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M. Adil Khan 1 and J. E. Pecaric 1,2
Recommended by Kunquan Lan
1, Abdus Salam School of Mathematical Sciences, GC University, Lahore 54000, Pakistan
2, Faculty of Textile Technology, University of Zagreb, Zagreb 10002, Croatia
Received 6 March 2010; Accepted 2 June 2010
1. Introduction
In 1981 Slater has proved an interesting companion inequality to Jensen's inequality [1].
Theorem 1.1.
Suppose that [varphi]:I⊆...[arrow right]... is increasing convex function on interval I , for x1 ,x2 ,...,xn ∈I[composite function] (where I[composite function] is the interior of the interval I ) and for p1 ,p2 ,...,pn ≥0 with Pn =∑i=1npi >0 , if ∑i=1npi[varphi]+[variant prime] (xi )>0 , then [figure omitted; refer to PDF] When [varphi] is strictly convex on I , inequality (1.1) becomes equality if and only if xi =c for some c∈I[composite function] and for all i with pi >0 .
It was noted in [2] that by using the same proof the following generalization of Slater's inequality (1981) can be given.
Theorem 1.2.
Suppose that [varphi]:I⊆...[arrow right]... is convex function on interval I , for x1 ,x2 ,...,xn ∈I[composite function] (where I[composite function] is the interior of the interval I ) and for p1 ,p2 ,...,pn ≥0 with Pn =∑i=1npi >0 . Let [figure omitted; refer to PDF] then inequality (1.1) holds.
When [varphi] is strictly convex on I , inequality (1.1) becomes equality if and only if xi =c for some c∈I[composite function] and for all i with pi >0 .
Remark 1.3.
For multidimensional version of Theorem 1.2 see [3].
Another companion inequality to Jensen's inequality is a converse proved by Dragomir and Goh in [4].
Theorem 1.4.
Let [varphi]:I⊆...[arrow right]... be differentiable convex function defined on interval I . If xi ∈I,i=1,2,...,n (n≥2) are arbitrary members and pi ≥0 (i=1,2,...,n) with Pn =∑i=1npi >0, and let [figure omitted; refer to PDF] Then the inequalities [figure omitted; refer to PDF] hold.
In the case when [varphi] is strictly convex, one has equalities in (1.4) if and only if there is some c∈I such that xi =c holds for all i with pi >0.
Matic and Pecaric in [5] proved more general inequality from which (1.1) and (1.4) can be obtained as special cases.
Theorem 1.5.
Let [varphi]:I⊆...[arrow right]... be differentiable convex function defined on interval I and let xi , pi , Pn , x¯, and y¯ be stated as in Theorem 1.4. If d∈I is arbitrary chosen number, then one has [figure omitted; refer to PDF] Also, when [varphi] is strictly convex, one has equality in (1.5) if and only if xi =d holds for all i with pi >0 .
Remark 1.6.
If [varphi], xi , pi , Pn , and x¯ are stated as in Theorem 1.4 and we let ∑i=1npi[varphi][variant prime] (xi )≠0 , also if x¯¯=∑i=1npixi[varphi][variant prime] (xi )/∑i=1npi[varphi][variant prime] (xi )∈I , then by setting d=x¯¯ in (1.5), we get Slater's inequality (1.1) and similarly by setting d=x¯ in (1.5), we get (1.4).
The following refinement of (1.4) is also valid [5].
Theorem 1.7.
Let [varphi]:I⊆...[arrow right]... be strictly convex differentiable function defined on interval I and let xi , pi , Pn , x¯, and y¯ be stated as in Theorem 1.4 and d¯=([varphi][variant prime])-1 ((1/Pn )∑i=1npi[varphi][variant prime] (xi )) , then the inequalities [figure omitted; refer to PDF] [figure omitted; refer to PDF] hold.
The equalities hold in (1.6) and in (1.7) if and only if x1 =x2 =...=xn .
