# The Inequalities for Quasiarithmetic Means

Authors: Jadranka Mićić; Zlatko Pavić; Josip Pečarić
Journal: Abstract and Applied Analysis
 (2012)

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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.


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## Body

## 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 Figure1).Figure 1
Graphic concept of Jensen’s inequality.The applications of convexity often used strictly monotone continuous functionsφ and ψ such that ψ is convex with respect to φ (ψ is φ convex); that is, f=ψ∘φ-1 is convex by [1, Definition 1.19]. Similar notation is used for concavity. We observe a monotonicity of quasiarithmetic means with these functions φ 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 thatI⊆ℝ is a nondegenerate interval, and φ,ψ:I→ℝ are strictly monotone continuous functions. It is assumed that the integer n≥2, wherever it appears in inequalities.


## 2. Results for Basic Case

Forn-tuple x=(x1,…,xn) with numbers xi∈ℝ, sometimes we will write x>0 if all xi>0, and x≠c if xi≠xj for some i≠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.
Letf:I→ℝ 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=1n‍pi=1.
A functionf is convex if and only if the following inequality
(2.1)f(∑i=1npixi)≤∑i=1npif(xi),
holds for all above n-tuples p and x.
Consequently, if∑i=1n‍pi=p>0, not necessarily equals 1, then f is convex if and only if
(2.2)f(1p∑i=1n‍pixi)≤1p∑i=1n‍pif(xi).
A functionf is concave if and only if the reverse inequality is valid in (2.1) and (2.2).A functionf is strictly convex if and only if the inequality in (2.1) and (2.2) is strict for all p>0 and x≠c.Letφ:I→ℝ 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=1n‍pi=1. The discrete basic φ-quasiarithmetic mean of points (particles) xi with coefficients (weights) pi is a number
(2.3)Mφ(px)=φ-1(∑i=1n‍piφ(xi)).
We understand that px=∑i=1n‍pixi. The φ-quasiarithmetic mean resulting, first by moving the convex combination px∈I into convex combination pφ(x)∈φ(I), then its return using φ-1 back in the interval I. So, the number ℳφ(px) is in the interval I, in fact in the closed interval [min{xi},max{xi}]. If φ is an identity function on I, that is, φ(x)=id(x)=x for x∈I, then the φ-quasiarithmetic mean is just a convex combination as follows:
(2.4)Mid(px)=∑i=1n‍pixi.Basic quasiarithmetic means have the property(2.5)Maφ+b(px)=Mφ(px),
for every pair of real numbers a and b with a≠0.Suppose that all coefficientspi=1/n. If we take φ1(x)=x, then ℳφ1(px) is the arithmetic mean of numbers xi. If all xi>0 and we take φ0(x)=lnx, then ℳφ0(px) is the geometric mean of numbers xi. If all xi>0 and we take φ-1(x)=1/x, then ℳφ-1(px) is the harmonic mean of numbers xi.Corollary 2.1.
Letφ,ψ:I→ℝ be strictly monotone continuous functions.
A functionψ is either φ-convex and increasing or φ-concave and decreasing if and only if following the inequality:
(2.6)Mφ(px)≤Mψ(px),
holds for all n-tuples p and x as in (2.3).
A functionψ is either φ-concave and increasing or φ-convex and decreasing if and only if the reverse inequality is valid in (2.6).A functionψ is strictly φ-convex if and only if the inequality in (2.6) is strict for all p>0 and x≠c (see Figure 2).Figure 2
Theψ-order among quasiarithmetic means ℳφ and ℳψ for φ-convex and increasing ψ.Suppose that allxi>0. If we apply Corollary 2.1 on three strictly monotone functions φ-1(x)=1/x, φ0(x)=lnx and φ1(x)=x (two by two in pairs), then we get the weighted harmonic-geometric-arithmetic inequality
(2.7)1(p1/x1)+⋯+(pn/xn)≤x1p1⋅…⋅xnpn≤p1x1+⋯+pnxn.Recall that a functionf:I→ℝ is convex if and only if the following inequality:
(2.8)f(y)-f(x)y-x≤f(z)-f(y)z-y,
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 φ-convex if and only if
(2.9)ψ(y)-ψ(x)φ(y)-φ(x)≤ψ(z)-ψ(y)φ(z)-φ(y).Letu,v:[a0,a1]→ℝ, where a0<a1, be nonnegative continuous functions so that v/u is a strictly monotone increasing positive function on an open interval 〈a0,a1〉, with boundary conditions u(a0)=v(a1)=1 and u(a1)=v(a0)=0. Let both φ and ψ be strictly monotone increasing or decreasing. For any t∈[a0,a1], we define a strictly monotone continuous function
(2.10)ϕt(x)=u(t)φ(x)+v(t)ψ(x)withx∈I.
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)=cost and v(t)=sint for t∈[0,π/2].Lemma 2.2.
Leta, b, α, β be real numbers.
Ifa≤b and α≤β, then
(2.11)1+βa1+αa≤1+βb1+αb,
provided that denominators are the same sign. The inequality in (2.11) is strict if a<b and α<β.
If eithera≥b and α≤β or a≤b and α≥β, then the reverse inequality is valid in (2.11).Proposition 2.3.
Letϕt=u(t)φ+v(t)ψ:I→ℝ for t∈[a0,a1] be functions as in (2.10). Let t0,t1∈[a0,a1] such that t0≤t1.
Ifψ is φ-convex (resp. φ-concave), then ϕt1 is ϕt0-convex (resp. ϕt0-concave).Proof.
Suppose thatψ is φ-convex. Show that the function ϕt1 is ϕt0-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,
(2.12)a=ψ(y)-ψ(x)φ(y)-φ(x)≤ψ(z)-ψ(y)φ(z)-φ(y)=b,α=v(t0)u(t0)≤v(t1)u(t1)=β.
