Jensen Functional, Quasi-Arithmetic Mean and Sharp Converses of Hölder’s Inequalities

Abstract

In this article, we give sharp two-sided bounds for the generalized Jensen functional $J_n(f,g,h;$p,x). Assuming convexity/concavity of the generating function h, we give exact bounds for the generalized quasi-arithmetic mean $A_n(h;$p,x). In particular, exact bounds are determined for the generalized power means in terms from the class of Stolarsky means. As a consequence, some sharp converses of the famous Hölder’s inequality are obtained.

1. Introduction

Recall that the Jensen functional $J_n(\varphi ;\mathit{p},\mathit{x})$ is defined on an interval $I\subseteq \mathbb{R}$ by

$$J_n(\varphi ;\mathit{p},\mathit{x}):=\sum _1^n{\mathit{p}}_i\varphi \left({\mathit{x}}_i\right)-\varphi \left(\sum _1^n{\mathit{p}}_i{\mathit{x}}_i\right),$$

where $\varphi :I\to \mathbb{R}$, $\mathit{x}=({\mathit{x}}_1,{\mathit{x}}_2,\cdots ,{\mathit{x}}_n)\in I^n$ and $\mathit{p}={\left\{{\mathit{p}}_i\right\}}_1^n,{\sum }_1^n{\mathit{p}}_i=1$, is a positive weight sequence.

If $\varphi$ is a convex function on I, then the inequality

$$0\le J_n(\varphi ;\mathit{p},\mathit{x})$$

holds for each $\mathit{x}\in I^n$ and any positive weight sequence p.

Jensen’s inequality plays a fundamental role in many parts of mathematical analysis and applications. For example, well known $\mathcal{A}-\mathcal{G}-\mathcal{H}$ inequality, Hölder’s inequality, Ky Fan inequality, etc., are proven by the help of Jensen’s inequality (cf. [1,2,3,4]).

Assuming that $\mathit{x}\in {[a,b]}^n\subset I^n$, our aim in this paper is to determine some sharp bounds for the generalized Jensen functional

$$J_n(f,g,h;\mathit{p},\mathit{x}):=f\left(\sum _1^n{\mathit{p}}_{i}h\left({\mathit{x}}_i\right)\right)-g\left(h\left(\sum _1^n{\mathit{p}}_i{\mathit{x}}_i\right)\right),$$

for suitably chosen functions $f,g$ and h, such that

$$c_{f,g,h}(a,b)\le J_n(f,g,h;\mathit{p},\mathit{x})\le C_{f,g,h}(a,b),$$

i.e., the bounds which does not depend on p or x, but only on $a,b$ and functions $f,g$ and h.

Our global bounds will be entirely presented in terms of elementary means.

Recall that the mean is a map $M:{\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$, with a property

$$min(\mathit{x},y)\le M(\mathit{x},y)\le max(\mathit{x},y),$$

for each $\mathit{x},y\in {\mathbb{R}}_{+}$.

In order to make our results condensed and applicable, we shall use in the sequel the class of so-called Stolarsky (or extended) two-parametric mean values, defined for positive values of $\mathit{x},y,\mathit{x}\ne y$ by the following:

$$E_{r,s}(\mathit{x},y)=\left\{\begin{array}{cc}{\left(\frac{r({\mathit{x}}^s-y^s)}{s({\mathit{x}}^r-y^r)}\right)}^{1/(s-r)},\hfill & rs(r-s)\ne 0\hfill \\ exp\left(\frac{-1}{s}+\frac{{\mathit{x}}^{s}log\mathit{x}-y^{s}logy}{{\mathit{x}}^s-y^s}\right),\hfill & r=s\ne 0\hfill \\ {\left(\frac{{\mathit{x}}^s-y^s}{s(log\mathit{x}-logy)}\right)}^{1/s},\hfill & s\ne 0,r=0\hfill \\ \sqrt{\mathit{x}y},\hfill & r=s=0,\hfill \\ \mathit{x},\hfill & y=\mathit{x}>0.\hfill \end{array}\right.$$

In this form, it was introduced by Keneth Stolarsky in [5].

Most of the classical two variable means are just special cases of the class E.

For example,

$$A(\mathit{x},y)=E_{1,2}(\mathit{x},y)=\frac{\mathit{x}+y}{2}$$

is the arithmetic mean;

$$G(\mathit{x},y)=E_{0,0}(\mathit{x},y)=E_{-r,r}(\mathit{x},y)=\sqrt{\mathit{x}y}$$

is the geometric mean;

$$L(\mathit{x},y)=E_{0,1}(\mathit{x},y)=\frac{\mathit{x}-y}{log\mathit{x}-logy}$$

is the logarithmic mean;

$$I(\mathit{x},y)=E_{1,1}(\mathit{x},y)={({\mathit{x}}^{\mathit{x}}/y^y)}^{\frac{1}{\mathit{x}-y}}/e$$

is the identric mean, etc.

More generally, the r-th power mean

$$A_r(\mathit{x},y)={\left(\frac{{\mathit{x}}^r+y^r}{2}\right)}^{1/r}$$

is equal to $E_{r,2r}(\mathit{x},y)$.

Theory of Stolarsky means is very well developed, cf. [6,7] and references therein.

Some basic properties are listed in the following:

Means $E_{r,s}(\mathit{x},y)$ are

a. symmetric in both parameters, i.e., $E_{r,s}(\mathit{x},y)=E_{s,r}(\mathit{x},y)$;

b. symmetric in both variables, i.e., $E_{r,s}(\mathit{x},y)=E_{r,s}(y,\mathit{x})$;

c. homogeneous of order one, that is $E_{r,s}(t\mathit{x},ty)=t{E}_{r,s}(\mathit{x},y),t>0$;

d. monotone increasing in either r or s;

e. monotone increasing in either x or y; and

f. logarithmically convex in either r or s for $r,s\in {\mathbb{R}}_{-}$ and logarithmically concave for $r,s\in {\mathbb{R}}_{+}$.