Remark 1.8.
In [6] Dragomir has also proved Theorem 1.7.
In this paper, we use an inequality given in [5] and derive two mean value theorems, exponential convexity, log-convexity, and Cauchy means. As applications, such results are also deduce for related inequality. We use some log-convexity criterion and prove improvement and reverse of Slater's and related inequalities. We also prove some determinantal inequalities.
2. Mean Value Theorems
Theorem 2.1.
Let [varphi]∈C2 (I) , where I is closed interval in ... , and let Pn =∑i=1npi , pi >0 , xi ,d∈I with xi ≠d (i=1,2,...,n) and y¯=(1/Pn )∑i=1npi [varphi](xi ) . Then there exists ξ∈I such that [figure omitted; refer to PDF]
Proof.
Since [varphi][variant prime][variant prime] (x) is continuous on I , m≤[varphi][variant prime][variant prime] (x)≤M for x∈I , where m=min x∈I[varphi][variant prime][variant prime] (x) and M=max x∈I[varphi][variant prime][variant prime] (x) .
Consider the functions [varphi]1 , [varphi]2 defined as [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] [varphi]i (x) for i=1,2 are convex.
Now by applying [varphi]1 for [varphi] in inequality (1.5), we have [figure omitted; refer to PDF] From (2.4) we get [figure omitted; refer to PDF] and similarly by applying [varphi]2 for [varphi] in (1.5), we get [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] by combining (2.5) and (2.6), we have [figure omitted; refer to PDF] Now using the fact that for m≤ρ≤M there exists ξ∈I such that [varphi][variant prime][variant prime] (ξ)=ρ , we get (2.1).
Corollary 2.2.
Let [varphi]∈C2 (I) , where I is closed interval in ... , and let xi , x¯, y¯ , and Pn be stated as in Theorem 1.4 with pi >0 and xi ≠x¯ (i=1,2,...,n) . Then there exists ξ∈I such that [figure omitted; refer to PDF]
Proof.
By setting d=x¯ in Theorem 2.1, we get (2.9).
Theorem 2.3.
Let [varphi],ψ∈C2 (I) , where I is closed interval in ... , and let Pn =∑i=1npi , pi >0 and xi ,d∈I with xi ≠d (i=1,2,...,n) . Then there exists ξ∈I such that [figure omitted; refer to PDF] provided that the denominators are nonzero.
Proof.
Let the function k∈C2 (I) be defined by [figure omitted; refer to PDF] where c1 and c2 are defined as [figure omitted; refer to PDF] Then, using Theorem 2.1 with [varphi]=k , we have [figure omitted; refer to PDF] because k(d)+(1/Pn )∑i=1npi (xi -d)k'(d)-(1/Pn )∑i=1npi k(xi )=0 .
Since (1/Pn )∑i=1npi (xi -d)2 >0 as xi ≠d and pi >0 (i=1,2,...,n) , therefore, (2.13) gives us [figure omitted; refer to PDF] After putting the values of c1 and c2 , we get (2.10).
Corollary 2.4.
Let [varphi],ψ∈C2 (I) , where I is closed interval in ... , and Pn =∑i=1npi , pi >0 and let xi ∈I , x¯=(1/Pn )∑i=1npixi with xi ≠x¯ (i=1,2,...,n) . Then there exists ξ∈I such that [figure omitted; refer to PDF] provided that the denominators are nonzero.
Proof.
By setting d=x¯ in Theorem 2.3, we get (2.15).
Corollary 2.5.
Let xi ,d∈I with xi ≠d and Pn =∑i=1npi , pi >0 (i=1,2,...,n) . Then for u,v∈...\{0,1} , u≠v , there exists ξ∈I , where I is positive closed interval, such that [figure omitted; refer to PDF]
Proof.
By setting [varphi](x)=xu and ψ(x)=xv , x∈I , in Theorem 2.3, we get (2.16).
Corollary 2.6.