Note that numbers a and b are positive because both φ and ψ are strictly monotone increasing or decreasing. Applying Lemma 2.2 with a, b, α, and β, we obtain that
(2.13)ϕt1(y)-ϕt1(x)ϕt0(y)-ϕt0(x)=u(t1)[φ(y)-φ(x)]+v(t1)[ψ(y)-ψ(x)]u(t0)[φ(y)-φ(x)]+v(t0)[ψ(y)-ψ(x)]=u(t1)u(t0)1+βa1+αa≤u(t1)u(t0)1+βb1+αb=u(t1)[φ(z)-φ(y)]+v(t1)[ψ(z)-ψ(y)]u(t0)[φ(z)-φ(y)]+v(t0)[ψ(z)-ψ(y)]=ϕt1(z)-ϕt1(y)ϕt0(z)-ϕt0(y),
which shows the required convexity by (2.9). Case of the concavity can be proved in a similar way.Ifψ is strictly φ-convex (resp. φ-concave), then ϕt1 is strictly ϕt0-convex (resp. ϕt0-concave).According to Proposition2.3, we can express refinements of the basic quasiarithmetic means.Theorem 2.4.
Letϕt=u(t)φ+v(t)ψ:I→ℝ for t∈[a0,a1] be functions as in (2.10). Let t0,t1∈[a0,a1] such that t0≤t1.
If eitherψ is φ-convex with both φ and ψ increasing or φ-concave with both φ and ψ decreasing, then the following inequality:
(2.14)Mφ(px)≤Mϕt0(px)≤Mϕt1(px)≤Mψ(px),
holds for all n-tuples p and x as in (2.3).
If eitherψ is φ-concave with both φ and ψ increasing or φ-convex with both φ and ψ decreasing, then the reverse inequality is valid in (2.14).Proof.
Ifψ is φ-convex with both φ and ψ increasing, then the function ϕt1 is increasing, and ϕt0-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(2.15)t↦Mϕt(px)witht∈[a0,a1],
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 φ-convex or φ-concave, p>0 and x≠c.Let us take strictly monotone decreasing functionsφ(x)=1/x and ψ(x)=-lnx with x>0. Then (ψ∘φ-1)(x)=lnx, so ψ is strictly φ-concave. Let
(2.16)ϕthg(x)=u(t)1x-v(t)ln⁡x=ln⁡eu(t)/xxv(t).
If we apply the inequality in (2.14) with t=t0=t1 on ϕthg, we get
(2.17)1∑i=1n(pi/xi)≤(ϕthg)-1(∑i=1n‍piln⁡eu(t)/xixiv(t))≤∏i=1n‍xipi.Let us take strictly monotone increasing functionsφ(x)=lnx and ψ(x)=x with x>0. Then (ψ∘φ-1)(x)=ex, so ψ is strictly φ-convex. Let
(2.18)ϕtga(x)=u(t)ln⁡x+v(t)x=ln⁡(xu(t)ev(t)x).
If we apply the inequality in (2.14) with t=t0=t1 on ϕtga, we get
(2.19)∏i=1n‍xipi≤(ϕtga)-1(∑i=1n‍piln⁡(xiu(t)ev(t)xi))≤∑i=1n‍pixi.Connecting two above inequalities results in(2.20)1∑i=1n(pi/xi)≤(ϕthg)-1(∑i=1n‍piln⁡eu(t)/xixiv(t))≤∏i=1n‍xipi≤(ϕtga)-1(∑i=1n‍piln⁡(xiu(t)ev(t)xi))≤∑i=1n‍pixi.The inequality in (2.20) is strict for a0<t<a1 if all pi>0 and xi≠xj for some i≠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 Corollary2.1 on power means. As a special case of the basic quasiarithmetic mean in (2.3) with I=〈0,+∞〉, φr(x)=xr for r≠0 and φ0(x)=lnx, we can observe the discrete basic power mean
(2.21)Mn[r](px)={(∑i=1npixir)1/rforr≠0exp⁡(∑i=1npiln⁡xi)forr=0.Very useful consequence of Corollary2.1 is a well-known property of monotonicity of basic power means.Corollary 2.5.
Ifr and s are real numbers such that r≤s, then the following inequality:
(2.22)Mn[r](px)≤Mn[s](px),
holds for all n-tuples p and x as in (2.3) with I=〈0,+∞〉.The inequality in (2.22) is strict for r<s if p>0 and x≠c.Let functionsϕt[r,s]:〈0,+∞〉→ℝ for t∈[a0,a1] be specially defined by
(2.23)ϕt[r,s](x)={u(t)xr+v(t)xsforr≠0,s≠0u(t)xr-v(t)ln⁡xforr≠0,s=0u(t)ln⁡x+v(t)xsforr=0,s≠0.
Then the functions t↦ℳϕt[r,s](px) with x>0 are monotone increasing in the next four cases.Caser<s<0.
Functionsφ(x)=xr and ψ(x)=xs are strictly monotone decreasing with strictly concave (ψ∘φ-1)(x)=xs/r because 0<(s/r)<1.Caser<0=s.
Functionsφ(x)=xr and ψ(x)=-lnx are strictly monotone decreasing with strictly concave (ψ∘φ-1)(x)=-(1/r)lnx because -(1/r)>0.Caser=0<s.
Functionsφ(x)=lnx and ψ(x)=xs are strictly monotone increasing with strictly convex (ψ∘φ-1)(x)=esx.Case0<r<s.
Functionsφ(x)=xr and ψ(x)=xs are strictly monotone increasing with strictly convex (ψ∘φ-1)(x)=xs/r because (s/r)>1.Given traditional signs of power means, we will markℳϕt[r,s](px) with ℳnϕt[r,s](px). The inequality in (2.22) can be refined using Theorem 2.4 with functions ϕt[r,s]. The following are refinements of power means.Corollary 2.6.
