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Volume 9, Issue 1

# Nonlinear Sherman-type inequalities

Marek Niezgoda
• Corresponding author
• Department of Applied Mathematics and Computer Science, University of Life Sciences in Lublin, Akademicka 15, 20-950 Lublin, Poland
• Email
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Published Online: 2018-09-21 | DOI: https://doi.org/10.1515/anona-2018-0098

## Abstract

An important class of Schur-convex functions is generated by convex functions via the well-known Hardy–Littlewood–Pólya–Karamata inequality. Sherman’s inequality is a natural generalization of the HLPK inequality. It can be viewed as a comparison of two special inner product expressions induced by a convex function of one variable. In the present note, we extend the Sherman inequality from the (bilinear) inner product to a (nonlinear) map of two vectorial variables satisfying the Leon–Proschan condition. Some applications are shown for directional derivatives and gradients of Schur-convex functions.

MSC 2010: 26B25; 26D15; 26D10

## 1 Introduction

We say that an n-tuple $𝐲={\left({y}_{1},\mathrm{\dots },{y}_{n}\right)}^{T}\in {ℝ}^{n}$ is majorized by an n-tuple $𝐱={\left({x}_{1},\mathrm{\dots },{x}_{n}\right)}^{T}\in {ℝ}^{n}$, and write $𝐲\prec 𝐱$, if

Here ${x}_{\left[1\right]}\ge \mathrm{\cdots }\ge {x}_{\left[n\right]}$ and ${y}_{\left[1\right]}\ge \mathrm{\cdots }\ge {y}_{\left[n\right]}$ are the entries of $𝐱$ and $𝐲$, respectively, arranged in decreasing order [13, p. 8].

It is known that for $𝐱,𝐲\in {ℝ}^{n}$,

$𝐲\prec 𝐱\mathit{ }\text{if and only if}\mathit{ }𝐲\in \mathrm{conv}{ℙ}_{n}𝐱,$(1.1)

where the symbol $\mathrm{conv}$ means “the convex hull of”, and ${ℙ}_{n}$ denotes the group of $n×n$ permutation matrices (see [5, p. 16], [6, p. 12] and [7]).

An $n×m$ real matrix $𝐒=\left({s}_{ij}\right)$ is called column stochastic (resp. row stochastic) if ${s}_{ij}\ge 0$ for $i=1,\mathrm{\dots },n$, $j=1,\mathrm{\dots },m$, and all column sums (resp. row sums) of $𝐒$ are equal to 1, i.e., ${\sum }_{i=1}^{n}{s}_{ij}=1$ for $j=1,\mathrm{\dots },m$ (resp. ${\sum }_{j=1}^{m}{s}_{ij}=1$ for $i=1,\mathrm{\dots },n$).

An $n×n$ real matrix $𝐒=\left({s}_{ij}\right)$ is said to be doubly stochastic if it is column stochastic and row stochastic [13, pp. 29–30]. The set of all $n×n$ doubly stochastic matrices is denoted by ${𝔻}_{n}$.

A doubly stochastic matrix is a convex combination of some permutation matrices, and vice versa [13, Theorem A.2.]. That is, ${𝔻}_{n}=\mathrm{conv}{ℙ}_{n}$. Therefore, (1.1) takes the following form: for $𝐱,𝐲\in {ℝ}^{n}$,

$𝐲\prec 𝐱\mathit{ }\text{if and only if}\mathit{ }𝐲=\mathrm{𝐒𝐱}$

for some doubly stochastic $n×n$ matrix $𝐒$ (see [13, p. 33]).

A function $F:{J}^{n}\to ℝ$ with an interval $J\subset ℝ$ is said to be Schur-convex on ${J}^{n}$ if for $𝐱,𝐲\in {J}^{n}$,

$𝐲\prec 𝐱\mathit{ }\text{implies}\mathit{ }F\left(𝐲\right)\le F\left(𝐱\right).$

See [13, pp. 79–154] for applications of Schur-convex functions.

Some important examples of Schur-convex functions are included in the following theorem.

#### Theorem A ([8, 11]).

Let $f\mathrm{:}J\mathrm{\to }\mathrm{R}$ be a real convex function defined on an interval $J\mathrm{\subset }\mathrm{R}$.