Let $h:I\to J$ be a continuous and strictly monotone function on an interval $I\subset \mathbb{R}$. Then, its inverse function $h^{-1}:J\to I$ exists and generates so-called $quasi-arithmetic$ mean ${\mathcal{A}}_h(\mathit{p},\mathit{x})$, given by

$${\mathcal{A}}_h(\mathit{p},\mathit{x}):=h^{-1}\left(\sum _1^n{\mathit{p}}_{i}h\left({\mathit{x}}_i\right)\right),$$

where $\mathit{x}=({\mathit{x}}_1,{\mathit{x}}_2,\cdots ,{\mathit{x}}_n)\in I^n$ and $\mathit{p}={\left\{{\mathit{p}}_i\right\}}_1^n,{\sum }_1^n{\mathit{p}}_i=1$ is a positive weight sequence.

Quasi-arithmetic means are introduced in [1] and then investigated by a plenty of researchers with most interesting results (cf. [8]). In this article, we shall give tight two-sided bounds for the difference

$${\mathcal{A}}_h(\mathit{p},\mathit{x})-\mathcal{A}(\mathit{p},\mathit{x}).$$

An important special case is the class of generalized power means ${\mathcal{B}}_s(\mathit{p},\mathit{x})$, generated by $h\left(\mathit{x}\right)={\mathit{x}}^s,s\in \mathbb{R}/\left\{0\right\}$,

$${\mathcal{B}}_s(\mathit{p},\mathit{x})={\left(\sum _1^n{\mathit{p}}_i{\mathit{x}}_i^s\right)}^{1/s}.$$

It is well known fact that power means are monotone increasing in $s\in \mathbb{R}$ (cf. [1]).

Some important particular cases are

$${\mathcal{B}}_{-1}(\mathit{p},\mathit{x})={(\sum _1^n{\mathit{p}}_i/{\mathit{x}}_i)}^{-1}:=\mathcal{H}(\mathit{p},\mathit{x});$$

$${\mathcal{B}}_0(\mathit{p},\mathit{x})=\underset{s\to 0}{lim}{\mathcal{B}}_s(\mathit{p},\mathit{x})=\prod _1^n{\mathit{x}}_i^{{\mathit{p}}_i}:=\mathcal{G}(\mathit{p},\mathit{x});$$

$${\mathcal{B}}_1(\mathit{p},\mathit{x})=\sum _1^n{\mathit{p}}_i{\mathit{x}}_i:=\mathcal{A}(\mathit{p},\mathit{x}),$$

that is, the generalized harmonic, geometric and arithmetic means, respectively.

Therefore,

$$\mathcal{H}(\mathit{p},\mathit{x})\le \mathcal{G}(\mathit{p},\mathit{x})\le \mathcal{A}(\mathit{p},\mathit{x}),$$

represents the celebrated $\mathcal{A}-\mathcal{G}-\mathcal{H}$ inequalities.

Some converses of these inequalities will be given in this paper.

For arbitrary positive sequences a and b and real numbers $s,t$ with $1/s+1/t=1$, the celebrated Hölder’s inequalities says that

$$\sum _1^n{a}_i{b}_i\le {\left(\sum _1^n{a}_i^s\right)}^{1/s}{\left(\sum _1^n{b}_i^t\right)}^{1/t},s>1;$$

and

$$\sum _1^n{a}_i{b}_i\ge {\left(\sum _1^n{a}_i^s\right)}^{1/s}{\left(\sum _1^n{b}_i^t\right)}^{1/t},0<s<1.$$

We shall give in the sequel precise estimations of the difference

$$\sum _1^n{a}_i{b}_i-{\left(\sum _1^n{a}_i^s\right)}^{1/s}{\left(\sum _1^n{b}_i^t\right)}^{1/t},$$

and the quotient

$${\left(\sum _1^n{a}_i^s\right)}^{1/s}{\left(\sum _1^n{b}_i^t\right)}^{1/t}/\sum _1^n{a}_i{b}_i,$$

that is,

$$\sum _1^n{a}_i{b}_i\le {\left(\sum _1^n{a}_i^s\right)}^{1/s}{\left(\sum _1^n{b}_i^t\right)}^{1/t}\le \frac{E_{s+t,s}(a,b)E_{s+t,t}(a,b)}{G^2(a,b)}\sum _1^n{a}_i{b}_i,$$

for $1/s+1/t=1,s,t>1;a\le a_i^{1/t}/b_i^{1/s}\le b,i=1,2,...,n$.

2. Results and Proofs

Our main result concerning the generalized Jensen functional $J_n(f,g,h;\mathit{p},\mathit{x})$ is given by the following:

Theorem 1.

Let $f:J\to \mathbb{R},g:J\to \mathbb{R},h:I\to J$ be continuous and eventually differentiable functions on their domains.

For $\mathit{x}\in {[a,b]}^n\subset I^n$, let h be convex on I and f be an increasing function on J.

Then,

$$c_{f,g,h}(a,b):=\underset{\mathit{p}}{min}\left[(f\circ h+g\circ h)(\mathit{p}a+(1-\mathit{p})b)\right]$$

$$\le J_n(f,g,h;\mathit{p},\mathit{x})\le$$

$$\underset{\mathit{p}}{max}[f(\mathit{p}h\left(a\right)+(1-\mathit{p})h\left(b\right))-g\left(h(\mathit{p}a+(1-\mathit{p})b)\right)]:=C_{f,g,h}(a,b).$$

Both bounds $c_{f,g,h}(a,b)$ and $C_{f,g,h}(a,b)$ are sharp.

Proof. 

Since $a\le {\mathit{x}}_i\le b$, there exist non-negative numbers ${\lambda }_i,{\mu }_i;{\lambda }_i+{\mu }_i=1$, such that ${\mathit{x}}_i={\lambda }_{i}a+{\mu }_{i}b,i=1,2,...,n$.

