(ProQuest: ... denotes non-US-ASCII text omitted.)
Jadranka Micic 1 and Zlatko Pavic 2 and Josip Pecaric 3
Recommended by Sergey V. Zelik
1, Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Ivana Lucica 5, 10000 Zagreb, Croatia
2, Mechanical Engineering Faculty, University of Osijek, Trg Ivane Brlic Mazuranic 2, 35000 Slavonski Brod, Croatia
3, Faculty of Textile Technology, University of Zagreb, Prilaz baruna Filipovica 30, 10000 Zagreb, Croatia
Received 15 March 2012; Accepted 9 June 2012
This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Quasiarithmetic means are very important because they are general and unavoidable in applications. This paper begins with the quasiarithmetic means of points, continues with the quasiarithmetic means of measurable function, through the quasiarithmetic means of functions with respect to linear functionals, and ends with the quasiarithmetic means of operators with respect to linear mappings. Conclusion of the paper is dedicated to the applications of operator quasiarithmetic means on power means with strictly positive operators. At this point, it should be emphasized that in all four of the next sections the basic and initial inequality was precisely the Jensen inequality (see Figure 1).
Figure 1: Graphic concept of Jensen's inequality.
[figure omitted; refer to PDF]
The applications of convexity often used strictly monotone continuous functions [straight phi] and ψ such that ψ is convex with respect to [straight phi] ( ψ is [straight phi] convex); that is, f=ψ[composite function][straight phi]-1 is convex by [1, Definition 1.19]. Similar notation is used for concavity. We observe a monotonicity of quasiarithmetic means with these functions [straight phi] and ψ . Good results for the monotonicity of quasiarithmetic means are obtained in [2] for the basic and integral case. The first results for the operator case without operator convexity are obtained in [3, 4]. Among other things, the paper gives some generalizations of the mentioned results.
Through this paper, we suppose that I⊆... is a nondegenerate interval, and [straight phi],ψ:I[arrow right]... are strictly monotone continuous functions. It is assumed that the integer n...5;2 , wherever it appears in inequalities.
2. Results for Basic Case
For n -tuple x=(x1 ,...,xn ) with numbers xi ∈... , sometimes we will write x>0 if all xi >0 , and x...0;c if xi ...0;xj for some i...0;j .
Below is a discrete basic form of Jensen's inequality for a convex function with respect to convex combinations points in interval.
Theorem A.
Let f:I[arrow right]... be a function. Let x=(x1 ,...,xn ) be n -tuple with points xi ∈I , and p=(p1 ,...,pn ) be n -tuple with numbers pi ∈[0,1] such that ∑i=1n ...pi =1 .
A function f is convex if and only if the following inequality [figure omitted; refer to PDF] holds for all above n -tuples p and x .
Consequently, if ∑i=1n ...pi =p>0 , not necessarily equals 1 , then f is convex if and only if [figure omitted; refer to PDF]
A function f is concave if and only if the reverse inequality is valid in (2.1) and (2.2).
A function f is strictly convex if and only if the inequality in (2.1) and (2.2) is strict for all p>0 and x...0;c .
Let [straight phi]:I[arrow right]... be a strictly monotone continuous function. Let x=(x1 ,...,xn ) be n -tuple with points xi ∈I , and p=(p1 ,...,pn ) be n -tuple with numbers pi ∈[0,1] such that ∑i=1n ...pi =1 . The discrete basic [straight phi] -quasiarithmetic mean of points (particles) xi with coefficients (weights) pi is a number [figure omitted; refer to PDF] We understand that px=∑i=1n ...pixi . The [straight phi] -quasiarithmetic mean resulting, first by moving the convex combination px∈I into convex combination p[straight phi](x)∈[straight phi](I) , then its return using [straight phi]-1 back in the interval I . So, the number [physics M-matrix][straight phi] (px) is in the interval I , in fact in the closed interval [min {xi },max {xi }] . If [straight phi] is an identity function on I , that is, [straight phi](x)=id(x)=x for x∈I , then the [straight phi] -quasiarithmetic mean is just a convex combination as follows: [figure omitted; refer to PDF]
Basic quasiarithmetic means have the property [figure omitted; refer to PDF] for every pair of real numbers a and b with a...0;0 .
Suppose that all coefficients pi =1/n . If we take [straight phi]1 (x)=x , then [physics M-matrix][straight phi]1 (px) is the arithmetic mean of numbers xi . If all xi >0 and we take [straight phi]0 (x)=ln x , then [physics M-matrix][straight phi]0 (px) is the geometric mean of numbers xi . If all xi >0 and we take [straight phi]-1 (x)=1/x , then [physics M-matrix][straight phi]-1 (px) is the harmonic mean of numbers xi .
Corollary 2.1.
Let [straight phi],ψ:I[arrow right]... be strictly monotone continuous functions.
A function ψ is either [straight phi] -convex and increasing or [straight phi] -concave and decreasing if and only if following the inequality: [figure omitted; refer to PDF] holds for all n -tuples p and x as in (2.3).
A function ψ is either [straight phi] -concave and increasing or [straight phi] -convex and decreasing if and only if the reverse inequality is valid in (2.6).
A function ψ is strictly [straight phi] -convex if and only if the inequality in (2.6) is strict for all p>0 and x...0;c (see Figure 2).
Figure 2: The ψ -order among quasiarithmetic means [physics M-matrix][straight phi] and [physics M-matrix]ψ for [straight phi] -convex and increasing ψ .