Letr,s∈ℝ such that r<s. Let ϕt[r,s]:〈0,+∞〉→ℝ for t∈[a0,a1] be functions as in (2.23). Let t0,t1∈[a0,a1] such that t0≤t1.
Ifr<s<0 or r<0=s or r=0<s or 0<r<s, then the inequality
(2.24)Mn[r](px)≤Mnϕt0[r,s](px)≤Mnϕt1[r,s](px)≤Mn[s](px),
holds for all n-tuples p and x as in (2.3) with I=〈0,+∞〉.
Ifr<0<s, then we can take the series of inequalities
(2.25)Mn[r](px)≤Mnϕt0[r,0](px)≤Mnϕt1[r,0](px)≤Mn[0](px)≤Mnϕt0[0,s](px)≤Mnϕt1[0,s](px)≤Mn[s](px).The inequalities in (2.24)-(2.25) are strict for a0<t0<t1<a1 if p>0 and x≠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→ℝ is nonnegative almost everywhere on Ω, that is, w(ω)≥0 for almost all ω∈Ω.Forn-tuple g=(g1,…,gn) with functions gi:Ω→ℝ, sometimes we will write g>0 if all gi>0 almost everywhere on Ω, and g≠c if gi≠gj almost everywhere on Ω for some i≠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.
Letf:I→ℝ be a function. Let (Ω,μ) be a probability measure space, g:Ω→I be a measurable function, and w∈L1(Ω,μ) be a weighted function with ∫Ω‍wdμ=1 such that w·g,w·(f∘g)∈L1(Ω,μ).
If a functionf is convex, then the inequality
(3.1)f(∫Ω‍w⋅gdμ)≤∫Ω‍w⋅(f∘g)dμ,
holds for all above w, g and μ.
Consequently, if∫Ω‍wdμ=p>0, not necessarily equals 1, then
(3.2)f(1p∫Ω‍w⋅gdμ)≤1p∫Ω‍w⋅(f∘g)dμ.
If a functionf is concave, then the reverse inequality is valid in (3.1) and (3.2).The assumption∫Ω‍wdμ=1 with nonnegative w almost everywhere on Ω, for the inequality in (3.1), assures that
(3.3)∫Ω‍w⋅gdμ∈I,∫Ω‍w⋅(f∘g)dμ∈f(I).Remark 3.1.
The reverse of Theorem B is valid if for anyp∈[0,1] a measurable set Ωp⊆Ω exists so that μ(Ωp)=p. In this case, we can determine a simple measurable function
(3.4)g=xχΩp+yχΩ∖Ωp,
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
(3.5)∫Ω‍w⋅gdμ=px+(1-p)y,∫Ω‍w⋅(f∘g)dμ=∫Ωp‍f(x)dμ+∫Ω∖Ωp‍f(y)dμ=pf(x)+(1-p)f(y).
If we include these integrals in the inequality in (3.1), we have the convexity of the function f.Theorem B can be generalized ton 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.
Letf:I→ℝ 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→I, and w=(w1,…,wn) be n-tuple with weighted functions wi∈L1(Ω,μi) with ∑i=1n‍∫Ω‍widμi=1 such that wi·gi,wi·(f∘gi)∈L1(Ω,μi).
A functionf is convex if and only if the inequality
(3.6)f(∑i=1n‍∫Ω‍wi⋅gidμi)≤∑i=1n‍∫Ω‍wi⋅(f∘gi)dμi,
holds for all above n-tuples w, g and μ.
Consequently, if∑i=1n‍∫Ω‍widμi=p>0, not necessarily equals 1, then f is convex if and only if
(3.7)f(1p∑i=1n‍∫Ω‍wi⋅gidμi)≤1p∑i=1n‍∫Ω‍wi⋅(f∘gi)dμi.
A functionf 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 takewi=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≥2 is coming to the fore.A functionf is strictly convex if and only if the inequality in (3.6) and (3.7) is strict for all w>0 and g≠c.Letφ:I→ℝ 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→I, and w=(w1,…,wn) be n-tuple with weighted functions wi∈L1(Ω,μi) with ∑i=1n‍∫Ω‍widμi=1 such that wi·(φ∘gi)∈L1(Ω,μi). The discrete integral φ-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
(3.8)Mφ(wg,μ)=φ-1(∑i=1n‍∫Ω‍wi⋅(φ∘gi)dμi).
This number belongs to I because the integral convex combination ∑i=1n‍∫Ω‍wi·(φ∘gi)dμi∈φ(I). Integral quasiarithmetic means also satisfy the property
(3.9)Maφ+b(wg,μ)=Mφ(wg,μ),
for every pair of real numbers a and b with a≠0.Bearing in mind Theorem3.2, the following corollary is valid.Corollary 3.3.
Letφ,ψ:I→ℝ be strictly monotone continuous functions.
A functionψ is either φ-convex and increasing or φ-concave and decreasing if and only if the inequality
(3.10)Mφ(wg,μ)≤Mψ(wg,μ),
holds for all n-tuples w, g, and μ as in (3.8).
A functionψ is either φ-concave and increasing or φ-convex and decreasing if and only if the reverse inequality is valid in (3.10).Combining basic and integral case by Corollaries2.1 and 3.3, we get the following.Proposition 3.4.
Letφ,ψ:I→ℝ be strictly monotone continuous functions. Then the inequality
(3.11)Mφ(px)≤Mψ(px),
holds for all n-tuples p and x as in (2.3) if and only if the inequality
(3.12)Mφ(wg,μ)≤Mψ(wg,μ),
holds for all n-tuples w, g, and μ as in (3.8).The one direction of Proposition3.4 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 Theorem2.4.Theorem 3.5.