Hence,

$$J_n(f,g,h;\mathit{p},\mathit{x})=f\left(\sum _1^n{\mathit{p}}_{i}h\left({\mathit{x}}_i\right)\right)-g\left(h\left(\sum _1^n{\mathit{p}}_i{\mathit{x}}_i\right)\right)=f\left(\sum _1^n{\mathit{p}}_{i}h({\lambda }_{i}a+{\mu }_{i}b)\right)-g\left(h\left(\sum _1^n{\mathit{p}}_i({\lambda }_{i}a+{\mu }_{i}b)\right)\right)$$

$$\le f\left(\sum _1^n{\mathit{p}}_i({\lambda }_{i}h\left(a\right)+{\mu }_{i}h\left(b\right))\right)-g\left(h(a\sum _1^n{\mathit{p}}_i{\lambda }_i+b\sum _1^n{\mathit{p}}_i{\mu }_i)\right))$$

$$=f(\mathit{p}h\left(a\right)+(1-\mathit{p})h\left(b\right))-g\left(h(\mathit{p}a+(1-\mathit{p})b)\right)\le \underset{\mathit{p}}{max}[f(\mathit{p}h\left(a\right)+(1-\mathit{p})h\left(b\right))-g\left(h(\mathit{p}a+(1-\mathit{p})b)\right)],$$

where we denoted ${\sum }_1^n{\mathit{p}}_i{\lambda }_i:=\mathit{p}\in [0,1]$.

The above estimate is valid for arbitrary sequences p and x. To prove its sharpness, suppose that the maxima is reached at the point $\mathit{p}={\mathit{p}}_0$, i.e.,

$$\underset{\mathit{p}}{max}[f(\mathit{p}h\left(a\right)+(1-\mathit{p})h\left(b\right))-g\left(h(\mathit{p}a+(1-\mathit{p})b)\right)]$$

$$=f({\mathit{p}}_{0}h\left(a\right)+(1-{\mathit{p}}_0)h\left(b\right))-g\left(h({\mathit{p}}_{0}a+(1-{\mathit{p}}_0)b)\right)=C_{f,g,h}(a,b).$$

Then

$$J_n(f,g,h;{\mathit{p}}_0,{\mathit{x}}_0)=C_{f,g,h}(a,b),$$

where

$${\mathit{p}}_0=({\mathit{p}}_0,{\mathit{p}}_2,...,{\mathit{p}}_n),{\mathit{x}}_0=(a,b,...,b).$$

On the other hand, since h is a convex function on I, by Jensen’s inequality we get

$$\sum _1^n{\mathit{p}}_{i}h\left({\mathit{x}}_i\right)\ge h\left(\sum _1^n{\mathit{p}}_i{\mathit{x}}_i\right).$$

Because f is an increasing function, it follows that

$$J_n(f,g,h;\mathit{p},\mathit{x})=f\left(\sum _1^n{\mathit{p}}_{i}h\left({\mathit{x}}_i\right)\right)-g\left(h\left(\sum _1^n{\mathit{p}}_i{\mathit{x}}_i\right)\right)\ge f\left(h\left(\sum _1^n{\mathit{p}}_i{\mathit{x}}_i\right)\right)-g\left(h\left(\sum _1^n{\mathit{p}}_i{\mathit{x}}_i\right)\right)$$

$$=f\left(h(\mathit{p}a+(1-\mathit{p})b)\right)-g\left(h(\mathit{p}a+(1-\mathit{p})b)\right)\ge \underset{\mathit{p}}{min}\left[(f\circ h+g\circ h)(\mathit{p}a+(1-\mathit{p})b)\right]:=c_{f,g,h}(a,b).$$

A simple analysis of the constant $c_{f,g,h}(a,b)$ reveals the next: if minima of the function $(f\circ h+g\circ h)\left(t\right)$ exists for $t=t_0\in [a,b]$, then $c_{f,g,h}(a,b)=(f\circ h+g\circ h)\left(t_0\right)$, taken for $\mathit{p}={\mathit{p}}_0=(b-t_0)/(b-a)$.

Otherwise, we have that $c_{f,g,h}(a,b)=min\{(f\circ h+g\circ h)\left(a\right),(f\circ h+g\circ h)\left(b\right)\}$.

Those results are evidently sharp, since

$$J_n(f,g,h;\mathit{p},{\mathit{x}}_0)=c_{f,g,h}(a,b),$$

with ${\mathit{x}}_0=(t_0,...,t_0),{\mathit{x}}_0=(a,...,a)$ or ${\mathit{x}}_0=(b,...,b)$, respectively. □

Theorem 1 with its variants (a decreasing function f, concave function h) is the source of a plenty of interesting inequalities. Further investigations are left to the reader.

Sometimes, it is a difficult problem to evaluate exact maxima in this theorem.

For this cause, we shall give in the sequel two estimations of $J_n(f,g,h;\mathit{p},\mathit{x})$ with the unique maxima, which could be easily calculated.

Theorem 2.

Under the conditions of Theorem 1, assume firstly that f is a convex function on J. Then,

$$J_n(f,g,h;\mathit{p},\mathit{x})\le \underset{\mathit{p}}{max}[\mathit{p}(f\circ h)\left(a\right)+(1-\mathit{p})(f\circ h)\left(b\right)-(g\circ h)(\mathit{p}a+(1-\mathit{p})b)].$$

Assuming that $g\circ h$ is a concave function, we obtain

$$J_n(f,g,h;\mathit{p},\mathit{x})\le \underset{\mathit{p}}{max}[f(\mathit{p}h\left(a\right)+(1-\mathit{p})h\left(b\right))-(\mathit{p}(g\circ h)\left(a\right)+(1-\mathit{p})(g\circ h)\left(b\right))].$$

Now, both maxima can be easily determined by the standard technique.

Proof. 