[figure omitted; refer to PDF]
Suppose that all xi >0 . If we apply Corollary 2.1 on three strictly monotone functions [straight phi]-1 (x)=1/x , [straight phi]0 (x)=ln x and [straight phi]1 (x)=x (two by two in pairs), then we get the weighted harmonic-geometric-arithmetic inequality [figure omitted; refer to PDF]
Recall that a function f:I[arrow right]... is convex if and only if the following inequality: [figure omitted; refer to PDF] holds for all triples x,y,z∈I such that x<y<z . A function f is strictly convex if and only if the above inequality is strict. So, the function ψ is [straight phi] -convex if and only if [figure omitted; refer to PDF]
Let u,v:[a0 ,a1 ][arrow right]... , where a0 <a1 , be nonnegative continuous functions so that v/u is a strictly monotone increasing positive function on an open interval Y9;a0 ,a1 YA; , with boundary conditions u(a0 )=v(a1 )=1 and u(a1 )=v(a0 )=0 . Let both [straight phi] and ψ be strictly monotone increasing or decreasing. For any t∈[a0 ,a1 ] , we define a strictly monotone continuous function [figure omitted; refer to PDF] For example, we can take u(t)=1-t and v(t)=t for t∈[0,1] , u(t)=1-t and v(t)=t2 for t∈[0,1] , u(t)=cos t and v(t)=sin t for t∈[0,π/2] .
Lemma 2.2.
Let a , b , α , β be real numbers.
If a...4;b and α...4;β , then [figure omitted; refer to PDF] provided that denominators are the same sign. The inequality in (2.11) is strict if a<b and α<β .
If either a...5;b and α...4;β or a...4;b and α...5;β , then the reverse inequality is valid in (2.11).
Proposition 2.3.
Let [varphi]t =u(t)[straight phi]+v(t)ψ:I[arrow right]... for t∈[a0 ,a1 ] be functions as in (2.10). Let t0 ,t1 ∈[a0 ,a1 ] such that t0 ...4;t1 .
If ψ is [straight phi] -convex (resp. [straight phi] -concave), then [varphi]t1 is [varphi]t0 -convex (resp. [varphi]t0 -concave).
Proof.
Suppose that ψ is [straight phi] -convex. Show that the function [varphi]t1 is [varphi]t0 -convex. If t0 =a1 , then t0 =t1 , so we can suppose that t0 <a1 . Let x,y,z∈I such that x<y<z . Let, with respect to (2.9) and definition of functions u and v , [figure omitted; refer to PDF] Note that numbers a and b are positive because both [straight phi] and ψ are strictly monotone increasing or decreasing. Applying Lemma 2.2 with a , b , α , and β , we obtain that [figure omitted; refer to PDF] which shows the required convexity by (2.9). Case of the concavity can be proved in a similar way.
If ψ is strictly [straight phi] -convex (resp. [straight phi] -concave), then [varphi]t1 is strictly [varphi]t0 -convex (resp. [varphi]t0 -concave).
According to Proposition 2.3, we can express refinements of the basic quasiarithmetic means.
Theorem 2.4.
Let [varphi]t =u(t)[straight phi]+v(t)ψ:I[arrow right]... for t∈[a0 ,a1 ] be functions as in (2.10). Let t0 ,t1 ∈[a0 ,a1 ] such that t0 ...4;t1 .
If either ψ is [straight phi] -convex with both [straight phi] and ψ increasing or [straight phi] -concave with both [straight phi] and ψ decreasing, then the following inequality: [figure omitted; refer to PDF] holds for all n -tuples p and x as in (2.3).
If either ψ is [straight phi] -concave with both [straight phi] and ψ increasing or [straight phi] -convex with both [straight phi] and ψ decreasing, then the reverse inequality is valid in (2.14).
Proof.
If ψ is [straight phi] -convex with both [straight phi] and ψ increasing, then the function [varphi]t1 is increasing, and [varphi]t0 -convex by Proposition 2.3, and according to Corollary 2.1 the inequality in (2.14) is valid. In the same way, we prove the concavity case.
In other words, the above theorem says that a function [figure omitted; refer to PDF] is monotone increasing for any fixed p and x as in (2.3). In the case n=2 it is proved in [2, Lemma 4] for functions u(t)=1-t and v(t)=t with t∈[0,1] .
We emphasize that the inequality in (2.14) is strict for a0 <t0 <t1 <a1 if ψ is strictly [straight phi] -convex or [straight phi] -concave, p>0 and x...0;c .
Let us take strictly monotone decreasing functions [straight phi](x)=1/x and ψ(x)=-ln x with x>0 . Then (ψ[composite function][straight phi]-1 )(x)=ln x , so ψ is strictly [straight phi] -concave. Let [figure omitted; refer to PDF] If we apply the inequality in (2.14) with t=t0 =t1 on [varphi]thg , we get [figure omitted; refer to PDF]
Let us take strictly monotone increasing functions [straight phi](x)=ln x and ψ(x)=x with x>0 . Then (ψ[composite function][straight phi]-1 )(x)=ex , so ψ is strictly [straight phi] -convex. Let [figure omitted; refer to PDF] If we apply the inequality in (2.14) with t=t0 =t1 on [varphi]tga , we get [figure omitted; refer to PDF]
Connecting two above inequalities results in [figure omitted; refer to PDF]
The inequality in (2.20) is strict for a0 <t<a1 if all pi >0 and xi ...0;xj for some i...0;j , so in this case, we have refinements of the weighted harmonic-geometric-arithmetic inequality.
The weighted harmonic-geometric-arithmetic inequality is only the special case of a whole collection of inequalities which can be derived by applying of Corollary 2.1 on power means. As a special case of the basic quasiarithmetic mean in (2.3) with I=Y9;0,+∞YA; , [straight phi]r (x)=xr for r...0;0 and [straight phi]0 (x)=ln x , we can observe the discrete basic power mean [figure omitted; refer to PDF]
Very useful consequence of Corollary 2.1 is a well-known property of monotonicity of basic power means.
Corollary 2.5.
If r and s are real numbers such that r...4;s , then the following inequality: [figure omitted; refer to PDF] holds for all n -tuples p and x as in (2.3) with I=Y9;0,+∞YA; .
The inequality in (2.22) is strict for r<s if p>0 and x...0;c .
