 2340011.

10. Samraiz, M.; Mehmood, A.; Iqbal, S.; Naheed, S.; Rehman, G.; Chu, Y.M. Generalized fractional operator with applications in mathematical physics. Chaos Solitons Fractals; 2022; 165, 4508139 112830.

11. Samraiz, M.; Mehmood, A.; Naheed, S.; Rehman, G.; Kashuri, A.; Nonlaopon, K. On novel fractional operators involving the multivariate M-L function. Mathematics; 2022; 10, 21 3991.

12. Huang, W.H.; Samraiz, M.; Mehmood, A.; Baleanu, D.; Rehman, G.; Naheed, S. Modified Atangana- Baleanu fractional operators involving generalized Mittag-Leffler function. Alex. Eng. J.; 2023; 75, pp. 639-648.

13. Cho, Y.; Kim, I.; Sheen, D. A fractional-order model for MINMOD. Millennium. Math. Biosci.; 2015; 262, pp. 36-45.3319783

14. Magin, R.L.; Ovadia, M. Modeling the cardiac tissue electrode interface using fractional calculus. J. Vib. Control; 2008; 14, pp. 1431-1442.

15. Chen, W.; Pang, G. A new definition of fractional Laplacian with application to modeling three-dimensional nonlocal heat conduction. J. Comput. Phys.; 2016; 309, pp. 350-367.3454438

16. Grahovac, N.M.; Zigic, M.M. Modelling of the hamstring muscle group by use of fractional derivatives. Comput. Math. Appl.; 2010; 59, pp. 1695-1700.2595942

17. Rossikhin, Y.A.; Shitikova, M.V. Analysis of two colliding fractionally damped spherical shells in modelling blunt human head impacts. Cent. Eur. J. Phys.; 2013; 11, pp. 760-778.

18. Balci, M.A. Fractional virus epidemic model on financial networks. Open Math.; 2016; 14, pp. 1074-1086.3590232

19. Roman, H.E.; Porto, M. Fractional derivatives of random walks: time series with long-time memory. Phys. Rev. E; 2008; 78, 031127.

20. Moreles, M.A.; Lainez, R. Mathematical modelling of fractional order circuit elements and bioimpedance applications. Commun. Nonlinear Sci. Numer. Simul.; 2017; 46, pp. 81-88.3573857

21. Reyes-Melo, E.; Martinez-Vega, J.; Guerrero-Salazar, C.; Ortiz-Mendez, U. Application of fractional calculus to the modeling of dielectric relaxation phenomena in polymeric materials. J. Appl. Polym. Sci.; 2005; 98, pp. 923-935.

22. Carcione, J.M. Theory and modeling of constant-QP- and S-waves using fractional time derivatives. Geophysics; 2009; 74, pp. 1-11.

23. Jensen, J.L.W.V. Sur les fonctions convexes et les inegalites entre les valeurs moyennes. Acta Math.; 1905; 30, pp. 175-193.1555027

24. Breckner, W.W. Stetigkeitsaussagen fur eine Klasse verallgemein konvexer Funktionen in topologischen linearen Raumen. Publ. Inst. Math.; 1978; 23, pp. 13-20.

25. Chen, H.; Katugampola, U.N. Hermite-Hadamard and Hermite-Hadamard-Fejer type inequalities for generalized fractional integrals. J. Math. Anal. Appl.; 2017; 446, pp. 1274-1291.3563034

26. Khan, T.U.; Khan, M.A. Generalized conformable fractional operators. J. Comput. Appl. Math.; 2019; 346, pp. 378-389.3864167

27. Jarad, F.; Ugurlu, E.; Abdeljawad, T.; Baleanu, D. On a new class of fractional operators. Adv. Differ. Equ.; 2017; 2017, 3688989 247.

28. Qi, Y.; Li, G. Katugampola fractional parameterized inequalities for s-convex function in the fourth sense. Fractals; 2024; 32, 06 pp. 1-16.

29. Qi, Y.; Li, G. Fractional Ostrowski type inequalities for $(s,m)$-convex function with applications. Fractals; 2023; 31, 09 2350128.

30. Qi, Y.; Wen, Q.; Li, G.; Xiao, K.; Wang, S. Discrete Hermite–Hadamard-type inequalities for $(s,m)$-convex function. Fractals; 2022; 30, 07 2250160.

31. Qi, Y.; Li, G.; Wang, S.; Wen, Q.Z. Hermite–Hadamard–Fejér-type inequalities via Katugampola fractional integrals for s-convex functions in the second sense. Fractals; 2022; 30, 07 2250131.

32. Set, E.; Iscan, I.; Sarikaya, M.Z.; Ozdemir, M.E. On new inequalities of Hermite-Hadamard-Fejer’ type for convex functions via fractional integrals. Appl. Math. Comput.; 2015; 259, pp. 875-881.3338419

33. Hakiki, M.T.; Wibowo, A. Hermite-Hadamard-Fejer type inequalities for s-convex functions in the ‘second sense via Riemann-Liouville fractional integral. J. Phys. Conf. Ser.; 2020; 1442, 012039.

34. Gradshteyn, I.S.; Ryzhik, I.M. Table of Integrals Series and Products; 1980; 7 San Diego, Academic Press:

35. Hölder, O.L.: Nachrichten von der Königl Gesel schaft der wissenschaften und der Georg-August-Universität zu Göttingen. 1889, 38–47 (1889)

36. Sarikaya, M.Z.; Dahmani, Z.; Kiris, M.E.; Ahmad, F. $(k,s)$-Riemann-Liouville fractional integral and applications. Hacet. J. Math. Stat.; 2016; 45, 1 pp. 77-89.3524811

37. Samko, S.G.; Kilbas, A.A.; Marichev, O.I. Fractional Integrals and Derivatives, Theory and Applications; 1993; Switzerland, Gordon and Breach:

38. Alqudah, M.A.; Mohammed, P.O.; Abdeljawad, T. Solution of singular integral equation via Riemann-Liouville fractional integral. Math. Probl. Eng.; 2020; 2020, 4160366 1250970.