Letϕt=u(t)φ+v(t)ψ:I→ℝ for t∈[a0,a1] be functions as in (2.10). Let t0,t1∈[a0,a1] such that t0≤t1.
If eitherψ is φ-convex with both φ and ψ increasing or φ-concave with both φ and ψ decreasing, then the inequality
(3.13)Mφ(wg,μ)≤Mϕt0(wg,μ)≤Mϕt1(wg,μ)≤Mψ(wg,μ),
holds for all n-tuples w, g, and μ as in (3.8).
If eitherψ is φ-concave with both φ and ψ increasing or φ-convex with both φ 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 φ-convex, w>0 and g≠c.An integral version of refinements of the harmonic-geometric-arithmetic inequality is also valid. So, the inequality(3.14)1∑i=1n∫Ω‍(wi/gi)dμi≤(ϕthg)-1(∑i=1n‍∫Ω‍wi⋅ln⁡eu(t)/gigiv(t)dμi)≤∏i=1n‍∫Ω‍giwidμi≤(ϕtga)-1(∑i=1n‍∫Ω‍wi⋅ln⁡(giu(t)ev(t)gi)dμi)≤∑i=1n‍∫Ω‍wi⋅gidμi,
holds for all n-tuples w=(w1,…,wn), g=(g1,…,gn) and μ=(μ1,…,μn) as in (3.8) with I=〈0,+∞〉. The above inequality is strict for a0<t<a1 if all wi>0 almost everywhere on Ω and gi≠gj almost everywhere on Ω for some i≠j.As a special case of the integral quasiarithmetic mean in (3.8) with I=〈0,+∞〉, φr(x)=xr for r≠0, and φ0(x)=lnx, we can observe the integral power mean
(3.15)Mn[r](wg,μ)={(∑i=1n∫Ω‍wi⋅girdμi)1/rforr≠0exp⁡(∑i=1n∫Ω‍wi⋅ln⁡gidμi)forr=0.We quote the integral analogy of Corollary2.6. The following is the property of monotonicity, with refinements, of integral power means.Corollary 3.6.
Letr,s∈ℝ such that r<s. Let ϕt[r,s]:〈0,+∞〉→ℝ for t∈[a0,a1] be functions as in (2.23). Let t0,t1∈[a0,a1] such that t0≤t1.
Ifr<s<0 or r<0=s or r=0<s or 0<r<s, then the inequality
(3.16)Mn[r](wg,μ)≤Mnϕt0[r,s](wg,μ)≤Mnϕt1[r,s](wg,μ)≤Mn[s](wg,μ),
holds for all n-tuples w, g, and μ as in (3.8) with I=〈0,+∞〉.
Ifr<0<s, then we can take the series of inequalities
(3.17)Mn[r](wg,μ)≤Mnϕt0[r,0](wg,μ)≤Mnϕt1[r,0](wg,μ)≤Mn[0](wg)≤Mnϕt0[0,s](wg,μ)≤Mnϕt1[0,s](wg,μ)≤Mn[s](wg,μ).The inequalities in (3.16)-(3.17) are strict for a0<t0<t1<a1 if w>0 and g≠c.All the observed integral cases are reduced to the corresponding basic cases when we take constantsgi=xi and wi=pi.


## 4. Applications on Functional Case

LetS be a nonempty set and 𝒮 be a vector space of real-valued functions g:S→ℝ. Linear functional P:𝒮→ℝ is positive (nonnegative) or monotone if P(g)≥0 for every nonnegative function g∈𝒮. If a space 𝒮 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 functionw:S→ℝ is nonnegative, that is, w(s)≥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.
Letf:I→ℝ be a continuous function where I is the closed interval. Let P:𝒮→ℝ be a positive linear functional, g:S→I be a function, and w∈𝒮 be a weighted function with P(w)=1 such that w·g,w·(f∘g)∈𝒮.
If a functionf is convex, then the inequality
(4.1)f(P(w⋅g))≤P(w⋅(f∘g)),
holds for all above w, g, and P.
Consequently, ifP(w)=p>0, not necessarily equals 1, then
(4.2)f(1pP(w⋅g))≤1pP(w⋅(f∘g)).
If a functionf is concave, then the reverse inequality is valid in (4.1) and (4.2).The inequality in (4.1) with w=1 (assuming 1∈𝒮 and P(1)=1) is usually called the Jessen functional form of Jensen’s inequality.The intervalI must be closed; otherwise, it could happen that P(w·g)∉I or P(w·(f∘g))∉f(I). The following example shows such an undesirable situation.Example 4.1.
LetS=I=〈0,1] and
(4.3)S={g:I⟶R∣lim⁡x→0+g(x)finite}.
If P:𝒮→ℝ is defined by
(4.4)P(g)=lim⁡x→0+g(x),
then P is positive linear functional. In that way, functional P is also unital because 1∈𝒮 and P(1)=1. If we take g(x)=x for x∈I, then g∈𝒮 and its image in I, but P(g)=0∉I.Remark 4.2.
Suppose that1∈𝒮 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∈𝒮 and P(χSp)=p. If we take g=xχSp+yχS∖Sp and w=1, then it follows that
(4.5)P(w⋅g)=px+(1-p)y,P(w⋅(f∘g))=P(f(x)χSp+f(y)χS∖Sp)=pf(x)+(1-p)f(y),
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 ton 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.
Letf:I→ℝ be a continuous function where I is the closed interval. Let P=(P1,…,Pn) be n-tuple with positive linear functionals Pi:𝒮→ℝ, g=(g1,…,gn) be n-tuple with functions gi:S→I, and w=(w1,…,wn) be n-tuple with weighted functions wi∈𝒮 with ∑i=1n‍Pi(wi)=1 such that wi·gi,wi·(f∘gi)∈𝒮.
If a functionf is convex, then the inequality
(4.6)f(∑i=1n‍Pi(wi⋅gi))≤∑i=1n‍Pi(wi⋅(f∘gi)),
holds for all above n-tuples w, g, and P.