Then, for $\mathrm{x}\mathrm{=}{\mathrm{\left(}{x}_{\mathrm{1}}\mathrm{,}{x}_{\mathrm{2}}\mathrm{,}\mathrm{\dots }\mathrm{,}{x}_{n}\mathrm{\right)}}^{T}\mathrm{\in }{J}^{n}$ and $\mathrm{y}\mathrm{=}{\mathrm{\left(}{y}_{\mathrm{1}}\mathrm{,}{y}_{\mathrm{2}}\mathrm{,}\mathrm{\dots }\mathrm{,}{y}_{n}\mathrm{\right)}}^{T}\mathrm{\in }{J}^{n}$,

$𝐲\prec 𝐱\mathit{ }\text{𝑖𝑚𝑝𝑙𝑖𝑒𝑠}\mathit{ }\sum _{i=1}^{n}f\left({y}_{i}\right)\le \sum _{i=1}^{n}f\left({x}_{i}\right).$

Throughout, the symbol ${\left(\cdot \right)}^{T}$ denotes the operation of taking the transpose of a matrix. So, $𝐒$ is a column stochastic matrix if and only if ${𝐒}^{T}$ is a row stochastic matrix.

A generalization of Theorem A is the following result (see [17], cf. also [4]).

#### Theorem B ([17]).

Let f be a real convex function defined on an interval $J\mathrm{\subset }\mathrm{R}$. Let $\mathrm{a}\mathrm{=}{\mathrm{\left(}{a}_{\mathrm{1}}\mathrm{,}\mathrm{\dots }\mathrm{,}{a}_{n}\mathrm{\right)}}^{T}\mathrm{\in }{\mathrm{R}}_{\mathrm{+}}^{n}$, $\mathrm{b}\mathrm{=}{\mathrm{\left(}{b}_{\mathrm{1}}\mathrm{,}\mathrm{\dots }\mathrm{,}{b}_{m}\mathrm{\right)}}^{T}\mathrm{\in }{\mathrm{R}}_{\mathrm{+}}^{m}$, $\mathrm{x}\mathrm{=}{\mathrm{\left(}{x}_{\mathrm{1}}\mathrm{,}\mathrm{\dots }\mathrm{,}{x}_{n}\mathrm{\right)}}^{T}\mathrm{\in }{J}^{n}$ and $\mathrm{y}\mathrm{=}{\mathrm{\left(}{y}_{\mathrm{1}}\mathrm{,}\mathrm{\dots }\mathrm{,}{y}_{m}\mathrm{\right)}}^{T}\mathrm{\in }{J}^{m}$.

If

$𝐲=\mathrm{𝐒𝐱}\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }𝐚={𝐒}^{T}𝐛$(1.2)

for some $n\mathrm{×}m$ row stochastic matrix $\mathrm{S}\mathrm{=}\mathrm{\left(}{s}_{i\mathit{}j}\mathrm{\right)}$, then

$\sum _{j=1}^{m}{b}_{j}f\left({y}_{j}\right)\le \sum _{i=1}^{n}{a}_{i}f\left({x}_{i}\right).$(1.3)

If f is concave, then inequality (1.3) is reversed.

Statements (1.2) and (1.3) are referred to as Sherman’s condition and Sherman’s inequality, respectively. Consult [1, 2, 3, 4, 9, 10, 14, 15, 16] for generalizations and applications of Theorem B.

Observe that, when $m=n$, inequality (1.3) can be rewritten as

$〈𝐛,f\left(𝐲\right)〉\le 〈𝐚,f\left(𝐱\right)〉,$(1.4)

where $〈\cdot ,\cdot 〉$ is the standard inner product on ${ℝ}^{n}$, and $f\left(𝐱\right)={\left(f\left({x}_{1}\right),\mathrm{\dots },f\left({x}_{n}\right)\right)}^{T}$ and $f\left(𝐲\right)={\left(f\left({y}_{1}\right),\mathrm{\dots },f\left({y}_{n}\right)\right)}^{T}$. Further, (1.4) can be restated as

$\mathrm{\Psi }\left(𝐛,f\left(𝐲\right)\right)\le \mathrm{\Psi }\left(𝐚,f\left(𝐱\right)\right),$(1.5)

where Ψ is the inner product map on ${ℝ}^{n}$, i.e.,

In the next section, we study inequalities of the form (1.5) for an arbitrary (nonlinear) map Ψ of two variables in ${ℝ}^{n}$.

## 2 Sherman-type inequality for nonlinear maps

In [12], Leon and Proschan gave some interesting inequalities for the Hadamard product map $\mathrm{\Psi }\left(𝐱,𝐲\right)=𝐱\circ 𝐲,$ where $𝐱\circ 𝐲={\left({x}_{1}{y}_{1},\mathrm{\dots },{x}_{n}{y}_{n}\right)}^{T}$ for $𝐱={\left({x}_{1},\mathrm{\dots },{x}_{n}\right)}^{T}\in {ℝ}^{n}$ and $𝐲={\left({y}_{1},\mathrm{\dots },{y}_{n}\right)}^{T}\in {ℝ}^{n}$. They applied a finite reflection group G acting on ${ℝ}^{n}$ with the property that for each $g\in G$ there exist $h,k\in G$ such that

(2.1)

#### Example 2.1.

Let $\mathrm{\Psi }:{ℝ}^{n}×{ℝ}^{n}\to {ℝ}^{n}$ be defined by $\mathrm{\Psi }\left(𝐱,𝐲\right)=𝐱\circ 𝐲$, the Hadamarad product on ${ℝ}^{n}$.