By Theorem 1, we know that there exists $\mathit{p}\in [0,1]$ such that

$$J_n(f,g,h;\mathit{p},\mathit{x})\le f(\mathit{p}h\left(a\right)+(1-\mathit{p})h\left(b\right))-g\left(h(\mathit{p}a+(1-\mathit{p})b)\right).$$

If additionally f is convex on J, then

$$f\left(\mathit{p}h\right(a)+(1-\mathit{p}\left)h\right(b\left)\right)\le \mathit{p}(f\circ h)\left(a\right)+(1-\mathit{p})(f\circ h)\left(b\right).$$

Hence,

$$J_n(f,g,h;\mathit{p},\mathit{x})\le \mathit{p}(f\circ h)\left(a\right)+(1-\mathit{p})(f\circ h)\left(b\right)-(g\circ h)(\mathit{p}a+(1-\mathit{p})b))$$

$$\le \underset{\mathit{p}}{max}[\mathit{p}(f\circ h)\left(a\right)+(1-\mathit{p})(f\circ h)\left(b\right)-(g\circ h)(\mathit{p}a+(1-\mathit{p})b))].$$

Similarly, if $g\circ h$ is a concave function on J, we have

$$g\left(h\right(\mathit{p}a+(1-\mathit{p})b)=(g\circ h\left)\right(\mathit{p}a+(1-\mathit{p})b)\ge \mathit{p}(g\circ h\left)\right(a)+(1-\mathit{p}\left)\right(g\circ h\left)\right(b),$$

and

$$J_n(f,g,h;\mathit{p},\mathit{x})\le \underset{\mathit{p}}{max}[f(\mathit{p}h\left(a\right)+(1-\mathit{p})h\left(b\right))-(\mathit{p}(g\circ h)\left(a\right)+(1-\mathit{p})(g\circ h)\left(b\right))].$$

□

An important special case is the converse of Jensen’s inequality.

Theorem 3.

Let ϕ be a convex function on $I\subset \mathbb{R}$ and, for $[\xi ,\eta ]\subset I$, let $\mathit{x}\in {[\xi ,\eta ]}^n$.

Then,

$$0\le J_n(\varphi ;\mathit{p},\mathit{x})\le \underset{\mathit{p}}{max}[\mathit{p}\varphi \left(\xi \right)+(1-\mathit{p})\varphi \left(\eta \right)-\varphi (\mathit{p}\xi +(1-\mathit{p})\eta )]:=T_{\varphi }(\xi ,\eta ).$$

If ϕ is a concave function, then

$$0\le -J_n(\varphi ;\mathit{p},\mathit{x})\le \underset{\mathit{p}}{max}[\varphi (\mathit{p}\xi +(1-\mathit{p})\eta )-(\mathit{p}\varphi \left(\xi \right)+(1-\mathit{p})\varphi \left(\eta \right))]=-T_{\varphi }(\xi ,\eta ).$$

The constant $T_{\varphi }(\xi ,\eta )$ is sharp since there exist sequences ${\mathit{p}}_0,{\mathit{x}}_0$ such that

$$J_n(\varphi ;{\mathit{p}}_0,{\mathit{x}}_0)=T_{\varphi }(\xi ,\eta ).$$

Proof. 

This is a simple consequence of Theorem 1 obtained for $f\left(\mathit{x}\right)=g\left(\mathit{x}\right)=\mathit{x};h=\varphi$. If $\varphi$ is a concave function, then $-\varphi$ is convex and the proof follows from the first part of this theorem. □

In this case, the bound $T_{\varphi }(\xi ,\eta )$ can be explicitly calculated.

Theorem 4.

For a differentiable convex mapping ϕ, we have that

$$T_{\varphi }(\xi ,\eta )=\frac{\varphi \left(\eta \right)-\varphi \left(\xi \right)}{\eta -\xi }{\Theta }_{\varphi }(\xi ,\eta )+\frac{\eta \varphi \left(\xi \right)-\xi \varphi \left(\eta \right)}{\eta -\xi }-\varphi \left({\Theta }_{\varphi }(\xi ,\eta )\right),$$

where ${\Theta }_{\varphi }(\xi ,\eta )$ is the Lagrange mean value of numbers ξ and η, defined by

$${\Theta }_{\varphi }(\xi ,\eta ):={\left({\varphi }^{\prime }\right)}^{-1}\left(\frac{\varphi \left(\eta \right)-\varphi \left(\xi \right)}{\eta -\xi }\right).$$

The function $T_{\varphi }$ is positive and symmetric, i.e., $T_{\varphi }(\xi ,\eta )=T_{\varphi }(\eta ,\xi )$ and ${lim}_{\eta \to \xi }{T}_{\varphi }(\xi ,\eta )=0$.

Proof. 

If the maximum is taken at the point $\mathit{p}={\mathit{p}}_0$, by the standard technique we get

$${\varphi }^{\prime }({\mathit{p}}_0\xi +(1-{\mathit{p}}_0)\eta )(\xi -\eta )=\varphi \left(\xi \right)-\varphi \left(\eta \right),$$

that is,

$${\mathit{p}}_0\xi +(1-{\mathit{p}}_0)\eta ={\Theta }_{\varphi }(\xi ,\eta ).$$

Therefore,

$${\mathit{p}}_0=\frac{{\Theta }_{\varphi }(\xi ,\eta )-\eta }{\xi -\eta };1-{\mathit{p}}_0=\frac{\xi -{\Theta }_{\varphi }(\xi ,\eta )}{\xi -\eta },$$

and

$$\underset{\mathit{p}}{max}[\mathit{p}\varphi \left(\xi \right)+(1-\mathit{p})\varphi \left(\eta \right)-\varphi (\mathit{p}\xi +(1-\mathit{p})\eta )]={\mathit{p}}_0\varphi \left(\xi \right)+(1-{\mathit{p}}_0)\varphi \left(\eta \right)-\varphi ({\mathit{p}}_0\xi +(1-{\mathit{p}}_0)\eta )$$

$$=\frac{{\Theta }_{\varphi }(\xi ,\eta )-\eta }{\xi -\eta }\varphi \left(\xi \right)+\frac{\xi -{\Theta }_{\varphi }(\xi ,\eta )}{\xi -\eta }\varphi \left(\eta \right)-\varphi \left({\Theta }_{\varphi }(\xi ,\eta )\right)=T_{\varphi }(\xi ,\eta ).$$

□

Now, some important inequalities concerning quasi-arithmetic mean can be easily obtained from Theorem 1 by putting $f=g=h^{-1}$. Nevertheless, in order to avoid unnecessary monotonicity issues, we turn another way.