Consequently, if∑i=1n‍Pi(wi)=p>0, not necessarily equals 1, then
(4.7)(1p∑i=1n‍Pi(wi⋅gi))≤1p∑i=1n‍Pi(wi⋅(f∘gi)).
If a functionf 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
(4.8)f(∑i=1n‍Pi(wi⋅gi))=f(∑i=1n‍pixi)≤∑i=1n‍pif(xi)=∑i=1n‍Pi(wi)f(1Pi(wi)Pi(wi⋅gi))≤∑i=1n‍Pi(wi⋅(f∘gi)).Iff is strictly convex, then the inequality in (4.6) and (4.7) is strict for all w>0 and g≠c.Remark 4.4.
Suppose that1∈𝒮 and all functionals Pi are unital; that is, Pi(1)=1 holds for all Pi. Then it is c·1∈𝒮 and Pi(c·1)=cPi(1)=c for every constant c∈ℝ. With the above assumptions, the reverse of Theorem 4.3 follows trivially if we take wi=pi and gi=xi.Letφ:I→ℝ be a strictly monotone continuous function where I is the closed interval. Let P=(P1,…,Pn) be n-tuple with positive linear functionals Pi:𝒮→ℝ, g=(g1,…,gn) be n-tuple with functions gi:S→I, and w=(w1,…,wn) be n-tuple with weighted functions wi∈𝒮 with ∑i=1n‍Pi(wi)=1 such that wi·(φ∘gi)∈𝒮. The discrete functional φ-quasiarithmetic mean of functions gi with weighted functions wi with respect to functionals Pi is a number
(4.9)Mφ(wg,P)=φ-1(∑i=1n‍Pi(wi⋅(φ∘gi))).
This number belongs to I because the functional convex combination ∑i=1n‍Pi(wi·(φ∘gi)) belongs to φ(I). Functional quasiarithmetic means also satisfy the property
(4.10)Maφ+b(wg,P)=Mφ(wg,P),
for every pair of real numbers a and b with a≠0. Indeed, if ϕ(x)=aφ(x)+b, then ϕ-1(x)=φ-1(x-b/a), and we have
(4.11)Mϕ(wg,P)=ϕ-1(∑i=1n‍Pi(wi⋅(ϕ∘gi)))=φ-1(∑i=1nPi(wi⋅(aφ∘gi+b))-ba)=φ-1(∑i=1n‍Pi(wi⋅(φ∘gi)))=Mφ(wg,P).Corollary 4.5.
Letφ,ψ:I→ℝ be strictly monotone continuous functions where I is the closed interval.
If a functionψ is either φ-convex and increasing or φ-concave and decreasing, then the inequality
(4.12)Mφ(wg,P)≤Mψ(wg,P),
holds for all n-tuples w, g, and P as in (4.9).
If a functionψ is either φ-concave and increasing or φ-convex and decreasing, then the reverse inequality is valid in (4.12).Proof.
Suppose thatψ is φ-convex and increasing. If we apply the inequality in (4.6) with f=ψ∘φ-1:f(I)→ℝ, and ψ∘gi:S→f(I) instead of gi, we get
(4.13)(ψ∘ψ-1)(∑i=1nPi(wi⋅(φ∘gi)))≤∑i=1nPi(wi⋅(ψ∘gi)).
After taking ψ-1 of the both sides, it follows that
(4.14)Mφ(wg,P)≤Mψ(wg,P).
In the same way, we can prove the case when ψ is φ-concave and decreasing.According to Remark4.2, the reverse of Corollary 4.5 is valid if 1∈𝒮 and all functionals Pi are unital. Then we connect the basic and functional case in the following proposition.Proposition 4.6.
Letφ,ψ:I→ℝ be strictly monotone continuous functions where I is the closed interval. Then the inequality
(4.15)Mφ(px)≤Mψ(px),
holds for all p and x as in (2.3) if and only if the inequality
(4.16)Mφ(wg,P)≤Mψ(wg,P),
holds for all n-tuples w, g, and P as in (4.9) with 1∈𝒮 and unital functionals Pi.Next in line is a functional analogy of refinements.Theorem 4.7.
Letϕt=u(t)φ+v(t)ψ:I→ℝ for t∈[a0,a1] be functions as in (2.10) where I is the closed interval. Let t0,t1∈[a0,a1] such that t0≤t1.
If eitherψ is φ-convex with both φ and ψ increasing or φ-concave with both φ and ψ decreasing, then the inequality
(4.17)Mφ(wg,P)≤Mϕt0(wg,P)≤Mϕt1(wg,P)≤Mψ(wg,P),
holds for all n-tuples w, g, and P as in (4.9) with 1∈𝒮 and unital functionals Pi.
If eitherψ is φ-concave with both φ and ψ increasing or φ-convex with both φ 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 φ-convex, w>0 and g≠c.A functional version of refinements of the harmonic-geometric-arithmetic inequality is also valid. So, the inequality(4.18)1∑i=1nPi(wi/gi)≤(ϕthg)-1(∑i=1n‍Pi(wi⋅ln⁡eu(t)/gigiv(t)))≤∏i=1n‍Pi(giwi)≤(ϕtga)-1(∑i=1n‍Pi(wi⋅ln⁡(giu(t)ev(t)gi)))≤∑i=1n‍Pi(wi⋅gi),
holds for all n-tuples w=(w1,…,wn), g=(g1,…,gn) and, P=(P1,…,Pn) as in (4.9) with I=[a,+∞〉 where a>0. The above inequality is strict for a0<t<a1 if all wi>0 and gi≠gj for some i≠j.As a special case of the functional quasiarithmetic mean in (4.9) with I=[a,+∞〉  where a>0, φr(x)=xr for r≠0 and φ0(x)=lnx, we can observe the functional power mean
(4.19)Mn[r](wg,P)={(∑i=1n‍Pi(wi⋅gir))1/rforr≠0exp⁡(∑i=1n‍Pi(wi⋅ln⁡gi))forr=0.The following is the property of monotonicity, with refinements, of functional power means.Corollary 4.8.