If $G={ℙ}_{n}$ (the permutation group acting on ${ℝ}^{n}$), then

So, (2.1) is met with $g=p$, $h={p}^{-1}={p}^{T}$ and $k=p$.

If $G={ℂ}_{n}$ (the sign changes group acting on ${ℝ}^{n}$), then

Therefore, (2.1) holds with $g=c$, $h=c$ and $k=\mathrm{id}$.

In what follows, we adapt this idea for any map Ψ of two vectorial variables and for the group $G={ℙ}_{n}$ of $n×n$ permutation matrices acting on ${ℝ}^{n}$.

We say that a map $\mathrm{\Psi }:{ℝ}^{n}×{ℝ}^{n}\to {ℝ}^{l}$ admits the Leon–Proschan property for the permutation group ${ℙ}_{n}$ if for each $g\in {ℙ}_{n}$ there exist $h\in {ℙ}_{n}$ and $k\in {ℙ}_{l}$ such that

(2.2)

With $k=\mathrm{id}$, statement (2.2) is called the simplified Leon–Proschan property.

#### Example 2.2.

Take $\mathrm{\Psi }:{ℝ}^{n}×{ℝ}^{n}\to {ℝ}^{l}$ to be given by $\mathrm{\Psi }\left(𝐱,𝐲\right)=\mathrm{\Phi }\left(𝐱+𝐲\right)$, where $\mathrm{\Phi }:{ℝ}^{n}\to {ℝ}^{l}$ is a permutation-invariant function.

We have

Therefore, (2.2) holds with $g=p$, $h={p}^{-1}={p}^{T}$ and $k=\mathrm{id}$.

More generally, let $\mathrm{\Psi }:{ℝ}^{n}×{ℝ}^{n}\to {ℝ}^{l}$ be a permutation-invariant function in the sense

Then it follows that

which is the simplified Leon–Proschan property with $g=p$ and $h={p}^{-1}={p}^{T}$.

Throughout, $\le$ stands for the componentwise order on ${ℝ}^{l}$ with $l\in ℕ$.

For a given function $f:ℝ\to ℝ$, we extend f to ${ℝ}^{n}$ by

(2.3)

In the sequel, for a map $\mathrm{\Psi }:{ℝ}^{n}×{ℝ}^{n}\to {ℝ}^{l}$, we consider the set

In other words,

(2.4)

For example, if Ψ is the inner product map, then

#### Theorem 2.3.

Let $\mathrm{\Psi }\mathrm{:}{\mathrm{R}}^{n}\mathrm{×}{\mathrm{R}}^{n}\mathrm{\to }{\mathrm{R}}^{l}$ be a map. Let $f\mathrm{:}\mathrm{R}\mathrm{\to }\mathrm{R}$ be a convex function. Assume the following conditions:

• (i)

The map Ψ admits the simplified Leon–Proschan property, that is, for each $g\in {ℙ}_{n}$ there exists $h\in {ℙ}_{n}$ such that

• (ii)

For each $𝐱\in {ℝ}^{n}$ the one-variable map $\mathrm{\Psi }\left(𝐱,\cdot \right)$ is convex (with respect to $\le$ ) on ${ℝ}^{n}$ , i.e., for ${𝐲}_{1},\mathrm{\dots },{𝐲}_{m}\in {ℝ}^{n}$, ${t}_{1},\mathrm{\dots },{t}_{m}\ge 0$, ${\sum }_{i=1}^{m}{t}_{i}=1$,

$\mathrm{\Psi }\left(𝐱,\sum _{i=1}^{m}{t}_{i}{𝐲}_{i}\right)\le \sum _{i=1}^{m}{t}_{i}\mathrm{\Psi }\left(𝐱,{𝐲}_{i}\right).$

• (iii)

For each $𝐲\in {ℝ}^{n}$ the one-variable map $\mathrm{\Psi }\left(\cdot ,𝐲\right)$ is concave (with respect to $\le$ ) on ${ℝ}^{n}$ , i.e., for ${𝐱}_{1},\mathrm{\dots },{𝐱}_{m}\in {ℝ}^{n}$, ${t}_{1},\mathrm{\dots },{t}_{m}\ge 0$, ${\sum }_{i=1}^{m}{t}_{i}=1$,