Our main result is contained in the following:

Theorem 5.

For $a\le {\mathit{x}}_i\le b,i=1,2,...,n;a,b\in I$, let $h:I\to J$ be continuous and strictly monotone function and assume that $h^{-1}:J\to I$ is convex. Then,

$$0\le \mathcal{A}(\mathit{p},\mathit{x})-{\mathcal{A}}_h(\mathit{p},\mathit{x})\le T_{h^{-1}}(h\left(a\right),h\left(b\right)),$$

where the constant $T_{\varphi }(\xi ,\eta )$ is defined in Theorems 3 and 4.

If $h^{-1}$ is a concave function, then

$$0\le {\mathcal{A}}_h(\mathit{p},\mathit{x})-\mathcal{A}(\mathit{p},\mathit{x})\le -T_{h^{-1}}(h\left(a\right),h\left(b\right)).$$

Proof. 

We shall give a simple proof of this theorem.

Namely, since $h^{-1}$ is a convex function, applying the first part of Theorem 3 with $\varphi =h^{-1}$, we obtain

$$0\le \sum _1^n{\mathit{p}}_i{h}^{-1}\left({\mathit{x}}_i\right)-h^{-1}(\sum _1^n{\mathit{p}}_i{\mathit{x}}_i)\le T_{h^{-1}}(a,b).$$

Now, by changing variables ${\mathit{x}}_i\to h\left({\mathit{x}}_i\right)i=1,2,...,n$, we get $h^{-1}\left({\mathit{x}}_i\right)\to h^{-1}\circ h\left({\mathit{x}}_i\right)={\mathit{x}}_i$ and $a\to h\left(a\right),b\to h\left(b\right)$.

Hence, $h\left(a\right)\le h\left({\mathit{x}}_i\right)\le h\left(b\right)$ or $h\left(b\right)\le h\left({\mathit{x}}_i\right)\le h\left(a\right)$ depending on the monotonicity of h. However, since $T_{\varphi }(\xi ,\eta )$ is symmetric in variables, we finally get

$$0\le \sum _1^n{\mathit{p}}_i{\mathit{x}}_i-h^{-1}(\sum _1^n{\mathit{p}}_{i}h\left({\mathit{x}}_i\right))\le T_{h^{-1}}(h\left(a\right),h\left(b\right)).$$

The second part of this theorem can be proved along the same lines. □

The most striking example of quasi-arithmetic means is the class of generalized power means ${\mathcal{B}}_s(\mathit{p},\mathit{x})$, generated by $h\left(\mathit{x}\right)={\mathit{x}}^s,h^{-1}\left(\mathit{x}\right)={\mathit{x}}^{1/s},s\in \mathbb{R}/\left\{0\right\}$, i.e.,

$${\mathcal{B}}_s(\mathit{p},\mathit{x})={\left(\sum _1^n{\mathit{p}}_i{\mathit{x}}_i^s\right)}^{1/s}.$$

As an application of Theorem 5, we shall estimate the difference ${\mathcal{B}}_s(\mathit{p},\mathit{x})-\mathcal{A}(\mathit{p},\mathit{x})$.

Theorem 6.

Let $a\le {\mathit{x}}_i\le b,i=1,2,...,n;0<a<b$.

Then,

$$0\le {\mathcal{B}}_s(\mathit{p},\mathit{x})-\mathcal{A}(\mathit{p},\mathit{x})\le \frac{s-1}{s}\left(E_{s,1}(a,b)-\frac{G^2(a,b)}{E_{s,s-1}(a,b)}\right),s>1;$$

$$0\le \mathcal{A}(\mathit{p},\mathit{x})-{\mathcal{B}}_s(\mathit{p},\mathit{x})\le \frac{1-s}{s}(E_{1,s}(a,b)-E_{1-s,-s}(a,b)),0<s<1;$$

$$0\le \mathcal{A}(\mathit{p},\mathit{x})-{\mathcal{B}}_s(\mathit{p},\mathit{x})\le \frac{s-1}{s}(E_{1-s,-s}(a,b)-E_{1,s}(a,b)),s<0.$$

Proof. 

Let $h\left(\mathit{x}\right)={\mathit{x}}^s,h^{-1}\left(\mathit{x}\right)={\mathit{x}}^{1/s},s\in \mathbb{R}/\left\{0\right\}$.

If $s>1$ then $h^{-1}$ is a concave function on ${\mathbb{R}}^{+}$. Hence, by the second part of Theorem 5, we get

$$0\le {\mathcal{B}}_s(\mathit{p},\mathit{x})-\mathcal{A}(\mathit{p},\mathit{x})\le -T_{{\mathit{x}}^{1/s}}(a^s,b^s).$$

Applying the result from Theorem 4, a simple calculation gives

$${\Theta }_{{\mathit{x}}^{1/s}}(a^s,b^s)={\left(\frac{b^s-a^s}{s(b-a)}\right)}^{s/(s-1)}=E_{s,1}^s(a,b)=\frac{b^s-a^s}{s(b-a)}{E}_{s,1}(a,b).$$

Hence,

$$-T_{{\mathit{x}}^{1/s}}(a^s,b^s)={(\Theta (·))}^{1/s}-\frac{b-a}{b^s-a^s}\Theta (·)-\frac{a{b}^s-b{a}^s}{b^s-a^s}$$

$$=E_{s,1}(a,b)-\frac{1}{s}{E}_{s,1}(a,b)-\frac{ab(b^{s-1}-a^{s-1})}{b^s-a^s}=\frac{s-1}{s}\left(E_{s,1}(a,b)-\frac{G^2(a,b)}{E_{s,s-1}(a,b)}\right).$$

In cases $0<s<1$ or $s<0$, one should apply the first part of Theorem 5, since $h^{-1}={\mathit{x}}^{1/s}$ is convex on ${\mathbb{R}}^{+}$. Proceeding as above, the result follows. □

As a consequence, we obtain some converses of the $\mathcal{A}(\mathit{p},\mathit{x})-\mathcal{G}(\mathit{p},\mathit{x})-\mathcal{H}(\mathit{p},\mathit{x})$ inequality.