Let functions [varphi]t[r,s] :Y9;0,+∞YA;[arrow right]... for t∈[a0 ,a1 ] be specially defined by [figure omitted; refer to PDF] Then the functions t...[physics M-matrix][varphi]t[r,s] (px) with x>0 are monotone increasing in the next four cases.
Case r<s<0 .
Functions [straight phi](x)=xr and ψ(x)=xs are strictly monotone decreasing with strictly concave (ψ[composite function][straight phi]-1 )(x)=xs/r because 0<(s/r)<1 .
Case r<0=s .
Functions [straight phi](x)=xr and ψ(x)=-ln x are strictly monotone decreasing with strictly concave (ψ[composite function][straight phi]-1 )(x)=-(1/r)ln x because -(1/r)>0 .
Case r=0<s .
Functions [straight phi](x)=ln x and ψ(x)=xs are strictly monotone increasing with strictly convex (ψ[composite function][straight phi]-1 )(x)=esx .
Case 0<r<s .
Functions [straight phi](x)=xr and ψ(x)=xs are strictly monotone increasing with strictly convex (ψ[composite function][straight phi]-1 )(x)=xs/r because (s/r)>1 .
Given traditional signs of power means, we will mark [physics M-matrix][varphi]t[r,s] (px) with [physics M-matrix]n[varphi]t[r,s] (px) . The inequality in (2.22) can be refined using Theorem 2.4 with functions [varphi]t[r,s] . The following are refinements of power means.
Corollary 2.6.
Let r,s∈... such that r<s . Let [varphi]t[r,s] :Y9;0,+∞YA;[arrow right]... for t∈[a0 ,a1 ] be functions as in (2.23). Let t0 ,t1 ∈[a0 ,a1 ] such that t0 ...4;t1 .
If r<s<0 or r<0=s or r=0<s or 0<r<s , then the inequality [figure omitted; refer to PDF] holds for all n -tuples p and x as in (2.3) with I=Y9;0,+∞YA; .
If r<0<s , then we can take the series of inequalities [figure omitted; refer to PDF]
The inequalities in (2.24)-(2.25) are strict for a0 <t0 <t1 <a1 if p>0 and x...0;c .
The inequality in (2.20) is a special case of the collection of inequalities in (2.24).
3. Applications on Integral Case
In this section, (Ω,μ) is a probability measure space. It is assumed that every weighted function w:Ω[arrow right]... is nonnegative almost everywhere on Ω , that is, w(ω)...5;0 for almost all ω∈Ω .
For n -tuple g=(g1 ,...,gn ) with functions gi :Ω[arrow right]... , sometimes we will write g>0 if all gi >0 almost everywhere on Ω , and g...0;c if gi ...0;gj almost everywhere on Ω for some i...0;j .
Here is an integral form of Jensen's inequality for a convex function with respect to measurable functions with weighted functions on the probability measure space.
Theorem B.
Let f:I[arrow right]... be a function. Let (Ω,μ) be a probability measure space, g:Ω[arrow right]I be a measurable function, and w∈L1 (Ω,μ) be a weighted function with ∫Ω ...w dμ=1 such that w·g,w·(f[composite function]g)∈L1 (Ω,μ) .
If a function f is convex, then the inequality [figure omitted; refer to PDF] holds for all above w , g and μ .
Consequently, if ∫Ω ...w dμ=p>0 , not necessarily equals 1 , then [figure omitted; refer to PDF]
If a function f is concave, then the reverse inequality is valid in (3.1) and (3.2).
The assumption ∫Ω ...w dμ=1 with nonnegative w almost everywhere on Ω , for the inequality in (3.1), assures that [figure omitted; refer to PDF]
Remark 3.1.
The reverse of Theorem B is valid if for any p∈[0,1] a measurable set Ωp ⊆Ω exists so that μ(Ωp )=p . In this case, we can determine a simple measurable function [figure omitted; refer to PDF] where χ is a characteristic set function, for every x,y∈I and p∈[0,1] . If we take w=1 at the same, then [figure omitted; refer to PDF] If we include these integrals in the inequality in (3.1), we have the convexity of the function f .
Theorem B can be generalized to n probability measures μi and n measurable functions gi with weighted functions wi . The following is a discrete integral form of Jensen's inequality.
Theorem 3.2.
Let f:I[arrow right]... be a function. Let μ=(μ1 ,...,μn ) be n -tuple with probability measures μi on Ω , g=(g1 ,...,gn ) be n -tuple with measurable functions gi :Ω[arrow right]I , and w=(w1 ,...,wn ) be n -tuple with weighted functions wi ∈L1 (Ω,μi ) with ∑i=1n ...∫Ω ...wi dμi =1 such that wi ·gi ,wi ·(f[composite function]gi )∈L1 (Ω,μi ) .
A function f is convex if and only if the inequality [figure omitted; refer to PDF] holds for all above n -tuples w , g and μ .
Consequently, if ∑i=1n ...∫Ω ...wi dμi =p>0 , not necessarily equals 1 , then f is convex if and only if [figure omitted; refer to PDF]
A function f is concave if and only if the reverse inequality is valid in (3.6) and (3.7).
In the proof of sufficiency theorem, we simply take wi =pi and gi =xi in which case the inequality in (3.6) and (3.7) becomes the basic inequality of convexity. The fact that n...5;2 is coming to the fore.
A function f is strictly convex if and only if the inequality in (3.6) and (3.7) is strict for all w>0 and g...0;c .