$\mathrm{\Psi }\left(\sum _{i=1}^{m}{t}_{i}{𝐱}_{i},𝐲\right)\ge \sum _{i=1}^{m}{t}_{i}\mathrm{\Psi }\left({𝐱}_{i},𝐲\right).$

Fix any $\mathrm{x}\mathrm{,}\mathrm{y}\mathrm{\in }{\mathrm{R}}^{n}$ and $\mathrm{a}\mathrm{,}\mathrm{b}\mathrm{\in }{\mathrm{R}}^{n}$ with $\mathrm{b}\mathrm{\in }{\mathcal{A}}_{\mathrm{\Psi }}$. If

$𝐲=\sum _{i=1}^{m}{t}_{i}{g}_{i}𝐱\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }𝐚=\sum _{i=1}^{m}{t}_{i}{h}_{i}𝐛$(2.5)

for some ${g}_{i}\mathrm{\in }{\mathrm{P}}_{n}$ and ${h}_{i}\mathrm{\in }{\mathrm{P}}_{n}$ such that $\mathrm{\Psi }\mathit{}\mathrm{\left(}{h}_{i}\mathit{}\mathrm{x}\mathrm{,}\mathrm{y}\mathrm{\right)}\mathrm{=}\mathrm{\Psi }\mathit{}\mathrm{\left(}\mathrm{x}\mathrm{,}{g}_{i}\mathit{}\mathrm{y}\mathrm{\right)}$, $i\mathrm{=}\mathrm{1}\mathrm{,}\mathrm{\dots }\mathrm{,}m$, then the following Sherman-type inequality holds:

$\mathrm{\Psi }\left(𝐛,f\left(𝐲\right)\right)\le \mathrm{\Psi }\left(𝐚,f\left(𝐱\right)\right).$(2.6)

#### Proof.

For any $𝐳\in {ℝ}^{n}$ we get

$\mathrm{\Psi }\left(𝐛,\sum _{i=1}^{m}{t}_{i}{g}_{i}𝐳\right)\le \sum _{i=1}^{m}{t}_{i}\mathrm{\Psi }\left(𝐛,{g}_{i}𝐳\right)=\sum _{i=1}^{m}{t}_{i}\mathrm{\Psi }\left({h}_{i}𝐛,𝐳\right)\le \mathrm{\Psi }\left(\sum _{i=1}^{m}{t}_{i}{h}_{i}𝐛,𝐳\right)=\mathrm{\Psi }\left(𝐚,𝐳\right).$(2.7)

In fact, the first inequality is due to (ii). The first equality follows from (i). The second inequality is a consequence of (iii). And the last equality is valid by (2.5).

By setting $𝐳=f\left(𝐱\right)$, from (2.7) we have

$\mathrm{\Psi }\left(𝐛,\sum _{i=1}^{m}{t}_{i}{g}_{i}f\left(𝐱\right)\right)\le \mathrm{\Psi }\left(𝐚,f\left(𝐱\right)\right).$(2.8)

Because the extension (2.3) of f is convex on ${ℝ}^{n}$ (with respect to $\le$), we find that

$f\left(\sum _{i=1}^{m}{t}_{i}{g}_{i}𝐱\right)\le \sum _{i=1}^{m}{t}_{i}f\left({g}_{i}𝐱\right).$

Hence, by the monotonicity of $\mathrm{\Psi }\left(𝐛,\cdot \right)$ on ${ℝ}^{n}$ (with respect to $\le$) (see (2.4)), we obtain

$\mathrm{\Psi }\left(𝐛,f\left(\sum _{i=1}^{m}{t}_{i}{g}_{i}𝐱\right)\right)\le \mathrm{\Psi }\left(𝐛,\sum _{i=1}^{m}{t}_{i}f\left({g}_{i}𝐱\right)\right).$

It is not hard to check that

$f\left({g}_{i}𝐱\right)={g}_{i}f\left(𝐱\right),i=1,\mathrm{\dots },m\text{.}$

Therefore, the last inequality becomes

$\mathrm{\Psi }\left(𝐛,f\left(\sum _{i=1}^{m}{t}_{i}{g}_{i}𝐱\right)\right)\le \mathrm{\Psi }\left(𝐛,\sum _{i=1}^{m}{t}_{i}{g}_{i}f\left(𝐱\right)\right).$(2.9)

Finally, by combining (2.5), (2.8) and (2.9) we derive a Sherman-type inequality as follows:

$\mathrm{\Psi }\left(𝐛,f\left(𝐲\right)\right)=\mathrm{\Psi }\left(𝐛,f\left(\sum _{i=1}^{m}{t}_{i}{g}_{i}𝐱\right)\right)\le \mathrm{\Psi }\left(𝐛,\sum _{i=1}^{m}{t}_{i}{g}_{i}f\left(𝐱\right)\right)\le \mathrm{\Psi }\left(𝐚,f\left(𝐱\right)\right).$

This completes the proof. ∎

By ${𝕄}_{n}$ we denote the space of all $n×n$ real matrices. Clearly, ${ℙ}_{n}\subset {𝔻}_{n}\subset {𝕄}_{n}$.