Let xi ∈I , Pn =∑i=1npi , pi >0 (i=1,2,...,n), and x¯=(1/Pn )∑i=1npixi with xi ≠x¯ . Then for u,v∈...\{0,1} , u≠v , there exists ξ∈I , where I is positive closed interval, such that [figure omitted; refer to PDF]
Proof.
By setting [varphi](x)=xu and ψ(x)=xv , x∈I , in (2.15), we get (2.17).
Remark 2.7.
Note that we can consider the interval I=[mx ,Mx ] , where mx =min i {xi ,d} , Mx =max i {xi ,d}.
Since the function ξ[arrow right]ξu-v with u≠v is invertible, then from (2.16) we have [figure omitted; refer to PDF] We will say that the expression in the middle is a mean of xi ,d .
From (2.17) we have [figure omitted; refer to PDF] The expression in the middle of (2.19) is a mean of xi .
In fact similar results can also be given for (2.10) and (2.15). Namely, suppose that [varphi][variant prime][variant prime] /ψ[variant prime][variant prime] has inverse function, then from (2.10) and (2.15) we have [figure omitted; refer to PDF] So, we have that the expression on the right-hand side of (2.20) is also means.
3. Improvements and Related Results
Definition 3.1 (see [7, page 2]).
A function [varphi]:I[arrow right]... is convex if [figure omitted; refer to PDF] holds for every s1 <s2 <s3 , s1 ,s2 ,s3 ∈I.
Lemma 3.2 (see [8]).
Let one define the function [figure omitted; refer to PDF] Then [straight phi]t[variant prime][variant prime] (x)=xt-2 , that is, [straight phi]t is convex for x>0 .
Definition 3.3 (see [9]).
A function [varphi]:I[arrow right]... is exponentially convex if it is continuous and [figure omitted; refer to PDF] for all n∈... , ak ∈..., and xk ∈I , k=1,2,...,n such that xk +xl ∈I,1≤k,l≤n, or equivalently [figure omitted; refer to PDF]
Corollary 3.4 (see [9]).
If [varphi] is exponentially convex function, then [figure omitted; refer to PDF] for every n∈... xk ∈I, k=1,2,...,n.
Corollary 3.5 (see [9]).
If [varphi]:I[arrow right](0,∞) is exponentially convex function, then [varphi] is a log-convex function that is [figure omitted; refer to PDF]
Theorem 3.6.
Let xi ,pi ,d∈...+ (i=1,2,...,n) , Pn =∑i=1npi . Consider Γt to be defined by [figure omitted; refer to PDF] Then
(i) for every m∈... and for every sk ∈..., k∈{1,2,3,...,m} , the matrix [Γ(sk +sl )/2]k,l=1m is a positive semidefinite matrix; particularly [figure omitted; refer to PDF]
(ii) the function t[arrow right]Γt is exponentially convex;
(iii): if Γt >0 , then the function t[arrow right]Γt is log-convex, that is, for -∞<r<s<t<∞ , one has [figure omitted; refer to PDF]
Proof.
(i) Let us consider the function defined by [figure omitted; refer to PDF] where skl =(sk +sl )/2,ak ∈... for all k∈{1,2,3,...,m}, x>0
Then we have [figure omitted; refer to PDF] Therefore, μ(x) is convex function for x>0 . Using μ(x) in inequality (1.5), we get [figure omitted; refer to PDF] so the matrix [Γ(sk +sl )/2]k,l=1m is positive semi-definite.
(ii) Since lim t[arrow right]0Γt =Γ0 and lim t[arrow right]1Γt =Γ1 , so Γt is continuous for all t∈..., x>0, and we have exponentially convexity of the function t[arrow right]Γt .
(iii) Let Γt >0 , then by Corollary 3.5 we have that Γt is log-convex, that is, t[arrow right]log Γt is convex, and by (3.1) for -∞<r<s <t<∞ and taking [varphi](t)=log Γt , we get [figure omitted; refer to PDF] which is equivalent to (3.9).