Let [straight phi]:I[arrow right]... be a strictly monotone continuous function. Let μ=(μ1 ,...,μn ) be n -tuple with probability measures μi on Ω , g=(g1 ,...,gn ) be n -tuple with measurable functions gi :Ω[arrow right]I , and w=(w1 ,...,wn ) be n -tuple with weighted functions wi ∈L1 (Ω,μi ) with ∑i=1n ...∫Ω ...wi dμi =1 such that wi ·([straight phi][composite function]gi )∈L1 (Ω,μi ) . The discrete integral [straight phi] -quasiarithmetic mean of measurable functions gi with weighted functions wi with respect to measures μi (namely, with respect to integrals ∫Ω ...· dμi ) is a number [figure omitted; refer to PDF] This number belongs to I because the integral convex combination ∑i=1n ...∫Ω ...wi ·([straight phi][composite function]gi )dμi ∈[straight phi](I) . Integral quasiarithmetic means also satisfy the property [figure omitted; refer to PDF] for every pair of real numbers a and b with a...0;0 .
Bearing in mind Theorem 3.2, the following corollary is valid.
Corollary 3.3.
Let [straight phi],ψ:I[arrow right]... be strictly monotone continuous functions.
A function ψ is either [straight phi] -convex and increasing or [straight phi] -concave and decreasing if and only if the inequality [figure omitted; refer to PDF] holds for all n -tuples w , g , and μ as in (3.8).
A function ψ is either [straight phi] -concave and increasing or [straight phi] -convex and decreasing if and only if the reverse inequality is valid in (3.10).
Combining basic and integral case by Corollaries 2.1 and 3.3, we get the following.
Proposition 3.4.
Let [straight phi],ψ:I[arrow right]... be strictly monotone continuous functions. Then the inequality [figure omitted; refer to PDF] holds for all n -tuples p and x as in (2.3) if and only if the inequality [figure omitted; refer to PDF] holds for all n -tuples w , g , and μ as in (3.8).
The one direction of Proposition 3.4 is proved in [2, Theorem 1]. It is proved that the inequality for basic case implies the inequality for integral case with one function g .
The following integral analogy of Theorem 2.4.
Theorem 3.5.
Let [varphi]t =u(t)[straight phi]+v(t)ψ:I[arrow right]... for t∈[a0 ,a1 ] be functions as in (2.10). Let t0 ,t1 ∈[a0 ,a1 ] such that t0 ...4;t1 .
If either ψ is [straight phi] -convex with both [straight phi] and ψ increasing or [straight phi] -concave with both [straight phi] and ψ decreasing, then the inequality [figure omitted; refer to PDF] holds for all n -tuples w , g , and μ as in (3.8).
If either ψ is [straight phi] -concave with both [straight phi] and ψ increasing or [straight phi] -convex with both [straight phi] and ψ decreasing, then the reverse inequality is valid in (3.13).
The inequality in (3.13) is strict for a0 <t0 <t1 <a1 if ψ is strictly [straight phi] -convex, w>0 and g...0;c .
An integral version of refinements of the harmonic-geometric-arithmetic inequality is also valid. So, the inequality [figure omitted; refer to PDF] holds for all n -tuples w=(w1 ,...,wn ) , g=(g1 ,...,gn ) and μ=(μ1 ,...,μn ) as in (3.8) with I=Y9;0,+∞YA; . The above inequality is strict for a0 <t<a1 if all wi >0 almost everywhere on Ω and gi ...0;gj almost everywhere on Ω for some i...0;j .
As a special case of the integral quasiarithmetic mean in (3.8) with I=Y9;0,+∞YA; , [straight phi]r (x)=xr for r...0;0 , and [straight phi]0 (x)=ln x , we can observe the integral power mean [figure omitted; refer to PDF]
We quote the integral analogy of Corollary 2.6. The following is the property of monotonicity, with refinements, of integral power means.
Corollary 3.6.
Let r,s∈... such that r<s . Let [varphi]t[r,s] :Y9;0,+∞YA;[arrow right]... for t∈[a0 ,a1 ] be functions as in (2.23). Let t0 ,t1 ∈[a0 ,a1 ] such that t0 ...4;t1 .
If r<s<0 or r<0=s or r=0<s or 0<r<s , then the inequality [figure omitted; refer to PDF] holds for all n -tuples w , g , and μ as in (3.8) with I=Y9;0,+∞YA; .
If r<0<s , then we can take the series of inequalities [figure omitted; refer to PDF]
The inequalities in (3.16)-(3.17) are strict for a0 <t0 <t1 <a1 if w>0 and g...0;c .
All the observed integral cases are reduced to the corresponding basic cases when we take constants gi =xi and wi =pi .
4. Applications on Functional Case
Let S be a nonempty set and ...AE; be a vector space of real-valued functions g:S[arrow right]... . Linear functional P:...AE;[arrow right]... is positive (nonnegative) or monotone if P(g)...5;0 for every nonnegative function g∈...AE; . If a space ...AE; contains a unit function 1 , by definition 1(s)=1 for every s∈S , and P(1)=1 , we say that functional P is unital or normalized.
In this section, it is assumed that every weighted function w:S[arrow right]... is nonnegative, that is, w(s)...5;0 for every s∈S .
Bellow is a functional form of Jensen's inequality for a convex function with respect to real-valued functions with weighted functions on the vector space of real-valued functions.
Theorem C.
Let f:I[arrow right]... be a continuous function where I is the closed interval. Let P:...AE;[arrow right]... be a positive linear functional, g:S[arrow right]I be a function, and w∈...AE; be a weighted function with P(w)=1 such that w·g,w·(f[composite function]g)∈...AE; .
If a function f is convex, then the inequality [figure omitted; refer to PDF] holds for all above w , g , and P .
Consequently, if P(w)=p>0 , not necessarily equals 1 , then [figure omitted; refer to PDF]
If a function f is concave, then the reverse inequality is valid in (4.1) and (4.2).
The inequality in (4.1) with w=1 (assuming 1∈...AE; and P(1)=1 ) is usually called the Jessen functional form of Jensen's inequality.
The interval I must be closed; otherwise, it could happen that P(w·g)∉I or P(w·(f[composite function]g))∉f(I) . The following example shows such an undesirable situation.
Example 4.1.
Let S=I=Y9;0,1] and [figure omitted; refer to PDF] If P:...AE;[arrow right]... is defined by [figure omitted; refer to PDF] then P is positive linear functional. In that way, functional P is also unital because 1∈...AE; and P(1)=1 . If we take g(x)=x for x∈I , then g∈...AE; and its image in I , but P(g)=0∉I .
Remark 4.2.
Suppose that 1∈...AE; and functional P is unital, that is, P(1)=1 . Then the reverse of Theorem C is valid if for any p∈[0,1] a subset Sp ⊆S exists so that χSp ∈...AE; and P(χSp )=p . If we take g=xχSp +yχS\Sp and w=1 , then it follows that [figure omitted; refer to PDF] for every x,y∈I and p∈[0,1] . If we include these expressions in the inequality in (4.1), we get the convexity of f .
Theorem C can be generalized to n linear functionals Pi and n functions gi with weighted functions wi . The following is a discrete functional form of Jensen's inequality.
Theorem 4.3.
Let f:I[arrow right]... be a continuous function where I is the closed interval. Let P=(P1 ,...,Pn ) be n -tuple with positive linear functionals Pi :...AE;[arrow right]... , g=(g1 ,...,gn ) be n -tuple with functions gi :S[arrow right]I , and w=(w1 ,...,wn ) be n -tuple with weighted functions wi ∈...AE; with ∑i=1n ...Pi (wi )=1 such that wi ·gi , wi ·(f[composite function]gi )∈...AE; .
If a function f is convex, then the inequality [figure omitted; refer to PDF] holds for all above n -tuples w , g , and P .
Consequently, if ∑i=1n ...Pi (wi )=p>0 , not necessarily equals 1 , then [figure omitted; refer to PDF]
If a function f is concave, then the reverse inequality is valid in (4.6) and (4.7).
Proof.
Let us prove the inequality in (4.6). If Pi (wi )=0 for some i , then Pi (wi ·gi )=0 . Without loss of generality, suppose that all pi =Pi (wi )>0 . Let xi =(1/pi )Pi (wi ·gi ) . All numbers xi belong to I . Then from the basic inequality in (2.1) and functional inequality in (4.2), it follows that [figure omitted; refer to PDF]
If f is strictly convex, then the inequality in (4.6) and (4.7) is strict for all w>0 and g...0;c .
Remark 4.4.
Suppose that 1∈...AE; and all functionals Pi are unital; that is, Pi (1)=1 holds for all Pi . Then it is c·1∈...AE; and Pi (c·1)=cPi (1)=c for every constant c∈... . With the above assumptions, the reverse of Theorem 4.3 follows trivially if we take wi =pi and gi =xi .
Let [straight phi]:I[arrow right]... be a strictly monotone continuous function where I is the closed interval. Let P=(P1 ,...,Pn ) be n -tuple with positive linear functionals Pi :...AE;[arrow right]... , g=(g1 ,...,gn ) be n -tuple with functions gi :S[arrow right]I , and w=(w1 ,...,wn ) be n -tuple with weighted functions wi ∈...AE; with ∑i=1n ...Pi (wi )=1 such that wi ·([straight phi][composite function]gi )∈...AE; . The discrete functional [straight phi] -quasiarithmetic mean of functions gi with weighted functions wi with respect to functionals Pi is a number [figure omitted; refer to PDF] This number belongs to I because the functional convex combination ∑i=1n ...Pi (wi ·([straight phi][composite function]gi )) belongs to [straight phi](I) . Functional quasiarithmetic means also satisfy the property [figure omitted; refer to PDF] for every pair of real numbers a and b with a...0;0 . Indeed, if [varphi](x)=a[straight phi](x)+b , then [varphi]-1 (x)=[straight phi]-1 (x-b/a) , and we have [figure omitted; refer to PDF]
Corollary 4.5.
Let [straight phi],ψ:I[arrow right]... be strictly monotone continuous functions where I is the closed interval.
If a function ψ is either [straight phi] -convex and increasing or [straight phi] -concave and decreasing, then the inequality [figure omitted; refer to PDF] holds for all n -tuples w , g , and P as in (4.9).
If a function ψ is either [straight phi] -concave and increasing or [straight phi] -convex and decreasing, then the reverse inequality is valid in (4.12).
Proof.
Suppose that ψ is [straight phi] -convex and increasing. If we apply the inequality in (4.6) with f=ψ[composite function][straight phi]-1 :f(I)[arrow right]... , and ψ[composite function]gi :S[arrow right]f(I) instead of gi , we get [figure omitted; refer to PDF] After taking ψ-1 of the both sides, it follows that [figure omitted; refer to PDF] In the same way, we can prove the case when ψ is [straight phi] -concave and decreasing.
According to Remark 4.2, the reverse of Corollary 4.5 is valid if 1∈...AE; and all functionals Pi are unital. Then we connect the basic and functional case in the following proposition.
Proposition 4.6.
Let [straight phi],ψ:I[arrow right]... be strictly monotone continuous functions where I is the closed interval. Then the inequality [figure omitted; refer to PDF] holds for all p and x as in (2.3) if and only if the inequality [figure omitted; refer to PDF] holds for all n -tuples w , g , and P as in (4.9) with 1∈...AE; and unital functionals Pi .
Next in line is a functional analogy of refinements.
Theorem 4.7.
Let [varphi]t =u(t)[straight phi]+v(t)ψ:I[arrow right]... for t∈[a0 ,a1 ] be functions as in (2.10) where I is the closed interval. Let t0 ,t1 ∈[a0 ,a1 ] such that t0 ...4;t1 .
If either ψ is [straight phi] -convex with both [straight phi] and ψ increasing or [straight phi] -concave with both [straight phi] and ψ decreasing, then the inequality [figure omitted; refer to PDF] holds for all n -tuples w , g , and P as in (4.9) with 1∈...AE; and unital functionals Pi .
If either ψ is [straight phi] -concave with both [straight phi] and ψ increasing or [straight phi] -convex with both [straight phi] and ψ decreasing, then the reverse inequality is valid in (4.17).