Letr,s∈ℝ such that r<s. Let ϕt[r,s]:[a,+∞〉→ℝ for t∈[a0,a1] be functions as in (2.23) where a>0. Let t0,t1∈[a0,a1] such that t0≤t1.
Ifr<s<0 or r<0=s or r=0<s or 0<r<s, then the inequality
(4.20)Mn[r](wg,P)≤Mnϕt0[r,s](wg,P)≤Mnϕt1[r,s](wg,P)≤Mn[s](wg,P),
holds for all n-tuples w, g, and P as in (4.9) with I=[a,+∞〉, 1∈𝒮 and unital functionals Pi.
Ifr<0<s, then we can take the series of inequalities
(4.21)Mn[r](wg,P)≤Mnϕt0[r,0](wg,P)≤Mnϕt1[r,0](wg,P)≤Mn[0](wg,P)≤Mnϕt0[0,s](wg,P)≤Mnϕt1[0,s](wg,P)≤Mn[s](wg,P).The inequalities in (4.20)-(4.21) are strict for a0<t0<t1<a1 if w>0 and g≠c.All the observed functional cases are reduced to the corresponding integral cases when we take(4.22)Pi(gi)=∫Ω‍gidμi,
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. LetH be a Hilbert space. We define the bounds of linear operator A:H→H with
(5.1)mA=inf⁡‖x‖=1〈Ax,x〉,MA=sup⁡‖x‖=1〈Ax,x〉.Letℬ(H) be the C*-algebra of all bounded linear operators A:H→H. If Sp(A) denotes the spectrum of a self-adjoint operator A∈ℬ(H), then it is well-known that Sp(A) is a subset of ℝ and Sp(A)⊆[mA,MA]. If 1H denotes the identity operator on H, then the following holds:
(5.2)mA1H≤A≤MA1H,‖A‖=sup⁡‖x‖=1|〈Ax,x〉|=max⁡{|mA|,|MA|}.A continuous functionf:I→ℝ is said to be operator increasing on I if
(5.3)A≤Bimpliesf(A)≤f(B),
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.Forn-tuple A=(A1,…,An) with operators Ai∈ℬ(H) sometimes, we will write A>0 if all Ai>0, and A≠C if Ai≠Aj for some i≠j.In this section, it is assumed that every weighted operatorW∈ℬ(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.
Letf:I→ℝ be a continuous function. Let Φ=(Φ1,…,Φn) be n-tuple with positive linear mappings Φi:ℬ(H)→ℬ(K), A=(A1,…,An) be n-tuple with self-adjoint operators Ai∈ℬ(H) with bounds mi≤Mi from I, and W=(W1,…,Wn) be n-tuple with weighted operators Wi∈ℬ(H) with ∑i=1n‍Φi(Wi)=1K. Let mB≤MB be bounds of an operator B=∑i=1nΦi(WiAi).
If a functionf is convex, then the inequality
(5.4)f(∑i=1n‍Φi(WiAi))≤∑i=1nΦi(Wif(Ai)),
holds for all above n-tuples W, A, and Φ provided spectral conditions
(5.5)[mB,MB]∩[mi,Mi]=∅or{endpoint}fori=1,…,n.
Consequently, if∑i=1n‍Φi(Wi)=WΦ is strictly positive, not necessarily equals 1K, then
(5.6)f(WΦ-1∑i=1n‍Φi(WiAi))≤WΦ-1∑i=1n‍Φ(Wif(Ai)).
If a functionf is concave, then the reverse inequality is valid in (5.4) and (5.6).Proof.
IfmB<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:
(5.7)f(∑i=1n‍Φi(WiAi))≤f[mB,MB]cho(∑i=1n‍Φi(WiAi))=k∑i=1n‍Φi(WiAi)+l1K=∑i=1n‍Φi(Wi(kAi+l1H))=∑i=1n‍Φi(Wif[mB,MB]cho(Ai))≤∑i=1n‍Φi(Wif(Ai)).
IfmB=MB, then we take any support line f[mB]sup(x)=kx+l instead of the chord line.Iff is strictly convex, then the inequality in (5.4) and (5.6) is strict for all W>0 and A≠C.Remark 5.2.
The reverse of Theorem5.1 is trivially valid if all Φi are unital. With this assumption, we can take Wi=pi1H and Ai=xi1H.Letφ:I→ℝ be a strictly monotone continuous function. Let Φ=(Φ1,…,Φn) be n-tuple with positive linear mappings Φi:ℬ(H)→ℬ(K), A=(A1,…,An) be n-tuple with self-adjoint operators Ai∈ℬ(H) with spectra in I, and W=(W1,…,Wn) be n-tuple with weighted operators Wi∈ℬ(H) with ∑i=1n‍Φi(Wi)=1K. The discrete operator φ-quasiarithmetic mean of operators Ai with weighted operators Wi with respect to mappings Φi is an operator
(5.8)Mφ(WA,Φ)=φ-1(∑i=1n‍Φi(Wiφ(Ai))).
The spectrum of operator ℳφ(WA,Φ) is contained in I because the spectrum of operator ∑i=1n‍Φi(Wiφ(Ai)) is contained in φ(I). Operator quasiarithmetic means also have the property
(5.9)Maφ+b(WA,Φ)=Mφ(WA,Φ),
for every pair of real numbers a and b with a≠0. To verify this equality, let us take ϕ=aφ+b, so ϕ-1(B)=φ-1((B-b1K)/a) if B∈ℬ(K), and we get
(5.10)Mϕ(WA,Φ)=ϕ-1(∑i=1n‍Φi(Wiϕ(Ai)))=φ-1(∑i=1nΦi(Wi(aφ(Ai)+b1H))-b1Ka)=φ-1(∑i=1n‍Φi(Wiφ(Ai)))=Mφ(WA,Φ).Corollary 5.3.