#### Corollary 2.4.

Let $f\mathrm{:}\mathrm{R}\mathrm{\to }\mathrm{R}$ be a convex function and let $\mathrm{\Psi }\mathrm{:}{\mathrm{R}}^{n}\mathrm{×}{\mathrm{R}}^{n}\mathrm{\to }{\mathrm{R}}^{l}$ be a map satisfying assumptions (ii) and (iii) of Theorem 2.3. Additionally, let $\theta \mathrm{:}{\mathrm{M}}_{n}\mathrm{\to }{\mathrm{M}}_{n}$ be a linear map such that

• (i’)

for each $g\in {ℙ}_{n}$,

(2.10)

Fix any $\mathrm{x}\mathrm{,}\mathrm{y}\mathrm{\in }{\mathrm{R}}^{n}$ and $\mathrm{a}\mathrm{,}\mathrm{b}\mathrm{\in }{\mathrm{R}}^{n}$ with $\mathrm{b}\mathrm{\in }{\mathcal{A}}_{\mathrm{\Psi }}$. If

$𝐲=\mathrm{𝐒𝐱}\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }𝐚=\theta \left(𝐒\right)𝐛$(2.11)

for some $\mathrm{S}\mathrm{\in }{\mathrm{D}}_{n}$, then inequality (2.6) holds.

#### Proof.

Since $𝐒\in \mathrm{conv}{ℙ}_{n}$, we obtain that $𝐒={\sum }_{i=1}^{m}{t}_{i}{g}_{i}$ for some ${g}_{1},\mathrm{\dots },{g}_{m}\in {ℙ}_{n}$ and ${t}_{1},\mathrm{\dots },{t}_{m}\ge 0$ with ${t}_{1}+\mathrm{\dots }+{t}_{m}=1$. Then (2.11) implies (2.5). So, it is enough to apply Theorem 2.3. ∎

#### Corollary 2.5.

Let $f\mathrm{:}\mathrm{R}\mathrm{\to }\mathrm{R}$ be a convex function and let $\mathrm{\Psi }\mathrm{:}{\mathrm{R}}^{n}\mathrm{×}{\mathrm{R}}^{n}\mathrm{\to }{\mathrm{R}}^{l}$ be a map satisfying assumptions (ii) and (iii) of Theorem 2.3. Additionally, assume that

• (i’)

a map Ψ is permutation-invariant in the sense that for each $g\in {ℙ}_{n}$

Fix any $\mathrm{x}\mathrm{,}\mathrm{y}\mathrm{\in }{\mathrm{R}}^{n}$ and $\mathrm{a}\mathrm{,}\mathrm{b}\mathrm{\in }{\mathrm{R}}^{n}$ with $\mathrm{b}\mathrm{\in }{\mathcal{A}}_{\mathrm{\Psi }}$. If

$𝐲=\mathrm{𝐒𝐱}\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }𝐚={𝐒}^{T}𝐛$(2.12)

for some $\mathrm{S}\mathrm{\in }{\mathrm{D}}_{n}$, then inequality (2.6) holds.

#### Proof.

It follows from (i’) that

However, for $g\in {ℙ}_{n}$ one has ${g}^{-1}={g}^{T}$. So, (2.10) is met with $\theta \left(g\right)={g}^{-1}={g}^{T}$ for $g\in {ℙ}_{n}$. Therefore, the usage of Corollary 2.4 with $\theta ={\left(\cdot \right)}^{T}$ leads us to (2.6) via (2.11) and (2.12), as desired. ∎

## 3 Sherman-type inequalities induced by directional derivative of a Schur-convex function

We remind that for a function $\psi :{ℝ}^{n}\to ℝ$ the directional derivative ${\nabla }_{𝐲}\psi \left(𝐱\right)$ of ψ at the point $𝐱$ in the direction $𝐲$ is given by

${\nabla }_{𝐲}\psi \left(𝐱\right)=\underset{t\to 0}{lim}\frac{\psi \left(𝐱+t𝐲\right)-\psi \left(𝐱\right)}{t}$

(provided the limit there exists).

It is readily seen that if ψ is permutation-invariant, i.e., $\psi \left(p𝐱\right)=\psi \left(𝐱\right)$ for $𝐱\in {ℝ}^{n}$ and $p\in {ℙ}_{n}$, then

(3.1)

Thus the directional derivative of ψ is a permutation-invariant map.