The inequality in (4.17) is strict for a0 <t0 <t1 <a1 if ψ is strictly [straight phi] -convex, w>0 and g...0;c .
A functional version of refinements of the harmonic-geometric-arithmetic inequality is also valid. So, the inequality [figure omitted; refer to PDF] holds for all n -tuples w=(w1 ,...,wn ) , g=(g1 ,...,gn ) and, P=(P1 ,...,Pn ) as in (4.9) with I=[a,+∞YA; where a>0 . The above inequality is strict for a0 <t<a1 if all wi >0 and gi ...0;gj for some i...0;j .
As a special case of the functional quasiarithmetic mean in (4.9) with I=[a,+∞YA; where a>0 , [straight phi]r (x)=xr for r...0;0 and [straight phi]0 (x)=ln x , we can observe the functional power mean [figure omitted; refer to PDF]
The following is the property of monotonicity, with refinements, of functional power means.
Corollary 4.8.
Let r,s∈... such that r<s . Let [varphi]t[r,s] :[a,+∞YA;[arrow right]... for t∈[a0 ,a1 ] be functions as in (2.23) where a>0 . Let t0 , t1 ∈[a0 ,a1 ] such that t0 ...4;t1 .
If r<s<0 or r<0=s or r=0<s or 0<r<s , then the inequality [figure omitted; refer to PDF] holds for all n -tuples w , g , and P as in (4.9) with I=[a,+∞YA; , 1∈...AE; and unital functionals Pi .
If r<0<s , then we can take the series of inequalities [figure omitted; refer to PDF]
The inequalities in (4.20)-(4.21) are strict for a0 <t0 <t1 <a1 if w>0 and g...0;c .
All the observed functional cases are reduced to the corresponding integral cases when we take [figure omitted; refer to PDF] for all functions gi ∈L1 (Ω,μi ) such that gi (ω)∈I for almost all ω∈Ω .
5. Results for Operator Case
We recall some notations and definitions. Let H be a Hilbert space. We define the bounds of linear operator A:H[arrow right]H with [figure omitted; refer to PDF]
Let [Bernoulli](H) be the C* -algebra of all bounded linear operators A:H[arrow right]H . If Sp(A) denotes the spectrum of a self-adjoint operator A∈[Bernoulli](H) , then it is well-known that Sp(A) is a subset of ... and Sp(A)⊆[mA ,MA ] . If 1H denotes the identity operator on H , then the following holds: [figure omitted; refer to PDF]
A continuous function f:I[arrow right]... is said to be operator increasing on I if [figure omitted; refer to PDF] for every pair of self-adjoint operators A,B on H with spectra in I . A function f is said to be operator decreasing if-- f is operator increasing. A function f is operator monotone if it is operator increasing or decreasing.
For n -tuple A=(A1 ,...,An ) with operators Ai ∈[Bernoulli](H) sometimes, we will write A>0 if all Ai >0 , and A...0;C if Ai ...0;Aj for some i...0;j .
In this section, it is assumed that every weighted operator W∈[Bernoulli](H) is positive.
From the second half of the last century, Jensen's inequality was formulated for operator convex functions, self-adjoint operators, and positive linear mappings (see [5-8]). Very recently, Jensen's inequality for operators without operator convexity is formulated in [3], and generalized in [4].
The following theorem essentially coincides with the main theorem in [3]. The only difference is that now we add the weighted operators. We also give a short proof of the theorem that relies on the geometric property of convexity and affinity of the chord line or support line. So, we start with an operator form of Jensen's inequality for a convex function with respect to self-adjoint operators with weighted operators on the Hilbert space, and positive linear mappings.
Theorem 5.1.
Let f:I[arrow right]... be a continuous function. Let Φ=(Φ1 ,...,Φn ) be n -tuple with positive linear mappings Φi :[Bernoulli](H)[arrow right][Bernoulli](K) , A=(A1 ,...,An ) be n -tuple with self-adjoint operators Ai ∈[Bernoulli](H) with bounds mi ...4;Mi from I , and W=(W1 ,...,Wn ) be n -tuple with weighted operators Wi ∈[Bernoulli](H) with ∑i=1n ...Φi (Wi )=1K . Let mB ...4;MB be bounds of an operator B=∑i=1nΦi (WiAi ) .
If a function f is convex, then the inequality [figure omitted; refer to PDF] holds for all above n -tuples W , A , and Φ provided spectral conditions [figure omitted; refer to PDF]
Consequently, if ∑i=1n ...Φi (Wi )=WΦ is strictly positive, not necessarily equals 1K , then [figure omitted; refer to PDF]
If a function f is concave, then the reverse inequality is valid in (5.4) and (5.6).
Proof.
If mB <MB , then we take the chord line f[mB ,MB ]cho (x)=kx+l through the points T1 (mB ,f(mB )) , and T2 (MB ,f(MB )) . It follows: [figure omitted; refer to PDF]
If mB =MB , then we take any support line f[mB ]sup (x)=kx+l instead of the chord line.
If f is strictly convex, then the inequality in (5.4) and (5.6) is strict for all W>0 and A...0;C .
Remark 5.2.
The reverse of Theorem 5.1 is trivially valid if all Φi are unital. With this assumption, we can take Wi =pi1H and Ai =xi1H .
Let [straight phi]:I[arrow right]... be a strictly monotone continuous function. Let Φ=(Φ1 ,...,Φn ) be n -tuple with positive linear mappings Φi :[Bernoulli](H)[arrow right][Bernoulli](K) , A=(A1 ,...,An ) be n -tuple with self-adjoint operators Ai ∈[Bernoulli](H) with spectra in I , and W=(W1 ,...,Wn ) be n -tuple with weighted operators Wi ∈[Bernoulli](H) with ∑i=1n ...Φi (Wi )=1K . The discrete operator [straight phi] -quasiarithmetic mean of operators Ai with weighted operators Wi with respect to mappings Φi is an operator [figure omitted; refer to PDF] The spectrum of operator [physics M-matrix][straight phi] (WA,Φ) is contained in I because the spectrum of operator ∑i=1n ...Φi (Wi [straight phi](Ai )) is contained in [straight phi](I) . Operator quasiarithmetic means also have the property [figure omitted; refer to PDF] for every pair of real numbers a and b with a...0;0 . To verify this equality, let us take [varphi]=a[straight phi]+b , so [varphi]-1 (B)=[straight phi]-1 ((B-b1K )/a) if B∈[Bernoulli](K) , and we get [figure omitted; refer to PDF]
Corollary 5.3.
Let [straight phi],ψ:I[arrow right]... be strictly monotone continuous functions with operator monotone ψ-1 .
Let W , A and Φ be as in (5.8). Let mi ...4;Mi and m[straight phi] ...4;M[straight phi] be bounds of operators Ai and [physics M-matrix][straight phi] (WA,Φ) , respectively.
If a function ψ is either [straight phi] -convex with operator increasing ψ-1 or [straight phi] -concave with operator decreasing ψ-1 , then the inequality [figure omitted; refer to PDF] holds for all above n -tuples W , A , and Φ provided spectral conditions [figure omitted; refer to PDF]
If a function ψ is either [straight phi] -concave with operator increasing ψ-1 or [straight phi] -convex with operator decreasing ψ-1 , then the reverse inequality is valid in (5.11).
The following is operator analogy of Theorem 2.4.
Theorem 5.4.
Let [varphi]t =u(t)[straight phi]+v(t)ψ:I[arrow right]... for t∈[a0 ,a1 ] be functions as in (2.10) with operator monotone ψ-1 . Let t0 ,t1 ∈[a0 ,a1 ] such that t0 ...4;t1 .
If either ψ is [straight phi] -convex with operator increasing [varphi]t0 -1 , [varphi]t1 -1 , and ψ-1 or [straight phi] -concave with operator decreasing [varphi]t0 -1 , [varphi]t1 -1 , and ψ-1 , then the inequality [figure omitted; refer to PDF] holds for all n -tuples W , A , and Φ as in (5.8) that provided the following spectral conditions: [figure omitted; refer to PDF]
If either ψ is [straight phi] -concave with operator increasing [varphi]t0 -1 , [varphi]t1 -1 , and ψ-1 or [straight phi] -convex with operator decreasing [varphi]t0 -1 , [varphi]t1 -1 , and ψ-1 , then the reverse inequality is valid in (5.13).
Proof.
Let us prove the middle part of the inequality in (5.13), one that refers to [varphi]t0 and [varphi]t1 . If ψ is [straight phi] -convex with both [straight phi] and ψ increasing, then [varphi]t1 is [varphi]t0 -convex by Proposition 2.3. If [varphi]t1 -1 is operator increasing, then by Corollary 5.3 the inequality [figure omitted; refer to PDF] is valid with spectral conditions [figure omitted; refer to PDF] Any part of the series of inequalities in (5.13) is proved similarly.
The inequality in (5.13) is strict for a0 <t0 <t1 <a1 if ψ is strictly [straight phi] -convex, [varphi]t0 -1 , [varphi]t1 -1 , and ψ-1 are strictly operator increasing, W>0 and A...0;C .
We are interested in sufficient conditions under which the functions [varphi]t will be operator increasing.
Lemma 5.5.
Let [varphi]=α[straight phi]+βψ be a convex combination (α,β...5;0 , α+β=1) of strictly monotone increasing or decreasing continuous functions [straight phi],ψ:I[arrow right]... such that [straight phi](I)=ψ(I)=J . Then [figure omitted; refer to PDF] where u,v:J[arrow right]... are nonnegative continuous functions such that u(y)+v(y)=1 for every y∈J .
Proof.
Take any x∈I . If [figure omitted; refer to PDF] then [figure omitted; refer to PDF] for some nonnegative numbers u(y) and v(y) such that u(y)+v(y)=1 (see Figure 3). Now, first replace v(y) with 1-u(y) in expression in (5.19), and then express u(y) . Realizing u(y) as a function of the variable y , we obtain that [figure omitted; refer to PDF] The above limit is onesided if [straight phi]-1 =ψ-1 on some subinterval of an interval J . The functions [straight phi]-1 ,ψ-1 , and [varphi]-1 are continuous on J , and the same is true for the function u . Thus, the expression in (5.19) is the required presentation of function [varphi]-1 as the convex combination of functions [straight phi]-1 and ψ-1 with coefficient functions u and v .
Figure 3: Convex combination of strictly monotone increasing functions [straight phi] and ψ .
[figure omitted; refer to PDF]
Theorem 5.4 can be simplified by using Lemma 5.5.
Corollary 5.6.
Let [varphi]t =u(t)[straight phi]+v(t)ψ:I[arrow right]... for t∈[a0 ,a1 ] be functions as in (2.10) with the addition of u(t)+v(t)=1 , [straight phi](I)=ψ(I) and operator monotone ψ-1 . Let t0 ,t1 ∈[a0 ,a1 ] such that t0 ...4;t1 .
If either ψ is [straight phi] -convex with operator increasing ψ-1 or [straight phi] -concave with operator decreasing ψ-1 , then the inequality in (5.13) holds for all n -tuples W , A , and Φ as in (5.8) with spectral conditions as in Theorem 5.4.
If either ψ is [straight phi] -concave with operator increasing ψ-1 or [straight phi] -convex with operator decreasing ψ-1 , then the reverse inequality is valid in (5.13).
Proof.
According to Lemma 5.5 continuous functions ut and vt , with ut +vt =1 , exist for every t∈[a0 ,a1 ] so that [figure omitted; refer to PDF] Let t>a0 ; otherwise, it is [varphi]a0 -1 =[straight phi]-1 . Then vt ...0;0 and [varphi]t-1 is operator increasing (resp. decreasing) if ψ-1 is operator increasing (resp. decreasing).
A special case of the operator quasiarithmetic mean in (5.8) with I=Y9;0,+∞YA; , [straight phi]r (x)=xr for r...0;0 and [straight phi]0 (x)=ln x , we can observe the operator power mean [figure omitted; refer to PDF]
The consequence of Corollary 5.3 for operator power means the following.
Corollary 5.7.
Let r and s be real numbers such that r...4;s .
Let W , A , and Φ be as in (5.8) with strictly positive A . Let mi ...4;Mi , m[r] ...4;M[r] , and m[s] ...4;M[s] be bounds of operators Ai , [physics M-matrix]n[r] (WA,Φ) , and [physics M-matrix]n[s] (WA,Φ) , respectively.
If s...4;-1 or s...5;1 , then the inequality [figure omitted; refer to PDF] holds for all above n -tuples W , A , and Φ provided spectral conditions [figure omitted; refer to PDF]
If r...4;-1 or r...5;1 , then the inequality in (5.23) holds provided spectral conditions [figure omitted; refer to PDF]
The inequality in (5.23) is strict for r<s if W>0 and A...0;C .
The proof of Corollary 5.7 is the same as the proof of [3, Corollary 7].
An operator version of the harmonic-geometric-arithmetic inequality is the consequence of Corollary 5.7. The inequality [physics M-matrix]n[-1] (WA,Φ)...4;[physics M-matrix]n[0] (WA,Φ)...4;[physics M-matrix]n[1] (WA,Φ) , that is, [figure omitted; refer to PDF] holds for all n -tuples W=(W1 ,...,Wn ) , A=(A1 ,...,An ) and Φ=(Φ1 ,...,Φn ) as in (5.8) with strictly positive Ai provided spectral conditions [figure omitted; refer to PDF]
There remain only the refinements of the operator power means by using Corollary 5.6.
Corollary 5.8.
Let r,s∈...\{0} such that r<s . Let [varphi]t[r,s] :Y9;0,+∞YA;[arrow right]... for t∈[a0 ,a1 ] be functions as in (2.23) with the addition of u(t)+v(t)=1 . Let t0 ,t1 ∈[a0 ,a1 ] such that t0 ...4;t1 .
Let W , A , and Φ be as in (5.8) with strictly positive A . Let mi ...4;Mi , m[r] ...4;M[r] , m[s] ...4;M[s] , m[[varphi]t0 [r,s] ] ...4;M[[varphi]t0 [r,s] ] , and m[[varphi]t1 [r,s] ] ...4;M[[varphi]t1 [r,s] ] be bounds of operators Ai , [physics M-matrix]n[r] (WA,Φ) , [physics M-matrix]n[s] (WA,Φ) , [physics M-matrix]n[varphi]t0 [r,s] (WA,Φ) , and [physics M-matrix]n[varphi]t1 [r,s] (WA,Φ) , respectively.
If r>0, s...5;1 or s=-1 , then the inequality [figure omitted; refer to PDF] holds for all above n -tuples W , A , and Φ provided spectral conditions: [figure omitted; refer to PDF]
If r=1 , then the inequality [figure omitted; refer to PDF] holds for all above n -tuples W , A , and Φ provided spectral conditions: [figure omitted; refer to PDF]
Proof.
Recall that [varphi]t[r,s] (x)=u(t)xr +v(t)xs for r...0;0, s...0;0 . The next is [straight phi](x)=xr and ψ(x)=xs with x∈I=Y9;0,+∞YA; , so [straight phi](I)=ψ(I)=I .
Case r>0, s...5;1 .
We have (ψ[composite function][straight phi]-1 )(x)=xs/r with (s/r)>1 , and ψ-1 (x)=x1/s with 0<(1/s)...4;1 . The function ψ is strictly [straight phi] -convex with strictly operator increasing ψ-1 . Then the inequality in (5.28) is valid with associated spectral conditions by Corollary 5.6.
Case s=-1 .
We have (ψ[composite function][straight phi]-1 )(x)=x-1/r with 0<-(1/r)<1 , and ψ-1 (x)=x-1 . The function ψ is strictly [straight phi] -concave with strictly operator decreasing ψ-1 . In this case, the inequality in (5.28) is also valid with associated spectral conditions by Corollary 5.6.
Case r=1 .
We use functions [varphi]t[s,1] (x)=u(t)xs +v(t)x . In this case [straight phi](x)=xs and ψ(x)=x , thus (ψ[composite function][straight phi]-1 )(x)=x1/s with 0<(1/s)<1 , and ψ-1 (x)=x . The function ψ is strictly [straight phi] -concave with strictly operator increasing ψ-1 . Then the inequality in (5.30) is valid with associated spectral conditions by Corollary 5.6.
The inequalities in (5.28)-(5.30) are strict for a0 <t0 <t1 <a1 if W>0 and A...0;C .
Unfortunately, we cannot use a logarithmic function because ln (I)=......0;I .
Remark 5.9.
Let r , s , [varphi]t , t0 , t1 , W , A , and Φ be as in Corollary 5.8.
If r...4;-1 and s<0 , then the problem remains the inequality [figure omitted; refer to PDF] valid for all above n -tuples W , A , and Φ provided spectral conditions: [figure omitted; refer to PDF]
The inequality in (5.32) is valid for r=-1 with associated spectral conditions because (ψ[composite function][straight phi]-1 )(x)=x-1/s with -(1/s)>1 , that is, ψ is [straight phi] -convex, and ψ-1 (x)=x-1 , that is, ψ-1 is operator decreasing.
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Abstract
Overview and refinements of the results are given for discrete, integral, functional and operator variants of inequalities for quasiarithmetic means. The general results are applied to further refinements of the power means. Jensen's inequalities have been systematically presented, from the older variants, to the most recent ones for the operators without operator convexity.
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