Corollary 3.7.
Let xi ,pi ∈...+ (i=1,2,...,n) , Pn =∑i=1npi and x¯=(1/Pn )∑i=1npixi . Consider Γ...t to be defined by [figure omitted; refer to PDF] Then
(i) for every m∈... and for every sk ∈..., k∈{1,2,3,...,m} , the matrix [Γ...(sk +sl )/2]k,l=1m is a positive semi-definite matrix. Particularly [figure omitted; refer to PDF]
(ii) the function t[arrow right]Γ...t is exponentially convex;
(iii): if Γ...t >0 , then the function t[arrow right]Γ...t is log-convex, that is, for -∞<r<s<t<∞ , one has [figure omitted; refer to PDF]
Proof.
To get the required results, set d=x¯ in Theorem 3.6.
Let x=(x1 ,x2 ,...,xn ) be positive n-tuple and p1 ,p2 ,...,pn positive real numbers, and let Pn =∑i=1npi . Let Mt (x) denote the power mean of order t (t∈...) , defined by [figure omitted; refer to PDF] Let us note that M1 (x)=x¯ .
By (2.18) we can give the following definition of Cauchy means.
Let xi ,d∈I with xi ≠d , I is positive closed interval, and Pn =∑i=1npi , pi >0 (i=1,2,...,n), [figure omitted; refer to PDF] for -∞<u≠v<+∞ are means of xi , d . Moreover we can extend these means to the other cases.
So by limit we have [figure omitted; refer to PDF] where log x=(log x1 ,log x2 ,...,log xn ) .
Theorem 3.8.
Let t,s,u,v∈... such that t≤u,s≤v , then the following inequality is valid: [figure omitted; refer to PDF]
Proof.
For convex function [varphi] it holds that ([7, page 2]) [figure omitted; refer to PDF] with x1 ≤y1 , x2 ≤y2 , x1 ≠x2 , y1 ≠y2 . Since by Theorem 3.6, Γt is log-convex, we can set in (3.21): [varphi](x)=log Γx , x1 =t , x2 =s , y1 =u , and y2 =v , then we get [figure omitted; refer to PDF] From (3.22) we get (3.20) for s≠t and u≠v .
For s=t and u=v we have limiting case.
Similarly by (2.19) we can give the following definition of Cauchy type means.
Let xi ∈I with xi ≠x¯ , I is positive closed interval, and Pn =∑i=1npi , pi >0 (i=1,2,...,n), [figure omitted; refer to PDF] for -∞<u≠v<+∞ are means of xi . Moreover we can extend these means to the other cases.
So by limit we have [figure omitted; refer to PDF] where log x=(log x1 ,log x2 ,...,log xn ) .
Theorem 3.9.
Let t,s,u,v∈... such that t≤u,s≤v , then the following inequality is valid: [figure omitted; refer to PDF]
Proof.
The proof is similar to the proof of Theorem 3.8.
Let Mt (x) be stated as above, define dt as [figure omitted; refer to PDF]
The following improvement and reverse of Slater's inequality are valid.
Theorem 3.10.
Let xi , pi , dt ∈...+ (i=1,2,...,n) , Pn =∑i=1npi . Let Ft be defined by [figure omitted; refer to PDF] Then
(i) [figure omitted; refer to PDF] for -∞<r<s<t<∞ and -∞<t<r<s<∞ .
(ii) [figure omitted; refer to PDF] for -∞<r<t<s<∞ .
where, [figure omitted; refer to PDF]
Proof.
(i) By setting d=dt in (3.7), Γt becomes Ft , and for -∞<r<s<t<∞ , setting d=dt in (3.9), we get [figure omitted; refer to PDF] that is, [figure omitted; refer to PDF] From (3.32) we get (3.28), and similarly for -∞<t<r<s<∞ (3.9) becomes [figure omitted; refer to PDF] by the same process we can get (3.28).