By taking $\mathrm{\Psi }\left(𝐱,𝐲\right)={\nabla }_{𝐲}\psi \left(𝐱\right)$ with $𝐱,𝐲\in {ℝ}^{n}$, we find that

#### Theorem 3.1.

Let $\psi \mathrm{:}{\mathrm{R}}^{n}\mathrm{\to }\mathrm{R}$ be a Schur-convex function. Assume that for any $\mathrm{x}\mathrm{,}\mathrm{y}\mathrm{\in }{\mathrm{R}}^{n}$ there exists the directional derivative ${\mathrm{\nabla }}_{\mathrm{y}}\mathit{}\psi \mathit{}\mathrm{\left(}\mathrm{x}\mathrm{\right)}$ of ψ at the point $\mathrm{x}$ in the direction $\mathrm{y}$. Let $f\mathrm{:}\mathrm{R}\mathrm{\to }\mathrm{R}$ be convex. Assume that assumptions (ii) and (iii) of Theorem 2.3 are satisfied for $\mathrm{\Psi }\mathit{}\mathrm{\left(}\mathrm{x}\mathrm{,}\mathrm{y}\mathrm{\right)}\mathrm{=}{\mathrm{\nabla }}_{\mathrm{y}}\mathit{}\psi \mathit{}\mathrm{\left(}\mathrm{x}\mathrm{\right)}$ with $\mathrm{x}\mathrm{,}\mathrm{y}\mathrm{\in }{\mathrm{R}}^{n}$.

Fix any $\mathrm{x}\mathrm{,}\mathrm{y}\mathrm{\in }{\mathrm{R}}^{n}$ and $\mathrm{a}\mathrm{,}\mathrm{b}\mathrm{\in }{\mathrm{R}}^{n}$ with $\mathrm{b}\mathrm{\in }{\mathcal{A}}_{\mathrm{\Psi }}$. If

$𝐲=\mathrm{𝐒𝐱}\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }𝐚={𝐒}^{T}𝐛$(3.2)

for some $\mathrm{S}\mathrm{\in }{\mathrm{D}}_{n}$, then the following Sherman-type inequality holds:

${\nabla }_{f\left(𝐲\right)}\psi \left(𝐛\right)\le {\nabla }_{f\left(𝐱\right)}\psi \left(𝐚\right).$(3.3)

#### Proof.

Since ψ is Schur-convex, it is permutation-invariant. By virtue of (3.1), the directional derivative of ψ is permutation-invariant. That is, Corollary 2.5 (i’) is fulfilled with $\mathrm{\Psi }\left(𝐱,𝐲\right)={\nabla }_{𝐲}\psi \left(𝐱\right)$ for $𝐱,𝐲\in {ℝ}^{n}$. Simultaneously, (3.2) gives (2.12) which implies (2.6). Thus we get (3.3), completing the proof. ∎

We now consider Theorem 3.1 in the context of ψ with Gâteaux differentiability. That is, we assume that the directional derivative ${\nabla }_{𝐲}\psi \left(𝐱\right)$, viewed as a function of a direction $𝐲$, is linear and continuous on ${ℝ}^{n}$. Then there exists the gradient $\nabla \psi \left(𝐱\right)$ of ψ at the point $𝐱$ such that

where $〈\cdot ,\cdot 〉$ is the standard inner product on ${ℝ}^{n}$.

For the map $\mathrm{\Psi }\left(𝐱,𝐲\right)=〈\nabla \psi \left(𝐱\right),𝐲〉$ with $𝐱,𝐲\in {ℝ}^{n}$, we find that

$=\left\{𝐱\in {ℝ}^{n}:\nabla \psi \left(𝐱\right)\in {ℝ}_{+}^{n}\right\}.$

Thus the condition $𝐛\in {\mathcal{𝒜}}_{\mathrm{\Psi }}$ means that $\nabla \psi \left(𝐛\right)\in {ℝ}_{+}^{n}$.

#### Corollary 3.2.

Let $\psi \mathrm{:}{\mathrm{R}}^{n}\mathrm{\to }\mathrm{R}$ be a Gâteaux differentiable Schur-convex function. Let $f\mathrm{:}\mathrm{R}\mathrm{\to }\mathrm{R}$ be convex. Assume that the assumption (iii) of Theorem 2.3 is satisfied for $\mathrm{\Psi }\mathit{}\mathrm{\left(}\mathrm{x}\mathrm{,}\mathrm{y}\mathrm{\right)}\mathrm{=}\mathrm{〈}\mathrm{\nabla }\mathit{}\psi \mathit{}\mathrm{\left(}\mathrm{x}\mathrm{\right)}\mathrm{,}\mathrm{y}\mathrm{〉}$ with $\mathrm{x}\mathrm{,}\mathrm{y}\mathrm{\in }{\mathrm{R}}^{n}$.