Letφ,ψ:I→ℝ be strictly monotone continuous functions with operator monotone ψ-1.
LetW, A and Φ be as in (5.8). Let mi≤Mi and mφ≤Mφ be bounds of operators Ai and ℳφ(WA,Φ), respectively.
If a functionψ is either φ-convex with operator increasing ψ-1 or φ-concave with operator decreasing ψ-1, then the inequality
(5.11)Mφ(WA,Φ)≤Mψ(WA,Φ),
holds for all above n-tuples W, A, and Φ provided spectral conditions
(5.12)[mφ,Mφ]∩[mi,Mi]=∅or{endpoint}fori=1,…,n.
If a functionψ is either φ-concave with operator increasing ψ-1 or φ-convex with operator decreasing ψ-1, then the reverse inequality is valid in (5.11).The following is operator analogy of Theorem2.4.Theorem 5.4.
Letϕt=u(t)φ+v(t)ψ:I→ℝ for t∈[a0,a1] be functions as in (2.10) with operator monotone ψ-1. Let t0,t1∈[a0,a1] such that t0≤t1.
If eitherψ is φ-convex with operator increasing ϕt0-1, ϕt1-1, and ψ-1 or φ-concave with operator decreasing ϕt0-1, ϕt1-1, and ψ-1, then the inequality
(5.13)Mφ(WA,Φ)≤Mϕt0(WA,Φ)≤Mϕt1(WA,Φ)≤Mψ(WA,Φ),
holds for all n-tuples W, A, and Φ as in (5.8) that provided the following spectral conditions:
(5.14)[mφ,Mφ]∩[mi,Mi]=∅or{endpoint}fori=1,…,n[mϕt0,Mϕt0]∩[mi,Mi]=∅or{endpoint}fori=1,…,n[mϕt1,Mϕt1]∩[mi,Mi]=∅or{endpoint}fori=1,…,n.
If eitherψ is φ-concave with operator increasing ϕt0-1, ϕt1-1, and ψ-1 or φ-convex with operator decreasing ϕt0-1, ϕt1-1, 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 ϕt0 and ϕt1. If ψ is φ-convex with both φ and ψ increasing, then ϕt1 is ϕt0-convex by Proposition 2.3. If ϕt1-1 is operator increasing, then by Corollary 5.3 the inequality
(5.15)Mϕt0(WA,Φ)≤Mϕt1(WA,Φ),
is valid with spectral conditions
(5.16)[mϕt0,Mϕt0]∩[mi,Mi]=∅or{endpoint}fori=1,…,n.
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 φ-convex, ϕt0-1,ϕt1-1,  andψ-1 are strictly operator increasing, W>0 and A≠C.We are interested in sufficient conditions under which the functionsϕt will be operator increasing.Lemma 5.5.
Letϕ=αφ+βψ be a convex combination (α,β≥0,α+β=1) of strictly monotone increasing or decreasing continuous functions φ,ψ:I→ℝ such that φ(I)=ψ(I)=J. Then
(5.17)ϕ-1=u⋅φ-1+v⋅ψ-1,
where u,v:J→ℝ are nonnegative continuous functions such that u(y)+v(y)=1 for every y∈J.Proof.
Take anyx∈I. If
(5.18)y=ϕ(x)=αφ(x)+βψ(x),
then
(5.19)x=ϕ-1(y)=u(y)⋅φ-1(y)+v(y)⋅ψ-1(y),
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
(5.20)u(y)={ϕ-1(y)-ψ-1(y)φ-1(y)-ψ-1(y)forφ-1(y)≠ψ-1(y)lim⁡y→y0ϕ-1(y)-ψ-1(y)φ-1(y)-ψ-1(y)forφ-1(y0)=ψ-1(y0).
The above limit is onesided if φ-1=ψ-1 on some subinterval of an interval J. The functions φ-1,ψ-1, and ϕ-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 ϕ-1 as the convex combination of functions φ-1 and ψ-1 with coefficient functions u and v.Figure 3
Convex combination of strictly monotone increasing functionsφ and ψ.Theorem5.4 can be simplified by using Lemma 5.5.Corollary 5.6.
Letϕt=u(t)φ+v(t)ψ:I→ℝ for t∈[a0,a1] be functions as in (2.10) with the addition of u(t)+v(t)=1, φ(I)=ψ(I) and operator monotone ψ-1. Let t0,t1∈[a0,a1] such that t0≤t1.
If eitherψ is φ-convex with operator increasing ψ-1 or φ-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 eitherψ is φ-concave with operator increasing ψ-1 or φ-convex with operator decreasing ψ-1, then the reverse inequality is valid in (5.13).Proof.
According to Lemma5.5 continuous functions ut and vt, with ut+vt=1, exist for every t∈[a0,a1] so that
(5.21)ϕt-1=ut⋅φ-1+vt⋅ψ-1=ut⋅(φ-1-ψ-1)+ψ-1.
Let t>a0; otherwise, it is ϕa0-1=φ-1. Then vt≠0 and ϕt-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=〈0,+∞〉,  φr(x)=xr for r≠0 and φ0(x)=lnx, we can observe the operator power mean
(5.22)Mn[r](WA,Φ)={(∑i=1nΦi(WiAir))1/rforr≠0exp⁡(∑i=1nΦi(Wiln⁡Ai))forr=0.The consequence of Corollary5.3 for operator power means the following.Corollary 5.7.
Letr and s be real numbers such that r≤s.