Corollary 1.

Let $a\le {\mathit{x}}_i\le b,i=1,2,...,n;0<a<b$.

Then,

$$0\le \mathcal{A}(\mathit{p},\mathit{x})-\mathcal{H}(\mathit{p},\mathit{x})\le 2\left(A\right(a,b)-G(a,b\left)\right).$$

Proof. 

Putting $s=-1$, we get

$$0\le \mathcal{A}(\mathit{p},\mathit{x})-{\mathcal{B}}_{-1}(\mathit{p},\mathit{x})=\mathcal{A}(\mathit{p},\mathit{x})-\mathcal{H}(\mathit{p},\mathit{x})$$

$$\le 2(E_{2,1}(a,b)-E_{1,-1}(a,b))=2(A(a,b)-G(a,b)).$$

□

Corollary 2.

Let $a\le {\mathit{x}}_i\le b,i=1,2,...,n;0<a<b$.

Then,

$$0\le \mathcal{A}(\mathit{p},\mathit{x})-\mathcal{G}(\mathit{p},\mathit{x})\le L(a,b)log\frac{L(a,b)I(a,b)}{G^2(a,b)}.$$

Proof. 

We have,

$$\mathcal{A}(\mathit{p},\mathit{x})-\mathcal{G}(\mathit{p},\mathit{x})=\underset{s\to 0}{lim}(\mathcal{A}(\mathit{p},\mathit{x})-{\mathcal{B}}_s(\mathit{p},\mathit{x}))$$

$$\le \underset{s\to 0}{lim}\left(\frac{1-s}{s}(E_{1,s}(a,b)-E_{1-s,-s}(a,b))\right)$$

$$=L(a,b)log\frac{L(a,b)I(a,b)}{G^2(a,b)}.$$

□

The sequences p and x in Theorem 6 are arbitrary. Specializing a little bit, we obtain sharp converses of slightly generalized Hölder’s inequalities.

Theorem 7.

Let ${\left\{t_i\right\}}_1^n,{\left\{u_i\right\}}_1^n,{\left\{v_i\right\}}_1^n$ be any sequences of positive real numbers with $a\le u_i{v}_i^{1-t}\le b$ for some constants $0<a<b$ and $1/s+1/t=1$ for some $s,t\in \mathbb{R}$.

Then,

$$0\le {\left(\sum _1^n{t}_i{u}_i^s\right)}^{1/s}{\left(\sum _1^n{t}_i{v}_i^t\right)}^{1/t}-\sum _1^n{t}_i{u}_i{v}_i\le C_s(a,b)\sum _1^n{t}_i{v}_i^t,$$

with

$$C_s(a,b)=\frac{s-1}{s}\left(E_{s,1}(a,b)-\frac{G^2(a,b)}{E_{s,s-1}(a,b)}\right),$$

and $s>1$;

$$0\le \sum _1^n{t}_i{u}_i{v}_i-{\left(\sum _1^n{t}_i{u}_i^s\right)}^{1/s}{\left(\sum _1^n{t}_i{v}_i^t\right)}^{1/t}\le D_s(a,b)\sum _1^n{t}_i{v}_i^t,$$

where

$$D_s(a,b)=\frac{1-s}{s}\left(E_{s,1}(a,b)-E_{1-s,-s}(a,b)\right),$$

and $0<s<1$.

Proof. 

Changing variables

$${\mathit{p}}_i=t_i{v}_i^t/\sum _i^n{t}_i{v}_i^t;{\mathit{x}}_i=u_i{v}_i^{1-t},i=1,2,...,n,$$

yields

$${\mathit{p}}_i{\mathit{x}}_i=t_i{u}_i{v}_i/\sum _1^n{t}_i{v}_i^t;$$

$${\mathit{p}}_i{\mathit{x}}_i^s=t_i{v}_i^t{\left(u_i{v}_i^{1-t}\right)}^s/\sum _1^n{t}_i{v}_i^t=t_i{u}_i^s{v}_i^{s+t-st}/\sum _1^n{t}_i{v}_i^t=t_i{u}_i^s/\sum _1^n{t}_i{v}_i^t.$$

Now, applying Theorem 6 for $s>1$, we get

$$0\le {\mathcal{B}}_s(\mathit{p},\mathit{x})-\mathcal{A}(\mathit{p},\mathit{x})={\left(\sum _1^n{\mathit{p}}_i{\mathit{x}}_i^s\right)}^{1/s}-\sum _1^n{\mathit{p}}_i{\mathit{x}}_i$$

$$=\frac{{\left({\sum }_1^n{t}_i{u}_i^t\right)}^{1/s}}{{\left({\sum }_1^n{t}_i{v}_i^t\right)}^{1/s}}-\frac{{\sum }_1^n{t}_i{u}_i{v}_i}{{\sum }_1^n{t}_i{v}_i^t}\le \frac{s-1}{s}\left(E_{s,1}(a,b)-\frac{G^2(a,b)}{E_{s,s-1}(a,b)}\right),$$

and the result clearly follows by multiplying both sides with ${\sum }_1^n{t}_i{v}_i^t$.

Applying the same procedure in the case $0<s<1$, we obtain the second part of this theorem. □

Finally, we prove another sharp converses of Hölder’s inequalities. For this cause, we shall estimate firstly the expression

$${\mathcal{F}}_{s,t}(\mathit{p},\mathit{x}):={\left(\sum _1^n{\mathit{p}}_i{\mathit{x}}_i^s\right)}^{1/s}{\left(\sum _1^n{\mathit{p}}_i{\mathit{x}}_i^{-t}\right)}^{1/t},1/s+1/t=1,s,t\in \mathbb{R}.$$

Lemma 1.

Let $a\le {\mathit{x}}_i\le b,i=1,2,...,n$ for some $0<a<b$.

If $s>1$, we have

$$1\le {\mathcal{F}}_{s,t}(\mathit{p},\mathit{x})\le \frac{E_{s,s+t}(a,b)E_{t,s+t}(a,b)}{G^2(a,b)},$$

and

$$\frac{E_{s,s+t}(a,b)E_{t,s+t}(a,b)}{G^2(a,b)}\le {\mathcal{F}}_{s,t}(\mathit{p},\mathit{x})\le 1,$$

for $0<s<1$.

Proof. 

Following the method from the proof of Theorem 1, we get

$${\mathit{x}}_i^s={\lambda }_i{a}^s+{\mu }_i{b}^s,{\lambda }_i+{\mu }_i=1,i=1,2,...,n.$$

If $s>1$, then also $t>1$, hence the function ${\mathit{x}}^{-t/s}$ is convex.

Therefore,

$${\mathit{x}}_i^{-t}={\left({\mathit{x}}_i^s\right)}^{-t/s}={({\lambda }_i{a}^s+{\mu }_i{b}^s)}^{-t/s}\le {\lambda }_i{\left(a^s\right)}^{-t/s}+{\mu }_i{\left(b^s\right)}^{-t/s}={\lambda }_i{a}^{-t}+{\mu }_i{b}^{-t},$$

and

$${\mathcal{F}}_{s,t}(\mathit{p},\mathit{x})={\left(\sum _1^n{\mathit{p}}_i{\mathit{x}}_i^s\right)}^{1/s}{\left(\sum _1^n{\mathit{p}}_i{\mathit{x}}_i^{-t}\right)}^{1/t}$$

$$\le {(a^s\sum _1^n{\mathit{p}}_i{\lambda }_i+b^s\sum _1^n{\mathit{p}}_i{\mu }_i)}^{1/s}{(a^{-t}\sum _1^n{\mathit{p}}_i{\lambda }_i+b^{-t}\sum _1^n{\mathit{p}}_i{\mu }_i)}^{1/t}$$

$$={(\mathit{p}{a}^s+q{b}^s)}^{1/s}{(\mathit{p}{a}^{-t}+q{b}^{-t})}^{1/t},$$

where we put

$$\sum _1^n{\mathit{p}}_i{\lambda }_i:=\mathit{p},\sum _1^n{\mathit{p}}_i{\mu }_i:=q;\mathit{p}+q=1.$$

Therefore, it follows that

$${\mathcal{F}}_{s,t}(\mathit{p},\mathit{x})\le \underset{\mathit{p}}{max}\left[{(\mathit{p}{a}^s+q{b}^s)}^{1/s}{(\mathit{p}{a}^{-t}+q{b}^{-t})}^{1/t}\right]$$

$$={({\mathit{p}}_0{a}^s+q_0{b}^s)}^{1/s}{({\mathit{p}}_0{a}^{-t}+q_0{b}^{-t})}^{1/t}.$$

By the standard technique we obtain that this maxima satisfy the equation

$$\frac{s({\mathit{p}}_0{a}^s+q_0{b}^s)}{a^s-b^s}=\frac{-t({\mathit{p}}_0{a}^{-t}+q_0{b}^{-t})}{a^{-t}-b^{-t}},$$

that is,

$${\mathit{p}}_0=\frac{1}{s+t}\left(\frac{s{b}^s}{b^s-a^s}-\frac{t{a}^t}{b^t-a^t}\right);q_0=\frac{1}{s+t}\left(\frac{t{b}^t}{b^t-a^t}-\frac{s{a}^s}{b^s-a^s}\right).$$

Henceforth,

$${\mathit{p}}_0{a}^{-t}+q_0{b}^{-t}=\frac{s}{s+t}\frac{a^{-t}{b}^s-a^s{b}^{-t}}{b^s-a^s}=\frac{s}{s+t}\frac{{\left(ab\right)}^{-t}(b^{s+t}-a^{s+t})}{b^s-a^s},$$

and

$${\mathit{p}}_0{a}^s+q_0{b}^s=\frac{t}{s+t}\frac{b^{s+t}-a^{s+t}}{b^t-a^t}.$$

Therefore,

$${({\mathit{p}}_0{a}^{-t}+q_0{b}^{-t})}^{1/t}=E_{s+t,s}(a,b)/G^2(a,b);$$

$${({\mathit{p}}_0{a}^s+q_0{b}^s)}^{1/s}=E_{s+t,t}(a,b),$$

and we finally obtain

$$\underset{\mathit{p}}{max}\left[{(\mathit{p}{a}^s+q{b}^s)}^{1/s}{(\mathit{p}{a}^{-t}+q{b}^{-t})}^{1/t}\right]={({\mathit{p}}_0{a}^s+q_0{b}^s)}^{1/s}{({\mathit{p}}_0{a}^{-t}+q_0{b}^{-t})}^{1/t}$$

$$=\frac{E_{s+t,s}(a,b)E_{s+t,t}(a,b)}{G^2(a,b)}.$$

On the other hand, by the monotonicity in s of ${\mathcal{B}}_s$, we get

$$1\le \frac{{\mathcal{B}}_s(\mathit{p},\mathit{x})}{{\mathcal{B}}_{-t}(\mathit{p},\mathit{x})}={\mathcal{F}}_{s,t}(\mathit{p},\mathit{x}),$$

since $s>-t$.

In the case $0<s<1$, we have that $t<0$. Therefore,

$$0>st=s+t,$$

and

$${\mathcal{F}}_{s,t}(\mathit{p},\mathit{x})\le 1,$$

since $s<-t$.

Additionally, $-t/s>1$, hence ${\mathit{x}}^{-t/s}$ is a convex function.

Therefore,

$${\mathit{x}}_i^{-t}={\left({\mathit{x}}_i^s\right)}^{-t/s}={({\lambda }_i{a}^s+{\mu }_i{b}^s)}^{-t/s}\le {\lambda }_i{\left(a^s\right)}^{-t/s}+{\mu }_i{\left(b^s\right)}^{-t/s}={\lambda }_i{a}^{-t}+{\mu }_i{b}^{-t}.$$

However, because the exponent $1/t$ is negative in this case, we obtain

$${\mathcal{F}}_{s,t}(\mathit{p},\mathit{x})={\left(\sum _1^n{\mathit{p}}_i{\mathit{x}}_i^s\right)}^{1/s}{\left(\sum _1^n{\mathit{p}}_i{\mathit{x}}_i^{-t}\right)}^{1/t}$$

$$\ge {(a^s\sum _1^n{\mathit{p}}_i{\lambda }_i+b^s\sum _1^n{\mathit{p}}_i{\mu }_i)}^{1/s}{(a^{-t}\sum _1^n{\mathit{p}}_i{\lambda }_i+b^{-t}\sum _1^n{\mathit{p}}_i{\mu }_i)}^{1/t}$$

$$={(\mathit{p}{a}^s+q{b}^s)}^{1/s}{(\mathit{p}{a}^{-t}+q{b}^{-t})}^{1/t}.$$

Therefore, we get

$${\mathcal{F}}_{s,t}(\mathit{p},\mathit{x})\ge \underset{\mathit{p}}{min}\left[{(\mathit{p}{a}^s+q{b}^s)}^{1/s}{(\mathit{p}{a}^{-t}+q{b}^{-t})}^{1/t}\right],$$

and, proceeding as above, the second part of this theorem follows. □

We are now able to formulate our main result.

Theorem 8.

Let ${\left\{t_i\right\}}_1^n,{\left\{u_i\right\}}_1^n,{\left\{v_i\right\}}_1^n$ be arbitrary sequences of positive numbers with $a\le u_i^{1/t}/v_i^{1/s}\le b$ for some constants $0<a<b$ and $1/s+1/t=1;s,t\in \mathbb{R}$.

For $s>1$, we have

$$\sum _1^n{t}_i{u}_i{v}_i\le {\left(\sum _1^n{t}_i{u}_i^s\right)}^{1/s}{\left(\sum _1^n{t}_i{v}_i^t\right)}^{1/t}\le \frac{E_{s,s+t}(a,b)E_{t,s+t}(a,b)}{G^2(a,b)}\sum _1^n{t}_i{u}_i{v}_i,$$

and

$$\frac{E_{s,s+t}(a,b)E_{t,s+t}(a,b)}{G^2(a,b)}\sum _1^n{t}_i{u}_i{v}_i\le {\left(\sum _1^n{t}_i{u}_i^s\right)}^{1/s}{\left(\sum _1^n{t}_i{v}_i^t\right)}^{1/t}\le \sum _1^n{t}_i{u}_i{v}_i,$$

for $0<s<1$.

Proof. 

Changing variables

$${\mathit{p}}_i=t_i{u}_i{v}_i/\sum _1^n{t}_i{u}_i{v}_i;{\mathit{x}}_i=u_i^{1/t}{v}_i^{-1/s},i=1,2,...,n,$$

we get

$${\mathit{p}}_i{\mathit{x}}_i^s=t_i{u}_i{v}_i{\left(u_i^{1/t}{v}_i^{-1/s}\right)}^s/\sum _1^n{t}_i{u}_i{v}_i=t_i{u}_i^{1+s/t}/\sum _1^n{t}_i{u}_i{v}_i=t_i{u}_i^s/\sum _1^n{t}_i{u}_i{v}_i;$$

$${\mathit{p}}_i{\mathit{x}}_i^{-t}=t_i{u}_i{v}_i{\left(u_i^{1/t}{v}_i^{-1/s}\right)}^{-t}/\sum _1^n{t}_i{u}_i{v}_i=t_i{v}_i^{1+t/s}/\sum _1^n{t}_i{u}_i{v}_i=t_i{v}_i^t/\sum _1^n{t}_i{u}_i{v}_i,$$

and

$${\mathcal{F}}_{s,t}(t,u,v)={\left(\frac{{\sum }_1^n{t}_i{u}_i^s}{{\sum }_1^n{t}_i{u}_i{v}_i}\right)}^{1/s}{\left(\frac{{\sum }_1^n{t}_i{v}_i^t}{{\sum }_1^n{t}_i{u}_i{v}_i}\right)}^{1/t}=\frac{{\left({\sum }_1^n{t}_i{u}_i^s\right)}^{1/s}{\left({\sum }_1^n{t}_i{v}_i^t\right)}^{1/t}}{{\sum }_1^n{t}_i{u}_i{v}_i}.$$

Now, an application of Lemma 1 gives the result. □

3. Conclusions

In this article, we give further development of our results from [3]. Sharp two-sided bounds are explicitly determined for the generalized Jensen functional $J_n(f,g,h;\mathit{p},\mathit{x})$ and, consequently, for Jensen’s inequality and quasi-arithmetic means. Exact converses of $\mathcal{A}-\mathcal{G}-\mathcal{H}$ inequalities and some forms of Hölder’s inequalities are also given. Since Theorem 1 achieved its definite form with very mild conditions posed on the generating functions $f,g$ and h, there remains a lot of work to apply its results in different areas of mathematics.