Fix any $\mathrm{x}\mathrm{,}\mathrm{y}\mathrm{\in }{\mathrm{R}}^{n}$ and $\mathrm{a}\mathrm{,}\mathrm{b}\mathrm{\in }{\mathrm{R}}^{n}$ with $\mathrm{\nabla }\mathit{}\psi \mathit{}\mathrm{\left(}\mathrm{b}\mathrm{\right)}\mathrm{\in }{\mathrm{R}}_{\mathrm{+}}^{n}$. If

$𝐲=\mathrm{𝐒𝐱}\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }𝐚={𝐒}^{T}𝐛$

for some $\mathrm{S}\mathrm{\in }{\mathrm{D}}_{n}$, then the following Sherman-type inequality holds:

$〈\nabla \psi \left(𝐛\right),f\left(𝐲\right)〉\le 〈\nabla \psi \left(𝐚\right),f\left(𝐱\right)〉.$(3.4)

#### Proof.

Condition (ii) of Theorem 2.3 holds by the linearity of $\mathrm{\Psi }\left(𝐱,𝐲\right)=〈\nabla \psi \left(𝐱\right),𝐲〉$ with respect to $𝐲$. Therefore, it is sufficient to apply Theorem 3.1. ∎

By putting

(3.5)

which is a Schur-convex function by virtue of its convexity and permutation-invariance, we have

(3.6)

In this case, condition $\nabla \psi \left(𝐛\right)\in {ℝ}_{+}^{n}$ means that $𝐛\in {ℝ}_{+}^{n}$.

It is interesting that under (3.5) and (3.6) inequality (3.4) in Corollary 3.2 holds in the form

$〈𝐛,f\left(𝐲\right)〉\le 〈𝐚,f\left(𝐱\right)〉,$

which is the classical Sherman’s inequality (see Theorem B). Thus Corollary 3.2 is a generalization of Theorem B (whenever $m=n$). Moreover, by setting $𝐚=\left(1,\mathrm{\dots },1\right)\in {ℝ}^{n}$ with a doubly stochastic $𝐒$, we can get Theorem A.

It is not hard to verify for the map $\mathrm{\Psi }\left(𝐱,𝐲\right)={\nabla }_{𝐱}\psi \left(𝐲\right)$ with $𝐱,𝐲\in {ℝ}^{n}$ that

#### Theorem 3.3.

Let $\psi \mathrm{:}{\mathrm{R}}^{n}\mathrm{\to }\mathrm{R}$ be a Schur-convex function. Assume that for any $\mathrm{x}\mathrm{,}\mathrm{y}\mathrm{\in }{\mathrm{R}}^{n}$ there exists the directional derivative ${\mathrm{\nabla }}_{\mathrm{x}}\mathit{}\psi \mathit{}\mathrm{\left(}\mathrm{y}\mathrm{\right)}$ of ψ at the point $\mathrm{x}$ in the direction $\mathrm{y}$. Let $f\mathrm{:}\mathrm{R}\mathrm{\to }\mathrm{R}$ be convex. Assume that assumptions (ii) and (iii) of Theorem 2.3 are satisfied for $\mathrm{\Psi }\mathit{}\mathrm{\left(}\mathrm{x}\mathrm{,}\mathrm{y}\mathrm{\right)}\mathrm{=}{\mathrm{\nabla }}_{\mathrm{x}}\mathit{}\psi \mathit{}\mathrm{\left(}\mathrm{y}\mathrm{\right)}$ with $\mathrm{x}\mathrm{,}\mathrm{y}\mathrm{\in }{\mathrm{R}}^{n}$.

Fix any $\mathrm{x}\mathrm{,}\mathrm{y}\mathrm{\in }{\mathrm{R}}^{n}$ and $\mathrm{a}\mathrm{,}\mathrm{b}\mathrm{\in }{\mathrm{R}}^{n}$ with $\mathrm{b}\mathrm{\in }{\mathcal{A}}_{\mathrm{\Psi }}$. If

$𝐲=\mathrm{𝐒𝐱}\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }𝐚={𝐒}^{T}𝐛$(3.7)

for some $\mathrm{S}\mathrm{\in }{\mathrm{D}}_{n}$, then the following Sherman-type inequality holds:

${\nabla }_{𝐛}\psi \left(f\left(𝐲\right)\right)\le {\nabla }_{𝐚}\psi \left(f\left(𝐱\right)\right).$(3.8)

#### Proof.

Because ψ is Schur-convex, its directional derivative is permutation-invariant (see (3.1)). For this reason, Corollary 2.5 (i’) is satisfied for the map $\mathrm{\Psi }\left(𝐱,𝐲\right)={\nabla }_{𝐱}\psi \left(𝐲\right)$, $𝐱,𝐲\in {ℝ}^{n}$. Furthermore, (3.7) gives (2.12), which implies (2.6). Finally, we get (3.8), as claimed. ∎

For the map $\mathrm{\Psi }\left(𝐱,𝐲\right)=〈\nabla \psi \left(𝐲\right),𝐱〉$ with $𝐱,𝐲\in {ℝ}^{n}$ we have

Condition $𝐛\in {\mathcal{𝒜}}_{\mathrm{\Psi }}$ means that the one-variable map $\mathrm{\Psi }\left(𝐛,\cdot \right)=〈\nabla \psi \left(\cdot \right),𝐛〉$ is nondecreasing on ${ℝ}^{n}$, that is, for $𝐲,𝐳\in {ℝ}^{n}$,

$𝐲\le 𝐳\mathit{ }\text{implies}\mathit{ }〈\nabla \psi \left(𝐲\right),𝐛〉\le 〈\nabla \psi \left(𝐳\right),𝐛〉.$(3.9)

#### Corollary 3.4.

Let $\psi \mathrm{:}{\mathrm{R}}^{n}\mathrm{\to }\mathrm{R}$ be a Gâteaux differentiable Schur-convex function. Let $f\mathrm{:}\mathrm{R}\mathrm{\to }\mathrm{R}$ be convex. Assume that assumptions (ii) and (iii) of Theorem 2.3 are satisfied for $\mathrm{\Psi }\mathit{}\mathrm{\left(}\mathrm{x}\mathrm{,}\mathrm{y}\mathrm{\right)}\mathrm{=}\mathrm{〈}\mathrm{\nabla }\mathit{}\psi \mathit{}\mathrm{\left(}\mathrm{y}\mathrm{\right)}\mathrm{,}\mathrm{x}\mathrm{〉}$ with $\mathrm{x}\mathrm{,}\mathrm{y}\mathrm{\in }{\mathrm{R}}^{n}$.

Fix any $\mathrm{x}\mathrm{,}\mathrm{y}\mathrm{\in }{\mathrm{R}}^{n}$ and $\mathrm{a}\mathrm{,}\mathrm{b}\mathrm{\in }{\mathrm{R}}^{n}$ with $\mathrm{b}\mathrm{\in }{\mathcal{A}}_{\mathrm{\Psi }}$. If

$𝐲=\mathrm{𝐒𝐱}\mathit{ }\text{𝑎𝑛𝑑}\mathit{ }𝐚={𝐒}^{T}𝐛$

for some $\mathrm{S}\mathrm{\in }{\mathrm{D}}_{n}$, then the following Sherman-type inequality holds:

$〈\nabla \psi \left(f\left(𝐲\right)\right),𝐛〉\le 〈\nabla \psi \left(f\left(𝐱\right)\right),𝐚〉.$(3.10)

#### Proof.

It is sufficient to apply Theorem 3.3. ∎

To illustrate the last result, choose

(3.11)

where $𝐞=\left(1,\mathrm{\dots },1\right)\in {ℝ}^{n}$. This is a Schur-convex function.

(3.12)

Here

$\mathrm{\Psi }\left(𝐱,𝐲\right)=〈\nabla \psi \left(𝐲\right),𝐱〉=〈\left(\mathrm{exp}〈𝐲,𝐞〉\right)𝐞,𝐱〉=〈𝐞,𝐱〉\mathrm{exp}〈𝐲,𝐞〉.$

Evidently, this map is convex with respect to $𝐲$ and concave with respect to $𝐱$, which proves the validity of conditions (ii) and (iii) in Theorem 2.3.

In this case, condition $𝐛\in {\mathcal{𝒜}}_{\mathrm{\Psi }}$ means that (3.9) holds in the form

$𝐲\le 𝐳\mathit{ }\text{implies}\mathit{ }〈𝐞,𝐛〉\mathrm{exp}〈𝐲,𝐞〉\le 〈𝐞,𝐛〉\mathrm{exp}〈𝐳,𝐞〉,$

which is true whenever $〈𝐞,𝐛〉\ge 0$.

It follows from (3.11) and (3.12) that inequality (3.10) in Corollary 3.4 holds in the form

$〈𝐞,𝐛〉\mathrm{exp}〈f\left(𝐲\right),𝐞〉\le 〈𝐞,𝐚〉\mathrm{exp}〈f\left(𝐱\right),𝐞〉.$

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Revised: 2018-06-28

Accepted: 2018-07-09

Published Online: 2018-09-21

Published in Print: 2019-03-01

Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 168–175, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496,

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