LetW, A, and Φ be as in (5.8) with strictly positive A. Let mi≤Mi, m[r]≤M[r], and m[s]≤M[s] be bounds of operators Ai, ℳn[r](WA,Φ), and ℳn[s](WA,Φ), respectively.
Ifs≤-1 or s≥1, then the inequality
(5.23)Mn[r](WA,Φ)≤Mn[s](WA,Φ),
holds for all above n-tuples W, A, and Φ provided spectral conditions
(5.24)[m[r],M[r]]∩[mi,Mi]=∅or{endpoint}fori=1,…,n.
Ifr≤-1 or r≥1, then the inequality in (5.23) holds provided spectral conditions
(5.25)[m[s],M[s]]∩[mi,Mi]=∅or{endpoint}fori=1,…,n.The inequality in (5.23) is strict for r<s if W>0 and A≠C.The proof of Corollary5.7 is the same as the proof of [3, Corollary 7].An operator version of the harmonic-geometric-arithmetic inequality is the consequence of Corollary5.7. The inequality ℳn[-1](WA,Φ)≤ℳn[0](WA,Φ)≤ℳn[1](WA,Φ), that is,
(5.26)(∑i=1n‍Φi(WiAi-1))-1≤exp⁡(∑i=1n‍Φi(Wiln⁡Ai))≤∑i=1n‍Φi(WiAi),
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
(5.27)[m[0],M[0]]∩[mi,Mi]=∅or{endpoint}fori=1,…,n.There remain only the refinements of the operator power means by using Corollary5.6.Corollary 5.8.
Letr,s∈ℝ∖{0} such that r<s. Let ϕt[r,s]:〈0,+∞〉→ℝ 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≤t1.
LetW, A, and Φ be as in (5.8) with strictly positive A. Let mi≤Mi, m[r]≤M[r], m[s]≤M[s], m[ϕt0[r,s]]≤M[ϕt0[r,s]], and m[ϕt1[r,s]]≤M[ϕt1[r,s]] be bounds of operators Ai, ℳn[r](WA,Φ), ℳn[s](WA,Φ), ℳnϕt0[r,s](WA,Φ), and ℳnϕt1[r,s](WA,Φ), respectively.
Ifr>0,s≥1 or s=-1, then the inequality
(5.28)Mn[r](WA,Φ)≤Mnϕt0[r,s](WA,Φ)≤Mnϕt1[r,s](WA,Φ)≤Mn[s](WA,Φ),
holds for all above n-tuples W, A, and Φ provided spectral conditions:
(5.29)[m[r],M[r]]∩[mi,Mi]=∅or{endpoint}fori=1,…,n,[m[ϕt0[r,s]],M[ϕt0[r,s]]]∩[mi,Mi]=∅or{endpoint}fori=1,…,n,[m[ϕt1[r,s]],M[ϕt1[r,s]]]∩[mi,Mi]=∅or{endpoint}fori=1,…,n.
Ifr=1, then the inequality
(5.30)Mn[1](WA,Φ)≤Mnϕt0[s,1](WA,Φ)≤Mnϕt1[s,1](WA,Φ)≤Mn[s](WA,Φ),
holds for all above n-tuples W, A, and Φ provided spectral conditions:
(5.31)[m[s],M[s]]∩[mi,Mi]=∅or{endpoint}fori=1,…,n,[m[ϕt0[s,1]],M[ϕt0[s,1]]]∩[mi,Mi]=∅or{endpoint}fori=1,…,n,[m[ϕt1[s,1]],M[ϕt1[s,1]]]∩[mi,Mi]=∅or{endpoint}fori=1,…,n.Proof.
Recall thatϕt[r,s](x)=u(t)xr+v(t)xs for r≠0,s≠0. The next is φ(x)=xr and ψ(x)=xs with x∈I=〈0,+∞〉, so φ(I)=ψ(I)=I.Caser>0,s≥1.
We have(ψ∘φ-1)(x)=xs/r with (s/r)>1, and ψ-1(x)=x1/s with 0<(1/s)≤1. The function ψ is strictly φ-convex with strictly operator increasing ψ-1. Then the inequality in (5.28) is valid with associated spectral conditions by Corollary 5.6.Cases=-1.
We have(ψ∘φ-1)(x)=x-1/r with 0<-(1/r)<1, and ψ-1(x)=x-1. The function ψ is strictly φ-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.Caser=1.
We use functionsϕt[s,1](x)=u(t)xs+v(t)x. In this case φ(x)=xs and ψ(x)=x, thus (ψ∘φ-1)(x)=x1/s with 0<(1/s)<1, and ψ-1(x)=x. The function ψ is strictly φ-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≠C.Unfortunately, we cannot use a logarithmic function becauseln(I)=ℝ≠I.Remark 5.9.
Letr, s, ϕt, t0, t1,W, A, and Φ be as in Corollary 5.8.
Ifr≤-1 and s<0, then the problem remains the inequality
(5.32)Mn[r](WA,Φ)≤Mnϕt0[s,r](WA,Φ)≤Mnϕt1[s,r](WA,Φ)≤Mn[s](WA,Φ),
valid for all above n-tuples W, A, and Φ provided spectral conditions:
(5.33)[m[s],M[s]]∩[mi,Mi]=∅or{endpoint}fori=1,…,n,[m[ϕt0[s,r]],M[ϕt0[s,r]]]∩[mi,Mi]=∅or{endpoint}fori=1,…,n,[m[ϕt1[s,r]],M[ϕt1[s,r]]]∩[mi,Mi]=∅or{endpoint}fori=1,…,n.
The inequality in (5.32) is valid for r=-1 with associated spectral conditions because (ψ∘φ-1)(x)=x-1/s with -(1/s)>1, that is, ψ is φ-convex, and ψ-1(x)=x-1, that is, ψ-1 is operator decreasing.



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*Source: 203145-2012-08-07.xml*