(ii) For -∞<r<t<s<∞ (3.9) becomes [figure omitted; refer to PDF] setting d=dt in (3.34), we get (3.29).
Theorem 3.11.
Let xi ,pi ,dt ∈...+ (i=1,2,...,n) , Pn =∑i=1npi .
Then for every m∈... and for every sk ∈...,k∈{1,2,3,...,m} , the matrices [H((sk +sl )/2,s1 )]k,l=1m , [H((sk +sl )/2,(s1 +s2 )/2)]k,l=1m are positive semi-definite matrices. Particularly [figure omitted; refer to PDF] [figure omitted; refer to PDF] where H(s,t) is defined by (3.30).
Proof.
By setting d=ds1 and d=d(s1 +s2 )/2 in Theorem 3.6(i), we get the required results.
Remark 3.12.
We note that H(t,t)=Ft . So by setting m=2 in (3.35), we have special case of (3.28) for t=s1 , s=s2 , and r=(s1 +s2 )/2 if s1 <s2 and for t=s1 , r=s2 , and s=(s1 +s2 )/2 if s2 <s1 . Similarly by setting m=2 in (3.36), we have special case of (3.29) for r=s1 , s=s2 , t=(s1 +s2 )/2 if s1 <s2 and for r=s2 , s=s1 , t=(s1 +s2 )/2 if s2 <s1 .
Let Mt (x) be stated as above, define d¯t as [figure omitted; refer to PDF]
The following improvement and reverse of inequality (1.6) are also valid.
Theorem 3.13.
Let xi ,pi ,dt ¯∈...+ for all i=1,2,...,n , Pn =∑i=1npi . Let Gt be defined by [figure omitted; refer to PDF] Then
(i) [figure omitted; refer to PDF] for -∞<r<s<t<∞ and -∞<t<r<s<∞ .
(ii) [figure omitted; refer to PDF] for -∞<r<t<s<∞ ,
where [figure omitted; refer to PDF]
Proof.
(i) By setting d=d¯t in (3.9), we get (3.39) for -∞<r<s<t<∞ , and similarly we can get (3.39) for the case -∞<t<r<s<∞ .
(ii) For -∞<r<t<s<∞ (3.9) becomes [figure omitted; refer to PDF] setting d=d¯t in (3.42), we get (3.40).
Theorem 3.14.
Let xi ,pi ,d¯t ∈...+ (i=1,2,...,n) , Pn =∑i=1npi .
Then for every m∈... and for every sk ∈..., k∈{1,2,3,...,m} , the matrices [K((sk +sl )/2,s1 )]k,l=1m , [K((sk +sl )/2,(s1 +s2 )/2)]k,l=1m are positive semi-definite matrices. Particularly [figure omitted; refer to PDF] [figure omitted; refer to PDF] where K(s,t) is defined by (3.41).
Proof.
By setting d=d¯s1 and d=d¯(s1 +s2 )/2 in Theorem 3.6(i), we get the required results.
Remark 3.15.
We note that K(t,t)=Gt . So by setting m=2 in (3.43), we have special case of (3.39) for t=s1 , s=s2 , r=(s1 +s2 )/2 if s1 <s2 and for t=s1 , r=s2 , and s=(s1 +s2 )/2 if s2 <s1 . Similarly by setting m=2 in (3.44), we have special case of (3.40) for r=s1 , s=s2 , and t=(s1 +s2 )/2 if s1 <s2 and for r=s2 , s=s1 , and t=(s1 +s2 )/2 if s2 <s1 .
Acknowledgments
The research of the first and second authors was funded by Higher Education Commission, Pakistan. The research of the second author was supported by the Croatian Ministry of Science, Education, and Sports under the Research Grant 117-1170889-0888.
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Abstract
We use an inequality given by Matic and Pecaric (2000) and obtain improvement and reverse of Slater's and related inequalities.
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer