# A new look at the Hardy-Littlewood-Pólya inequality of majorization

Constantin P. Niculescu  University of Craiova, Department of Mathematics,
A.I. Cuza Street 13, Craiova 200585, ROMANIA<EMAIL_ADDRESS>
###### Abstract.

The Hardy-Littlewood-Pólya inequality of majorization is extended to the
framework of ordered Banach spaces. Several applications illustrating our main
results are also included.

###### Key words and phrases:

$\omega$-convex function, strongly smooth function, majorization theory,
ordered Banach space, isotone operator

###### 2000 Mathematics Subject Classification:

Primary 26B25; Secondary 26D10, 46B40, 47B60, 47H07

Published in _J. Math. Anal. Appl._ 501 (2021), Issue 2, paper 125211\. DOI:
10.1016/j.jmaa.2021.125211

##  1\. Introduction

In their celebrated book on _Inequalities_ , G. H. Hardy, J. E. Littlewood and
G. Pólya [11] have proved an important characterization of convex functions in
terms of a preorder of vectors in $\mathbb{R}^{N}$ called _majorization_.
Given two vectors $\mathbf{x}$ and $\mathbf{y}$ in $\mathbb{R}^{N}$, we say
that $\mathbf{x}$ is _weakly majorized_ by $\mathbf{y}$ $($denoted
$\mathbf{x}\prec_{wHLP}\mathbf{y})$ if their decreasing rearrangements,
respectively $x_{1}^{\downarrow}\geq\cdots\geq x_{N}^{\downarrow}$ and
$y_{1}^{\downarrow}\geq\cdots\geq y_{N}^{\downarrow}$ verify the inequalities

(1.1)
$\sum_{i=1}^{k}x_{i}^{\downarrow}\leq\sum_{i=1}^{k}y_{i}^{\downarrow}\quad\text{for
}k=1,\dots,N;$

we say that $\mathbf{x}$ is _majorized_ by $\mathbf{y}$ $($denoted
$\mathbf{x}\prec_{HLP}\mathbf{y})$ if in addition

(1.2) $\sum_{i=1}^{N}x_{i}^{\downarrow}=\sum_{i=1}^{N}y_{i}^{\downarrow}$

The basic result relating majorization to convexity is the _Hardy-Littlewood-
Pólya inequality_ _of_ _majorization_ :

######  Theorem 1.

$($Hardy-Littlewood-Pólya _[11]_$)$ If $\mathbf{x}\prec_{HLP}\mathbf{y},$ then

(1.3) $\sum_{k=1}^{N}f(x_{k})\leq\sum_{k=1}^{N}f(y_{k})$

for every real-valued continuous convex function $f$ defined on an interval
that contains the components of $\mathbf{x}$ and
$\mathbf{y}\mathrm{.}\vspace{1mm}$ Conversely, if the inequality $($_1.3_ $)$
holds for every real-valued continuous convex function defined on an interval
including the components of $\mathbf{x}$ and $\mathbf{y},$ then
$\mathbf{x}\prec_{HLP}\mathbf{y}.$

The inequality (1.3) still works when $f$ is a nondecreasing convex function
and $\mathbf{x}\prec_{wHLP}\mathbf{y}.$ This important remark, due
independently to Tomić and Weyl, can be derived directly from Theorem 1. See
[15] and [19].

As we briefly noticed in [18], the Hardy-Littlewood-Pólya inequality can be
extended to the framework of ordered Banach spaces alongside an argument that
can be traced back to [14]. The aim of the present paper is to prove much more
general results and to show that they are best possible (that is, no such
theorems exist with less restrictions than ours).

The necessary background on ordered Banach spaces can be covered from [18].
Additional information is available in the classical books of Aliprantis and
Tourky [1] and Meyer-Nieberg [16].

According to Choquet’s theory (see [19] and [22]), the right framework for
developing the majorization theory is that of probability measures. In the
case of the Hardy-Littlewood-Pólya preorder relation $\prec_{HLP}~{}$this can
be done simply by identifying each vector $\mathbf{x}=(x_{1},...,x_{N})$ in
$\mathbb{R}^{N}$ with the discrete probability measure
$\left(1/N\right)\sum_{k=1}^{N}\delta_{x_{k}}$ acting on $\mathbb{R};$ as
usually $\delta_{x_{k}}$ denotes the Dirac measure concentrated at $x_{k}$. We
put

$\frac{1}{N}\sum_{k=1}^{N}\delta_{x_{k}}\prec_{HLP}\frac{1}{N}\sum_{k=1}^{N}\delta_{y_{k}}$

with the same understanding as $\mathbf{x}\prec_{HLP}\mathbf{y}.$ Under these
terms, the Hardy-Littlewood-Pólya inequality of majorization can be rephrased
as

$\mu\prec_{HLP}\nu\text{ if and only if
}\int_{I}f\mathrm{d}\mu\leq\int_{I}f\mathrm{d}\nu$

for every real-valued continuous and convex function $f$ whose domain of
definition is an interval $I$ that includes the supports of the discrete
probability measures$\ \mu$ and $\nu\mathrm{.}\vspace{1mm}$

Since in an ordered Banach space not every string of elements admits a
decreasing rearrangement, in this paper we will concentrate to the case of
pairs of discrete probability measures of which at least one of them is
supported by a monotone string of points. The case where the support of the
left measure consists of a decreasing string is defined as follows.

######  Definition 1.

Suppose that $\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{x}_{k}}$ and
$\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{y}_{k}}$ are two discrete Borel
probability measures that act on the ordered Banach space $E$. We say that
$\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{x}_{k}}$ is weakly
$L^{\downarrow}$-majorized by
$\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{y}_{k}}$ _(_ denoted
$\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{x}_{k}}\prec_{wL^{\downarrow}}\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{y}_{k}}$_)_
if the left hand measure is supported by a decreasing string of points

(1.4) $\mathbf{x}_{1}\geq\cdots\geq\mathbf{x}_{N}$

and

(1.5)
$\sum_{k=1}^{n}\lambda_{k}\mathbf{x}_{k}\leq\sum_{k=1}^{n}\lambda_{k}\mathbf{y}_{k}\quad\text{for
all }n\in\\{1,\dots,N\\}.$

We say that $\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{x}_{k}}$ is
$L^{\downarrow}$-majorized by
$\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{y}_{k}}$ $($denoted
$\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{x}_{k}}\prec_{L^{\downarrow}}\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{y}_{k}})$
if in addition

(1.6)
$\sum_{k=1}^{N}\lambda_{k}\mathbf{x}_{k}=\sum_{k=1}^{N}\lambda_{k}\mathbf{y}_{k}.$

Notice that the context of Definition 1 makes necessary that all weights
$\lambda_{1},...,\lambda_{N}$ belong to $(0,1]$ and
$\sum_{k=1}^{N}\lambda_{k}=1.$

The three conditions (1.4), (1.5) and (1.6) imply
$\mathbf{y}_{N}\leq\mathbf{x}_{N}\leq\mathbf{x}_{1}\leq$ $\mathbf{y}_{1}$ but
not the ordering $\mathbf{y}_{1}\geq\cdots\geq\mathbf{y}_{N}.$ For example,
when $N=3,$ one may choose

$\lambda_{1}=\lambda_{2}=\lambda_{3}=1/3,\text{
}\mathbf{x}_{1}=\mathbf{x}_{2}=\mathbf{x}_{3}=\mathbf{x}$

and

$\mathbf{y}_{1}=\mathbf{x},\text{ }\mathbf{y}_{2}=\mathbf{x+z,}\text{
}\mathbf{y}_{3}=\mathbf{x-z}$

where $\mathbf{z}$ is any positive element.

Under these circumstances it is natural to introduce the following companion
to Definition 1, involving the ascending strings of elements as support for
the right hand measure.

######  Definition 2.

The relation of _weak_ $R^{\uparrow}$-_majorization_ ,

$\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{x}_{k}}\prec_{wR^{\uparrow}}\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{y}_{k}},$

between two discrete Borel probability measures means the fulfillment of the
condition $($_1.5_ $)$ under the presence of the ordering

(1.7) $\mathbf{y}_{1}\leq\cdots\leq\mathbf{y}_{N};$

assuming in addition the condition $($_1.6_ $)$_,_ we say that
$\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{x}_{k}}$ is $R^{\uparrow}$-majorized
by $\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{y}_{k}}$ $($denoted
$\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{x}_{k}}\prec_{R^{\uparrow}}\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{y}_{k}})$.

When every element of $E$ is the difference of two positive elements, the weak
majorization relations $\prec_{wL^{\downarrow}}$and $\prec_{wR^{\uparrow}}$
can be augmented so to obtain majorization relations.

The corresponding extensions of the Hardy-Littlewood-Pólya inequality of
majorization for
$\prec_{wL^{\downarrow}},\prec_{L^{\downarrow}},\prec_{wR^{\uparrow}}$and
$\prec_{R^{\uparrow}}$make the objective of two theorems in Section 4. The
first one, Theorem 4, deals with Gâteaux differentiable convex functions whose
differentials are isotone (that is, order preserving). The second one, Theorem
6, extends the conclusion of the preceding theorem to a nondifferentiable
framework involving convex functions defined on open $N$-dimensional box of
$\mathbb{R}^{N}$ which verify a condition of monotonicity à la Popoviciu [23]
(called by us $2$-box monotonicity). This is done via the approximation
Theorem 3, whose proof makes the objective of Section 3.

Unlike the case of functions of one real variable, when the isotonicity of the
differential is automatic, for several variables, while this is not
necessarily true in the case of a differentiable convex function of a vector
variable. See Remark 3. Remarkably, the isotonicity of the differential is not
only a sufficient condition for the validity of our differentiable
generalization of the Hardy-Littlewood-Pólya theorem, but also a necessary
one. See Remark 5.

For the convenience of the reader we review in Section 2 some very basic
results concerning the various classes of convex or convex like functions and
the gradient inequalities they generate. This section also includes several
significant examples of differentiable convex functions with isotone
differentials.

Not entirely surprising, the inequalities of majorization may occur outside
the class of convex functions. This is illustrated by Theorem 6, that deals
with the case of strongly smooth functions.

Applications of the above majorization theorems include the isotonicity of
Jensen’s gap, a general form of the parallelogram law and also the extension
of several classical inequalities to the setting of convex functions of a
vector variable. They are all presented in Section 5.

##  2\. Classes of convex functions

In what follows $E$ and $F$ are two ordered Banach spaces and
$\Phi:C\rightarrow F$ is a function defined on a convex subset of $E.$

The function $\Phi$ is said to be a _perturbed_ _convex function_ _with
modulus_ $\omega:\mathbb{[}0,\infty\mathbb{)}\rightarrow F$ (abbreviated,
$\omega$-_convex function_) if it verifies an estimate of the form

(2.1)
$\Phi((1-\lambda)\mathbf{x}+\lambda\mathbf{y})\leq(1-\lambda)\Phi(\mathbf{x})+\lambda\Phi(\mathbf{y})-\lambda(1-\lambda)\omega\left(\left\|\mathbf{x}-\mathbf{y}\right\|\right),\text{\quad}$

for all $\mathbf{x},\mathbf{y}$ in $C$ and $\lambda\in(0,1).$ The usual convex
functions represent the particular case when $\omega$ is identically 0. Every
$\omega$-convex function associated to a modulus __ $\omega\geq 0$ is
necessarily convex. When $\omega$ is allowed to take negative values, there
are $\omega$-convex functions which are not convex. See the case of semiconvex
functions, described below.

The $\omega$-convex functions whose moduli $\omega$ are strictly positive
except at the origin (where $\omega(0)=0)$ are usually called _uniformly
convex_.__ In their case the inequality (2.1) is strict whenever
$\mathbf{x}\neq\mathbf{y}$ and $\lambda\in(0,1).$__ More information on this
important class of convex functions is available in [3], [5], [6], [26] and
[27].

By changing $\Phi$ to $-\Phi$ one obtain the notions of $\omega$-_concave
function_ and _uniformly concave function._

######  Remark 1.

Much of the study of perturbed convex functions with vector values can be
reduced to that of real-valued functions. Indeed, in any ordered Banach space
$F$ with generating cone, any inequality of the form$\ u\leq v$ is equivalent
to $w^{\ast}(u)\leq w^{\ast}(v)$ for all $w^{\ast}\in F_{+}^{\ast}.$ See
_[18]_, Lemma $1$ $(c)$. As a consequence, a function $\Phi:C\rightarrow F$ is
$\omega$-convex if and only if $w^{\ast}\circ\Phi$ is
$(w^{\ast}\circ\omega)$-convex whenever $w^{\ast}\in F_{+}^{\ast}.$

There are several variants of convexity that play a prominent role in convex
optimization, calculus of variations, isoperimetric inequalities,
Monge–Kantorovich theory of transport etc. Some of them are mentioned in what
follows.

A real-valued function $\Phi$ defined on a convex subset $C$ of
$\mathbb{R}^{N}$ is called $\alpha$-_strongly convex_ functions (that is,
strongly convex with parameter $\alpha>0)$ if $\Phi-$
$(\alpha/2)\left\|\cdot\right\|^{2}$ is convex. The function $\Phi$ is called
$\beta$-_semiconvex_ (that is, semiconvex with parameter $\beta>0)$ if it
becomes convex after the addition of $(\beta/2)\left\|\cdot\right\|^{2}.$
Equivalently, these are the functions that verify respectively estimates of
the form

(2.2)
$\Phi((1-\lambda)\mathbf{x}+\lambda\mathbf{y})\leq(1-\lambda)\Phi(\mathbf{x})+\lambda\Phi(\mathbf{y})-\frac{1}{2}\lambda(1-\lambda)\alpha\left\|\mathbf{x}-\mathbf{y}\right\|^{2}$

and

(2.3)
$\Phi((1-\lambda)\mathbf{x}+\lambda\mathbf{y})\leq(1-\lambda)\Phi(\mathbf{x})+\lambda\Phi(\mathbf{y})+\frac{1}{2}\lambda(1-\lambda)\beta\left\|\mathbf{x}-\mathbf{y}\right\|^{2},\text{\quad}$

for all $\mathbf{x},\mathbf{y}$ in $C$ and $\lambda\in(0,1)$. By changing
$\Phi$ to $-\Phi$ one obtain the notions of $\alpha$-_strong concavity_ and
$\beta$-_semiconcavity_.

Under the presence of Gâteaux differentiability, each of the above classes of
functions generates specific gradient inequalities that play a prominent role
in our generalization of the Hardy-Littlewood-Pólya inequality of
majorization.

######  Lemma 1.

Suppose that $C$ is an open convex subset of $E$ and $\Phi:C\rightarrow F$ is
a function both Gâteaux differentiable and $\omega$-convex. Then __

(2.4)
$\Phi(\mathbf{x})-\Phi(\mathbf{a})\geq\Phi^{\prime}(\mathbf{a})(\mathbf{x}-\mathbf{a})+\omega\left(\left\|\mathbf{x}-\mathbf{a}\right\|\right)$

for all points $\mathbf{a}\in C$ and $\mathbf{x}\in C$.

As is well known, if $C$ is an open convex subset of $\mathbb{R}^{N},$ then a
twice continuously differentiable function $\Phi:C\rightarrow\mathbb{R}$ is
$\alpha$-strongly convex (respectively $\beta$-semiconvex) if and only if its
Hessian matrix verifies the inequality $\nabla^{2}\Phi\geq\alpha I$
(respectively $\nabla^{2}\Phi\geq-\beta I).$ However, valuable
characterizations are possible with less smoothness.

A continuously differentiable function $\Phi:C\rightarrow\mathbb{R}$ defined
on an open convex subset of $\mathbb{R}^{N}$ is said to be $\sigma$-_strongly_
_smooth_ if its gradient is $\sigma$-Lipschitz, that is,

$\left\|\nabla\Phi(\mathbf{x})-\nabla\Phi(\mathbf{y})\right\|\leq\sigma\left\|\mathbf{x}-\mathbf{y}\right\|\text{\quad
for all }\mathbf{x},\mathbf{y}\in C.$

Notice that every $\sigma$-strongly smooth function $\Phi$ verifies the
following variant of the gradient inequality:

(2.5)
$\Phi(\mathbf{y})-\Phi(\mathbf{x})\leq\Phi^{\prime}(\mathbf{x})(\mathbf{y}-\mathbf{x})+\frac{1}{2}\sigma\left\|\mathbf{y}-\mathbf{x}\right\|^{2}$

for all $\mathbf{x},\mathbf{y}$ in $C.$ See [8], Lemma 3.4, p. 267.

######  Lemma 2.

If $\Phi$ is simultaneously convex and $\sigma$-strongly smooth, then

$\frac{1}{2\sigma}\left\|\Phi^{\prime}(\mathbf{y})-\Phi^{\prime}(\mathbf{x})\right\|\leq\Phi(\mathbf{y})-\Phi(\mathbf{x})-\Phi^{\prime}(\mathbf{x})(\mathbf{y}-\mathbf{x})\leq\frac{1}{2}\sigma\left\|\mathbf{y}-\mathbf{x}\right\|^{2}.$

###### Proof.

The left-hand side inequality is just Lemma 3.5 in [8], p. 268, while the
right-hand side inequality is a restatement of the inequality (2.5). ∎

An important source of strongly smooth convex functions is offered by the
following result:

######  Lemma 3.

If $\Phi$ is an $\alpha$-strongly convex function, then its Legendre-Fenchel
conjugate

$\Phi^{\ast}(\mathbf{x}^{\ast})=\sup\left\\{\mathbf{x}^{\ast}(\mathbf{x})-\Phi(\mathbf{x}):\mathbf{x}\in
C\right\\}$

is an $(1/\alpha)$-strongly smooth function and also a convex function. In
particular, $\Phi^{\ast}$ is defined and differentiable on the whole dual
space $E^{\ast}.$

For details, see [25], Lemma 15, p. 126. The converse also works. See [13],
Theorem 6.

The connection between $\sigma$-strong smoothness and semiconvexity is
outlined by the following theorem.

######  Theorem 2.

$(a)$ Suppose that $C$ is an open convex subset of $\mathbb{R}^{N}.$ If
$\Phi:C\rightarrow\mathbb{R}$ is a $\sigma$-strongly smooth function, then
$\Phi+\frac{\sigma}{2}\left\|\cdot\right\|^{2}$ is convex and
$\Phi-\frac{\sigma}{2}\left\|\cdot\right\|^{2}$ is concave.

$(b)$ Conversely, if $\Phi:C\rightarrow\mathbb{R}$ is a function
simultaneously semiconvex and semiconcave with parameter $\sigma>0,$ then
$\Phi$ is $\sigma$-strongly smooth.

The details are available in the book of Cannarsa and Sinestrari [9]; the
assertion $(a)$ follows from Proposition 2.1.2, p. 30, while the assertion
$(b)$ is motivated by Corollary 3.3.8, p. 61.

As was noticed by Amann [2], Proposition 3.2, p. 184, the Gâteaux
differentiability offers a convenient way to recognize the property of
isotonicity of functions acting on ordered Banach spaces: the positivity of
the differential. In the case of convex functions his result can be stated as
follows:

######  Lemma 4.

Suppose that $E$ and $F$ are two ordered Banach space, $C$ is a convex subset
of $E$ with nonempty interior $\operatorname{int}C$ and $\Phi:C\rightarrow F$
is a __ convex function, continuous on $C$ and Gâteaux differentiable on
$\operatorname{int}C.$ Then $\Phi$ is isotone on $C$ if and only if
$\Phi^{\prime}(\mathbf{a})\geq 0$ for all $\mathbf{a}\in\operatorname{int}C.$

###### Proof.

The ”only if” part follows immediately from the definition of the Gâteaux
derivative. For the other implication, notice that the gradient inequality
mentioned by Lemma $2$ shows that $\Phi$ is isotone on $\operatorname{int}C$
if $\Phi^{\prime}(\mathbf{a})\geq 0$ for all
$\mathbf{a}\in\operatorname{int}C$. As concerns the isotonicity on $C,$ that
follows by an approximation argument. Suppose that $\mathbf{x},\mathbf{y}\in
C$ and $\mathbf{x}\leq\mathbf{y}$. For $\mathbf{x}_{0}\in\operatorname*{int}C$
arbitrarily fixed and $t\in[0,1),$ both elements
$\mathbf{u}_{t}=\mathbf{x}_{0}+t(\mathbf{x}-\mathbf{x}_{0})$ and
$\mathbf{v}_{t}=\mathbf{x}_{0}+t(\mathbf{y}-\mathbf{x}_{0})$ belong to
$\operatorname*{int}C$ and $\mathbf{u}_{t}\leq\mathbf{v}_{t}.$ Moreover,
$\mathbf{u}_{t}\rightarrow\mathbf{x}$ and
$\mathbf{v}_{t}\rightarrow\mathbf{y}$ as $t\rightarrow 1.$ Passing to the
limit in the inequality $\Phi(\mathbf{u}_{t})\leq\Phi(\mathbf{v}_{t})$ we
conclude that $\Phi(\mathbf{x})\leq\Phi(\mathbf{y}).$ ∎

######  Remark 2.

If the ordered Banach space $E$ has finite dimension, then the statement of
Lemma _4_ remains valid by replacing the interior of $C$ by the relative
interior of $C$. See _[19]_, Exercise $6$, p. $81$.

A key ingredient in our extension of the Hardy-Littlewood-Pólya inequality is
the isotonicity of the differentials of the functions involved. Unlike the
case of differentiable convex functions of one variable, the isotonicity of
the differential is not mandatory for the differentiable convex functions of
several variables.

######  Remark 3.

$($A difference between the differentiable convex functions of one real
variable and those of several variables$)$ The twice continuously
differentiable function

$\Phi(x,y)=-2\left(xy\right)^{1/2},\quad\left(x,y\right)\in\mathbb{R}_{++}^{2},$

is convex due to the fact that its Hessian,

$H=\frac{1}{2}\left(\begin{array}[c]{cc}x^{-3/2}y^{1/2}&-x^{-1/2}y^{-1/2}\\\
-x^{-1/2}y^{-1/2}&x^{1/2}y^{-3/2}\end{array}\right),$

is a positive semidefinite matrix. However, unlike the case of convex
functions of one real variable, the differential of $\Phi$,

$d\Phi:\mathbb{R}_{++}^{2}\rightarrow\mathbb{R}^{2},\text{\quad}d\Phi(x,y)=-(x^{-1/2}y^{1/2},x^{1/2}y^{-1/2}),$

is not isotone. Indeed, at the points $\left(1,1\right)<(2,1)$ in
$\mathbb{R}_{++}^{2}$ we have

$d\Phi(1,1)=-\left(1,1\right)\text{ and }d\Phi(2,1)=-(1/\sqrt{2},\sqrt{2})$

and these values are not comparable.

On the other hand, a simple example of nonconvex differentiable function whose
differential is isotone is provided by the function

$H(x,y)=(2x-1)(2y-1),\quad\left(x,y\right)\in\mathbb{R}^{2}.$

Using the aforementioned result of Amann, one can easily prove the following
criterion of isotonicity of the differentials.

######  Lemma 5.

Suppose that $C$ is an open convex subset of the Banach lattice
$\mathbb{R}^{N}$ and $\Phi:C\rightarrow\mathbb{R}$ is a continuous function
which is twice Gâteaux differentiable. Then $\Phi^{\prime}$ is isotone on $C$
if $($and only if$)$ all partial derivatives of second order of $\Phi$ are
nonnegative.

When $\Phi$ is also convex, the isotonicity of $\Phi^{\prime}$ is equivalent
to the condition that all mixed derivatives $\frac{\partial^{2}\Phi}{\partial
x_{i}\partial x_{j}}$ are nonnegative.

In the light of Lemma 5, the example exhibited in Remark 3, shows that the
property of positive definiteness of the Hessian matrix does not necessarily
imply its positivity as a linear map from $\mathbb{R}^{2}$ to
$\mathbb{R}^{2}$.

Several examples of differentiable functions which are isotone and/or admit
isotone differentials are presented in the Appendix of this paper.

##  3\. An approximation result

One can characterize the isotonicity of the differential of a convex function
by using the concept of $2$-box monotonicity, first noticed by Popoviciu [23]
in the case when $N=2.$ See also [10], where the $2$-box monotonicity is
described in its relationship with another concept due to Popoviciu, $2$-box
convexity. The natural domains of such functions are the open $N$-dimensional
boxes, that is, the products $\prod\nolimits_{k=1}^{N}(a_{k},b_{k})$ of $N$
open intervals.

######  Definition 3.

A real-valued function $\Phi$ defined on an open and solid subset $C$ of the
Banach lattice $\mathbb{R}^{N}$ $(N\geq 2)$ is $2$-box monotone if the
increment of $\Phi$ over every nondegenerate $2$-dimensional box

$B_{ij}=\\{u_{1}\\}\times\cdots\times[v_{i},w_{i}]\times\cdots\times\left\\{u_{k}\right\\}\times\cdots\times[v_{j},w_{j}]\times\cdots\times\left\\{u_{N}\right\\},\quad
1\leq i<j\leq N,$

included in $C$ and parallel to one of the planes of coordinates is
nonnegative, that is,

$\Delta(\Phi;B_{ij})=\Phi(u_{1},...,v_{i},...,v_{j},...,u_{N})-\Phi(u_{1},...,v_{i},...,w_{j},...,u_{N})\\\
-\Phi(u_{1},...,w_{i},...,v_{j},...,u_{N})+\Phi(u_{1},...,w_{i},...,w_{j},...,u_{N})\geq
0.$

The property of isotonicity of the differential of a convex function (of two
or more variables) is equivalent to the property of 2-box monotonicity for the
given function. When $\Phi$ is twice continuously differentiable, this follows
directly from Lemma 5. Indeed, for $i<j,$

$\int\nolimits_{v_{i}}^{w_{i}}\int\nolimits_{v_{j}}^{w_{j}}\frac{\partial^{2}\Phi}{\partial
x_{i}\partial
x_{j}}(u_{1},...,x_{i},...,x_{j},...,u_{N})\mathrm{d}x_{i}\mathrm{d}x_{j}\\\
=\Phi(u_{1},...,v_{i},...,v_{j},...,u_{N})-\Phi(u_{1},...,v_{i},...,w_{j},...,u_{N})\\\
-\Phi(u_{1},...,w_{i},...,v_{j},...,u_{N})+\Phi(u_{1},...,w_{i},...,w_{j},...,u_{N}).$

Remarkably, the continuous differentiability of $\Phi$ suffices as well.

######  Lemma 6.

Suppose that $C$ is an open box of the Banach lattice $\mathbb{R}^{N}$ and
$\Phi:C\rightarrow\mathbb{R}$ is a continuously differentiable convex
function. Then $\Phi^{\prime}$ is isotone on $C$ if $($and only if$)$ $\Phi$
is $2$-box monotone.

###### Proof.

The fact that $\Phi^{\prime}$ is isotone is equivalent to

(3.1) $\frac{\partial\Phi}{\partial
x_{k}}(u_{1},...,u_{N})\leq\frac{\partial\Phi}{\partial
x_{k}}(v_{1},...,v_{N})$

for all indices $k\in\\{1,...,N\\}$ and all pairs of points
$\mathbf{u}=(u_{1},...,u_{N})\leq\mathbf{v=}(v_{1},...,v_{N})$ in $C.$

Since $\Phi$ is differentiable and convex in each variable, we have

$\frac{\partial\Phi}{\partial
x_{k}}(u_{1},...,u_{k-1},x_{k},u_{k+1},...,u_{N})\leq\frac{\partial\Phi}{\partial
x_{k}}(u_{1},...,u_{k-1},y_{k},u_{k+1},...,u_{N})$

whenever
$(u_{1},...,u_{k-1},x_{k},u_{k+1},...,u_{N})\leq(u_{1},...,u_{k-1},y_{k},u_{k+1},...,u_{N})$
in $C.$

Using the identity

$\displaystyle\int_{x_{j}}^{y_{j}}\left(\frac{\partial\Phi}{\partial
x_{j}}(u_{1},...,y_{i},...,t,...,u_{N})-\frac{\partial\Phi}{\partial
x_{j}}(u_{1},...,x_{i},...,t,...,u_{N})\right)\mathrm{d}t$
$\displaystyle=\Phi(x_{1},...,y_{i},...,y_{j},...,x_{N})-\Phi(x_{1},...,y_{i},...,x_{j},...,x_{N})$
$\displaystyle-\Phi(x_{1},...,x_{i},...,y_{j},...,x_{N})+\Phi(x_{1},...,x_{i},...,x_{j},...,x_{N}),$

which works for every nondegenerate $2$-dimensional box

$\\{u_{1}\\}\times\cdots\times[x_{i},y_{i}]\times\cdots\times\left\\{u_{k}\right\\}\times\cdots\times[x_{j},y_{j}]\times\cdots\times\left\\{u_{N}\right\\}$

included in $C,$ we can easily infer that the $2$-box monotonicity is
equivalent with the isotonicity of each partial derivative
$\frac{\partial\Phi}{\partial x_{k}}$ in each variable distinct from the
$k$th, when the others are kept fixed. By mathematical induction one can put
together all these facts to obtain the inequalities (3.1). ∎

The reader can verify easily that the following two nondifferentiable
functions,

$\min\left\\{x_{1},x_{2}\right\\}\text{ and
}\max\\{x_{1}+x_{2}-1,0\\},\text{\quad}x_{1},x_{2}\in[0,1],$

are 2-box monotone. They are known in the theory of copulas as the _Fréchet-
Hoeffding bounds_. See [17]. The second function is convex but the first one
is not convex. This fact combined with with Remark 3 shows that the notions of
convexity and 2-box monotonicity are independent in dimension $N\geq 2.$

The analogue of $2$-box monotonicity for functions $f$ defined on an interval
$[a,b]$ is the property of _equal increasing increments_ (which is equivalent
to convexity):

$f(x+z)-f(x)\leq f(y+z)-f(y)$

whenever $x\leq y,$ $z>0$ and $x,y,y+z\in[a,b].$ See [19], Remark 1.4.1, p. 25
and Corollary 1.4.6, p. 29. An immediate consequence of this property is the
fact that every function of the form

$\Phi(\mathbf{x})=f\left(\langle\mathbf{x},\mathbf{v}\rangle\right),\quad\mathbf{x}\in\mathbb{R}^{N},$

associated to a convex function $f:\mathbb{R\rightarrow R}$ and a vector
$\mathbf{w}\in\mathbb{R}_{+}^{N}$ is $2$-box convex.

The increment of the log-sum-exp function over the box

$[0,1]\times[0,1]\times\\{0\\}\times\cdots\times\\{0\\}$

equals

$\log N-2\log(e+N-1)+\log\left(2e+N-2\right)<0$

so that, this function is not $2$-box monotone. According to Lemma 6, the
differential of the log-sum-exp function is not isotone.

The usefulness of the concept of 2-box monotonicity is made clear by the
following approximation result.

######  Theorem 3.

Suppose that $\Phi$ is a $2$-box monotone convex function defined on an open
box $C$ included in $\mathbb{R}^{N}.$ Then on every compact box $K\subset C,$
$\Phi$ is the uniform limit of a sequence of infinitely differentiable
strongly convex functions with isotone differentials.

When $C\subset\mathbb{R}_{++}^{N}$ and the function $\Phi$ is also isotone,
then the approximants $\Phi_{n}$ can be chosen to be isotone.

###### Proof.

We use the usual convolution based smooth approximation. Let $\varepsilon>0$
be arbitrarily fixed. Then the function
$\Psi=\Phi+\varepsilon\left\|\cdot\right\|^{2}$ is $2$-box monotone and
$\varepsilon$-strongly convex. Besides,

$\left\|\Psi(\mathbf{x})-\Phi(\mathbf{x})\right\|\leq\varepsilon\sup\left\\{\left\|\mathbf{x}\right\|^{2}:\mathbf{x}\in
K\right\\}\text{\quad for all }\mathbf{x}\in K.$

According to the method of mollifiers, the convolution

$(\Psi\ast\varphi)(\mathbf{x})=\int_{\mathbb{R}^{N}}\Psi(\mathbf{x}-\mathbf{y})\varphi(\mathbf{y})\mathrm{d}\mathbf{y,}$

of $\Psi$ with any infinitely differentiable function
$\varphi:\mathbb{R}^{N}\rightarrow[0,\infty)$ such that $\varphi=0$ on
$\mathbb{R}^{N}\backslash K$ and
$\int_{\mathbb{R}^{N}}\varphi(\mathbf{y})\mathrm{d}\mathbf{y}=1,$ is an
infinitely differentiable function that provides a regularization of $\Psi$
since $\Psi\ast\varphi\rightarrow\Psi$ uniformly on $K$ as the support of
$\varphi$ shrinks to $\left\\{0\right\\}.$ An easy computation shows that
$\Psi\ast\varphi$ is also a $2$-box monotone and $\varepsilon$-strongly convex
function. Indeed, with the notation in Definition 3, we have

$\Delta(\Psi\ast\varphi;B_{ij})=\int_{\mathbb{R}^{N}}\Delta(\Psi(\mathbf{x}-\mathbf{y});B_{ij})\varphi(\mathbf{y})\mathrm{d}\mathbf{y\geq
0}$

and

$(\Psi\ast\varphi)((1-\lambda)\mathbf{u+\lambda
v})=\int_{\mathbb{R}^{N}}\Psi((1-\lambda)\left(\mathbf{u-\mathbf{y}}\right)\mathbf{+\lambda}\left(\mathbf{v}-\mathbf{y}\right))\varphi(\mathbf{y})\mathrm{d}\mathbf{y}\\\
\leq\int_{\mathbb{R}^{N}}\left[\left(1-\lambda\right)\Psi\left(\mathbf{u-\mathbf{y}}\right)\mathbf{+}\lambda\Psi\left(\mathbf{v}-\mathbf{y}\right))-\lambda(1-\lambda)\left\|\mathbf{u}-\mathbf{v}\right\|^{2}\right]\varphi(\mathbf{y})\mathrm{d}\mathbf{y}\\\
=\left(1-\lambda\right)\left(\Psi\ast\varphi\right)(\mathbf{u})+\lambda\left(\Psi\ast\varphi\right)(\mathbf{v})-\varepsilon\lambda(1-\lambda)\left\|\mathbf{u}-\mathbf{v}\right\|^{2}.$

Then the conclusion of Theorem 3 follows from Lemma 6. ∎

##  4\. The majorization inequality in the context of ordered Banach spaces

We start with the case of differentiable convex functions.

######  Theorem 4.

Suppose that $\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{x}_{k}}$ and
$\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{y}_{k}}$ are two discrete
probability measures whose supports are included in an open convex subset $C$
of the ordered Banach space $E$. If
$\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{x}_{k}}\prec_{L^{\downarrow}}\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{y}_{k}},$
then

(4.1)
$\sum_{k=1}^{N}\lambda_{k}\Phi(\mathbf{y}_{k})\geq\sum_{k=1}^{N}\lambda_{k}\Phi(\mathbf{x}_{k})+\sum_{k=1}^{N}\lambda_{k}\omega(\left\|\mathbf{x}_{k}-\mathbf{y}_{k}\right\|)$

for every Gâteaux differentiable $\omega$-convex function $\Phi:C\rightarrow
F$ whose differential is isotone, while if
$\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{x}_{k}}\prec_{R^{\uparrow}}\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{y}_{k}},$
the inequality $(\ref{Cons1})$ works in the reversed sense.

If
$\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{x}_{k}}\prec_{wL^{\downarrow}}\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{y}_{k}},$
then

(4.2)
$\sum_{k=1}^{n}\lambda_{k}\Phi(\mathbf{y}_{k})\geq\sum_{k=1}^{n}\lambda_{k}\Phi(\mathbf{x}_{k})+\sum_{k=1}^{n}\lambda_{k}\omega(\left\|\mathbf{x}_{k}-\mathbf{y}_{k}\right\|)\text{\quad
for }n\in\\{1,...,N\\}$

whenever $\Phi:C\rightarrow F$ is an isotone and Gâteaux differentiable
$\omega$-convex function whose differential is isotone. Under the same
hypotheses on $\Phi,$ the inequality $(\ref{Cons2})$ works in the reverse way
when
$\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{x}_{k}}\prec_{wR^{\uparrow}}\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{y}_{k}}.$

###### Proof.

According to the gradient inequality (2.4),

$\sum_{k=1}^{N}\lambda_{k}\Phi(\mathbf{y}_{k})-\sum_{k=1}^{N}\lambda_{k}\Phi(\mathbf{x}_{k})=\sum_{k=1}^{N}\lambda_{k}\left(\Phi(\mathbf{y}_{k})-\Phi(\mathbf{x}_{k})\right)\\\
\geq\sum_{k=1}^{N}\Phi^{\prime}(\mathbf{x}_{k})(\lambda_{k}\mathbf{y}_{k}-\lambda_{k}\mathbf{x}_{k})+\sum_{k=1}^{N}\lambda_{k}\omega\left(\left\|\mathbf{x}_{k}-\mathbf{y}_{k}\right\|\right),$

whence, by using Abel’s trick of interchanging the order of summation ([19],
Theorem 1.9.5, p. 57), one obtains

$D=\sum_{k=1}^{N}\lambda_{k}\Phi(\mathbf{y}_{k})-\sum_{k=1}^{N}\lambda_{k}\Phi(\mathbf{x}_{k})-\sum_{k=1}^{N}\lambda_{k}\omega\left(\left\|\mathbf{x}_{k}-\mathbf{y}_{k}\right\|\right)\\\
\geq\Phi^{\prime}(\mathbf{x}_{1})(\lambda_{1}\mathbf{y}_{1}-\lambda_{1}\mathbf{x}_{1})+\sum_{m=2}^{N}\Phi^{\prime}(\mathbf{x}_{m})\Bigl{[}\sum_{k=1}^{m}(\lambda_{k}\mathbf{y}_{k}-\lambda_{k}\mathbf{x}_{k})-\sum_{k=1}^{m-1}(\lambda_{k}\mathbf{y}_{k}-\lambda_{k}\mathbf{x}_{k})\Bigr{]}\\\
=\sum_{m=1}^{N-1}\Bigl{[}(\Phi^{\prime}(\mathbf{x}_{m})-\Phi^{\prime}(\mathbf{x}_{m+1}))\sum_{k=1}^{m}(\lambda_{k}\mathbf{y}_{k}-\lambda_{k}\mathbf{x}_{k})\Bigr{]}+\Phi^{\prime}(\mathbf{x}_{N})\left(\sum_{k=1}^{N}(\lambda_{k}\mathbf{y}_{k}-\lambda_{k}\mathbf{x}_{k})\right).$

When
$\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{x}_{k}}\prec_{L^{\downarrow}}\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{y}_{k}},$
the last term vanishes and the fact that $D\geq 0$ is a consequence of the
isotonicity of $\Phi^{\prime}.$ When
$\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{x}_{k}}\prec_{wL^{\downarrow}}\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{y}_{k}}$
and $\Phi$ is isotone, one applies Lemma 4 $(a)$ to infer that

$\Phi^{\prime}(\mathbf{x}_{N})\left(\sum_{k=1}^{N}(\lambda_{k}\mathbf{y}_{k}-\lambda_{k}\mathbf{x}_{k})\right)\geq
0.$

The other cases can be treated in a similar way. ∎

The specific statement of Theorem 4 for the class of strongly convex
functions, the class of semiconvex functions as well as its translation in the
case of strongly concave functions and of semiconcave functions is left to the
reader as an exercise. We will detail here only the case of $\sigma$-smooth
functions, which in the light of Lemma 3 appears as a Legendre-Fenchel dual of
the majorization inequality.

######  Theorem 5.

Suppose that $\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{x}_{k}}$ and
$\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{y}_{k}}$ are two discrete
probability measures whose supports are included in an open convex subset $C$
of the ordered Banach space $E$. If
$\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{x}_{k}}\prec_{L^{\downarrow}}\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{y}_{k}},$
then

(4.3)
$\sum_{k=1}^{N}\lambda_{k}\Phi(\mathbf{y}_{k})\leq\sum_{k=1}^{N}\lambda_{k}\Phi(\mathbf{x}_{k})+\frac{\sigma}{2}\sum_{k=1}^{N}\lambda_{k}\left\|\mathbf{x}_{k}-\mathbf{y}_{k}\right\|^{2}$

for every Gâteaux differentiable and $\sigma$-smooth function
$\Phi:C\rightarrow F$ whose differential is antitone on $C.$

If
$\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{x}_{k}}\prec_{R^{\uparrow}}\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{y}_{k}},$
then the conclusion $(\ref{maj1sm})$ should be replaced by

(4.4)
$\sum_{k=1}^{N}\lambda_{k}\Phi(\mathbf{x}_{k})+\sum_{k=1}^{N}\lambda_{k}\omega(\left\|\mathbf{x}_{k}-\mathbf{y}_{k}\right\|)\geq\sum_{k=1}^{N}\lambda_{k}\Phi(\mathbf{y}_{k}).$

Moreover, if the majorization relations $\prec_{L^{\downarrow}}$and
$\prec_{R^{\uparrow}}$are replaced respectively by
$\prec_{wL^{\downarrow}}$and $\prec_{wR^{\uparrow}},$ then the inequalities
$(\ref{maj1sm})$ and $(\ref{maj2sm})$ still work for those Gâteaux
differentiable and $\sigma$-smooth functions $\Phi:C\rightarrow F$ which are
antitone and have antitone differentials.

One might wonder if the majorization relations $\prec_{L^{\downarrow}}$and
$\prec_{R^{\uparrow}}$ can be reformulated in terms of doubly stochastic
matrices (as, for example, $\mathbf{x}\prec_{L^{\downarrow}}\mathbf{y}$ if and
only if $\mathbf{x}=P\mathbf{y}$ for some doubly stochastic matrix $P$). The
answer is negative as shows the case of two pairs of elements

$\mathbf{x}_{1}\geq\mathbf{x}_{2}\text{ and
}\mathbf{y}_{1}\geq\mathbf{y}_{2}=0$

such that

$\mathbf{x}_{1}\leq\mathbf{y}_{1}\text{ and
}\left(\mathbf{x}_{1}+\mathbf{x}_{2}\right)/2=\left(\mathbf{y}_{1}+\mathbf{y}_{2}\right)/2.$

Clearly, no $2\times 2$-dimensional real matrix $A$ could exist such that

$\left(\begin{array}[c]{c}\mathbf{x}_{1}\\\
\mathbf{x}_{2}\end{array}\right)=A\left(\begin{array}[c]{c}\mathbf{y}_{1}\\\
\mathbf{y}_{2}\end{array}\right).$

This would imply that $\mathbf{x}_{1},\mathbf{x}_{2}$ belong necessarily to
the segment $[\mathbf{y}_{2}.\mathbf{y}_{1}],$ which is not the case.

However, one implication is true.

######  Remark 4.

If $\mathbf{x}_{1}\geq\cdots\geq\mathbf{x}_{N}$ and
$\mathbf{y}_{1}\geq\dots\geq\mathbf{y}_{N}$ are two families of points in the
ordered Banach space $E$ such that

$P\left(\begin{array}[c]{c}\mathbf{y}_{1}\\\ \vdots\\\
\mathbf{y}_{N}\end{array}\right)=\left(\begin{array}[c]{c}\mathbf{x}_{1}\\\
\vdots\\\ \mathbf{x}_{N}\end{array}\right)$

for a suitable doubly stochastic matrix $P$, then

$\frac{1}{N}\sum_{k=1}^{N}\delta_{\mathbf{x}_{k}}\prec_{L^{\downarrow}}\frac{1}{N}\sum_{k=1}^{N}\delta_{\mathbf{y}_{k}}.$

Indeed, the argument used by Ostrowski $($see _[15]_, Theorem $A.4$, p. $31)$
to settle the case $E=\mathbb{R}$ extends verbatim to the case of ordered
Banach spaces.

In the context of functions of several variables, one can take advantage of
the approximation Theorem 3 to prove the following variant of Theorem 4, where
the assumption on differentiability is discarded.

######  Theorem 6.

Suppose that $C$ is an open box included in $\mathbb{R}_{++}^{N}$ and
$\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{x}_{k}}$ and
$\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{y}_{k}}$ are two discrete
probability measures supported at points in $C.$

If
$\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{x}_{k}}\prec_{L^{\downarrow}}\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{y}_{k}},$
then

$\sum\nolimits_{k=1}^{n}\lambda_{k}\Phi(\mathbf{y}_{k})\geq\sum\nolimits_{k=1}^{n}\lambda_{k}\Phi(\mathbf{x}_{k})$

for every $2$-box monotone convex function $\Phi:C\rightarrow F,$ while if

$\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{x}_{k}}\prec_{R^{\uparrow}}\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{y}_{k}},$
the latter inequality works in the opposite direction.

###### Proof.

Suppose that
$\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{x}_{k}}\prec_{L^{\downarrow}}\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{y}_{k}}$
and choose a compact box $K\subset C$ that contains all points
$\mathbf{x}_{k}$ and $\mathbf{y}_{k}.$ According to Theorem 3, for
$\varepsilon>0$ arbitrarily fixed, there is an infinitely differentiable,
convex and isotone function $\Psi_{\varepsilon}$ with isotone differential,
such that $\sup_{\mathbf{x}\in
K}\left|\Phi(\mathbf{x})-\Psi_{\varepsilon}(\mathbf{x})\right|<\varepsilon.$
Taking into account Theorem 4, we infer that

$\sum_{k=1}^{N}\lambda_{k}\Psi_{\varepsilon}(\mathbf{y}_{k})\geq\sum_{k=1}^{N}\lambda_{k}\Psi_{\varepsilon}(\mathbf{x}_{k}).$

Then

$\sum_{k=1}^{N}\lambda_{k}\Phi(\mathbf{y}_{k})\geq\sum_{k=1}^{N}\lambda_{k}\Phi(\mathbf{x}_{k})-2\varepsilon.$

As $\varepsilon>0$ was arbitrarily fixed, we conclude that

$\sum_{k=1}^{N}\lambda_{k}\Phi(\mathbf{y}_{k})\geq\sum_{k=1}^{N}\lambda_{k}\Phi(\mathbf{x}_{k}).$

The case when
$\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{x}_{k}}\prec_{R^{\uparrow}}\sum_{k=1}^{N}\lambda_{k}\delta_{\mathbf{y}_{k}}$
can be treated similarly. ∎

######  Remark 5.

$($The isotonicity of the differential is not only sufficient but also
necessary for the validity of Theorem _4 _and Theorem _6_ $)$ As was already
noticed in Remark _3_ , the infinitely differentiable function

$\Phi(x,y)=-2\left(xy\right)^{1/2},\quad x,y\in(0,\infty),$

is convex and its differential

$d\Phi:\mathbb{R}_{++}^{2}\rightarrow\mathbb{R}^{2},\text{\quad}d\Phi(x,y)=-(x^{-1/2}y^{1/2},x^{1/2}y^{-1/2})$

is not isotone. Therefore $($see Lemma _ 6_$)$, the function $\Phi$ is not
$2$-box monotone. Consider the points
$\mathbf{x}_{1}=(3/2,1)>\mathbf{x}_{2}=(1/2,1)$ and
$\mathbf{y}_{1}=(2,2)>\mathbf{y}_{2}=(0,0)$. Then

$\mathbf{x}_{1}\leq\mathbf{y}_{1}\text{ and
}\left(\mathbf{x}_{1}+\mathbf{x}_{2}\right)/2=\left(\mathbf{y}_{1}+\mathbf{y}_{2}\right)/2,$

but

$\Phi(\mathbf{x}_{1})+\Phi(\mathbf{x}_{2})=-2\left(3/2\right)^{1/2}-2\left(1/2\right)^{1/2}\approx-3.\,\allowbreak
863\,7>\Phi(\mathbf{y}_{1})+\Phi(\mathbf{y}_{2})=\allowbreak-4.$

Therefore $\Phi$ fails the conclusion of Theorem _4_.

######  Remark 6.

In the variant of weak majorization, the assertions of Theorem _6_ remain
valid for the $2$-box monotone, isotone and convex functions defined on an
open box included in $\mathbb{R}^{N}.$ Indeed, in this case the approximants
$\Psi_{\varepsilon}$ $($that appear in the proof of Theorem _3_ $)$ are not
only $2$-box monotone and strictly convex but also isotone.

##  5\. Applications

The following consequence of Theorem 4 shows that the gap in Jensen’s
inequality (the difference of the two sides of this inequality) decreases when
the order interval under attention is shrinking.

######  Theorem 7.

$($The contractibility of Jensen’s gap$)$ Suppose that $E$ and $F$ are ordered
Banach spaces, $C$ is an open convex subset of $E$ and $\Phi:C\rightarrow F$
is a differentiable convex function whose differential is isotone on $C$. Then
for every family of points
$\mathbf{x}_{1},\mathbf{x}_{2},\mathbf{y}_{1},\mathbf{y}_{2}$ in $C$ and any
$\lambda\in(0,1)$ such that

$\mathbf{y}_{2}\leq\mathbf{x}_{2}\leq(1-\lambda)\mathbf{y}_{1}+\lambda\mathbf{y}_{2}\leq\mathbf{x}_{1}\leq\mathbf{y}_{1}.$

we have

$0\leq(1-\lambda)\Phi(\mathbf{x}_{1})+\lambda\Phi(\mathbf{x}_{2})-\Phi\left((1-\lambda)\mathbf{x}_{1}+\lambda\mathbf{x}_{2}\right)\\\
\leq(1-\lambda)\Phi(\mathbf{y}_{1})+\lambda\Phi(\mathbf{y}_{2})-\Phi\left((1-\lambda)\mathbf{y}_{1}+\lambda\mathbf{y}_{2}\right).$

###### Proof.

Indeed, under the above hypotheses, we have
$\mathbf{x}_{1}\geq(1-\lambda)\mathbf{y}_{1}+\lambda\mathbf{y}_{2}\geq\mathbf{x}_{2}$
and also

$\displaystyle\frac{1-\lambda}{2}\mathbf{x}_{1}$
$\displaystyle\leq\frac{1-\lambda}{2}\mathbf{y}_{1}$
$\displaystyle\frac{1-\lambda}{2}\mathbf{x}_{1}+\frac{1}{2}\left((1-\lambda)\mathbf{y}_{1}+\lambda\mathbf{y}_{2}\right)$
$\displaystyle\leq\frac{1-\lambda}{2}\mathbf{y}_{1}+\frac{1}{2}\left((1-\lambda)\mathbf{x}_{1}+\lambda\mathbf{x}_{2}\right)$
$\displaystyle\frac{1-\lambda}{2}\mathbf{x}_{1}+\frac{1}{2}\left((1-\lambda)\mathbf{y}_{1}+\lambda\mathbf{y}_{2}\right)+\frac{\lambda}{2}\mathbf{x}_{2}$
$\displaystyle=\frac{1-\lambda}{2}\mathbf{y}_{1}+\left((1-\lambda)\mathbf{x}_{1}+\lambda\mathbf{x}_{2}\right)+\lambda\mathbf{y}_{2}.$

Therefore

$\frac{1-\lambda}{2}\delta_{\mathbf{x}_{1}}+\frac{1}{2}\delta_{(1-\lambda)\mathbf{y}_{1}+\lambda\mathbf{y}_{2}}+\frac{\lambda}{2}\delta_{\mathbf{x}_{2}}\prec_{\mathbf{L^{\downarrow}}}\frac{1-\lambda}{2}\delta_{\mathbf{y}_{1}}+\frac{1}{2}\delta_{(1-\lambda)\mathbf{x}_{1}+\lambda\mathbf{x}_{2}}+\frac{\lambda}{2}\delta_{\mathbf{y}_{2}}$

so that, taking into account Theorem 4, we infer that

$(1-\lambda)\Phi(\mathbf{x}_{1})+\Phi\left((1-\lambda)\mathbf{y}_{1}+\lambda\mathbf{y}_{2}\right)+\lambda\Phi(\mathbf{x}_{2})\\\
\leq(1-\lambda)\Phi(\mathbf{y}_{1})+\Phi\left((1-\lambda)\mathbf{x}_{1}+\lambda\mathbf{x}_{2}\right)+\lambda\Phi(\mathbf{y}_{2}),$

an inequality that is equivalent to the conclusion of Theorem 7. ∎

A particular case of Theorem 7 is as follows:

######  Corollary 1.

$($The parallelogram rule$)$ Suppose that $\Phi:C\rightarrow F$ is as in the
statement of Theorem 7 and $\mathbf{x}_{1},\mathbf{x}_{2},\mathbf{y}_{1}$ and
$\mathbf{y}_{2}~{}$are points in $C$ such that
$\mathbf{y}_{2}\leq\mathbf{x}_{2}\leq\mathbf{x}_{1}\leq\mathbf{y}_{1}$ and
$\left(\mathbf{x}_{1}+\mathbf{x}_{2}\right)/2=\left(\mathbf{y}_{1}+\mathbf{y}_{2}\right)/2$
then the following extension of the _parallelogram law_ takes place:

$\Phi(\mathbf{x}_{1})+\Phi(\mathbf{x}_{2})\leq\Phi(\mathbf{y}_{1})+\Phi(\mathbf{y}_{2}).$

######  Remark 7.

$($A multiplicative version of the generalized parallelogram law$)$ Suppose
that $A_{1},A_{2},B_{1},B_{2}~{}$are positively definite matrices from
$\operatorname*{Sym}(N,\mathbb{R})$ such that

$B_{2}\leq A_{2}\leq A_{1}\leq
B_{1},\text{\quad}A_{1}A_{2}=A_{2}A_{1},\text{\quad}B_{1}B_{2}=B_{2}B_{1},\text{
}$

and $\left(A_{1}A_{2}\right)^{1/2}=\left(B_{1}B_{2}\right)^{1/2}.$ Since the
logarithm is an operator monotone function $($see _[12]_$)$, we have

$\log B_{2}\leq\log A_{2}\leq\log A_{1}\leq\log B_{1}\text{ and }\log
A_{1}+\log A_{2}=\log B_{1}+\log B_{2}.$

From Example _5 (_presented in Appendix _)_ and Corollary _1 _$($applied to __
$\operatorname*{trace}f(\exp(A))$ we infer that

$\operatorname*{trace}f(A_{1})+\operatorname*{trace}f(A_{2})\leq\operatorname*{trace}f(B_{1})+\operatorname*{trace}f(B_{2}),$

whenever $f:(0,\infty)\mathbb{\rightarrow R}$ is a continuously differentiable
and nondecreasing function such that $f\circ\exp$ is convex.

######  Remark 8.

$($Another variant of the generalized parallelogram law$)$ Suppose that $E$
and $F$ are ordered Banach spaces, $C$ is an open convex subset of $E$ and
$\Phi:C\rightarrow F$ is a differentiable, isotone and convex function whose
differential is isotone on $C$. Then for every family of points
$\mathbf{x}_{1},\mathbf{x}_{2},\mathbf{y}_{1},\mathbf{y}_{2}$ in $E_{+}$ such
that

$\mathbf{x}_{2}\leq\mathbf{x}_{1}\leq\mathbf{y}_{1}\text{ and
}\mathbf{x}_{1}+\mathbf{x}_{2}\leq\mathbf{y}_{1}+\mathbf{y}_{2}$

we have

$\Phi(\mathbf{x}_{1})+\Phi(\mathbf{x}_{2})\leq\Phi(\mathbf{y}_{1})+\Phi(\mathbf{y}_{2}).$

Indeed, in this case $\mathbf{x}_{1}\leq\mathbf{y}_{1}$ and
$\mathbf{x}_{1}+\mathbf{x}_{2}\leq\mathbf{y}_{1}+\mathbf{y}_{2}.$ Though
$x_{1}+x_{2}=y_{1}+y_{2}$ could fail, Theorem _4_ still applies because
$\Phi^{\prime}(\mathbf{x}_{2})\geq 0$ $($see Lemma _4_ $)$.

Numerous classical inequalities from real analysis can be extended to the
context of ordered Banach spaces via Theorems 4-6. Here are three examples
based on Theorem 4.

######  Theorem 8.

$($The extension of Szegö and Bellman inequalities$)$ Suppose that $E$ and $F$
are two ordered Banach spaces, $C$ is an open convex subset of $E$ that
contains the origin and $\Phi:C\rightarrow F$ is a Gâteaux differentiable
$\omega$-convex function whose differential is isotone. Then for every finite
family $\mathbf{x}_{1}\geq\mathbf{x}_{2}\geq\cdots\geq\mathbf{x}_{n}\geq 0$ of
points in $C$ we have

$(1-\sum\nolimits_{k=1}^{n}\left(-1\right)^{k+1})\Phi(0)+\sum\nolimits_{k=1}^{n}\left(-1\right)^{k+1}\Phi(\mathbf{x}_{k})\geq\Phi(\sum\nolimits_{k=1}^{n}\left(-1\right)^{k+1}\mathbf{x}_{k})\\\
+\sum\nolimits_{k=1}^{n}\omega\left(\left\|x_{k}-x_{k+1}\right\|\right)+\omega(||\sum\nolimits_{k=1}^{n}\left(-1\right)^{k+1}\mathbf{x}_{k}||).$

The proof is immediate, by considering separately the cases where $n$ is odd
or even. The weighted case of Theorem 8 can be easily deduced from it
following the argument of Olkin [20] for the strings of real numbers.

######  Theorem 9.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a nondecreasing, differentiable and
convex function. If $A_{1},A_{2},...,A_{n}$ and $B_{1},B_{2},...,B_{n}$ are
two families of elements in $\operatorname*{Sym}(N,\mathbb{R})$ such that

$A_{1}\geq A_{2}\geq\cdots\geq A_{n}\geq 0\text{ and
}\sum\nolimits_{k=1}^{j}A_{k}\leq\sum\nolimits_{k=1}^{j}B_{k}\text{ for
}j\in\\{1,2,...,n\\},$

then

$\sum\nolimits_{k=1}^{n}\operatorname*{Trace}f\left(A_{k}\right)\leq\sum\nolimits_{k=1}^{n}\operatorname*{Trace}f\left(B_{k}\right).$

This is a consequence of Theorem 4 when combined with Example 5 in the
Appendix. The particular case where $f(x)=x^{2}$ is attributed by Petz [21] to
K. L. Chung.

The third example concerns the case of Popoviciu’s inequality. In its simplest
form this inequality asserts that every convex function $\Phi$ defined on a
real interval $I$ verifies the inequality

$\frac{\Phi(x)+\Phi(y)+\Phi(z)}{3}-\Phi\left(\frac{x+y+z}{3}\right)\\\ \geq
2\,\left[\frac{\Phi\left(\frac{x+y}{2}\right)+\Phi\left(\frac{y+z}{2}\right)+\Phi\left(\frac{z+x}{2}\right)}{3}-\Phi\left(\frac{x+y+z}{3}\right)\right],$

whenever $x,y,z\in I$ (which is an illustration of the contractibility of
Jensen gap in the case of triplets of elements. See [24] and [19] for details.
While Popoviciu’s inequality makes sense in any Banach space, it was shown in
[4] that it actually works only for a special class of convex functions
(including the norm of a Hilbert space). Based on Theorem 4, we will show that
the class of useful functions can be enlarged at the cost of limiting the
triplets of elements under consideration.

######  Theorem 10.

Suppose that $E$ and $F$ are two ordered Banach spaces, $C$ is an open convex
subset of $E$ and $\mathbf{x}\geq\mathbf{y}\geq\,\mathbf{z}$ is a triplet of
points in $C.$ In addition, $\Phi:C\rightarrow F$ is a Gâteaux differentiable
$\omega$-convex function whose differential is isotone.

$(a)$ If
$\mathbf{x}\geq(\mathbf{x}+\mathbf{y}+\mathbf{z})/3\geq$$y$$\geq\,\mathbf{z},$
then

$\frac{\Phi(\mathbf{x})+\Phi(\mathbf{y})+\Phi(\mathbf{z})}{3}+\Phi\left(\frac{\mathbf{x}+\mathbf{y}+\mathbf{z}}{3}\right)\\\
\geq\frac{2}{3}\,\left[\Phi\left(\frac{\mathbf{x}+\mathbf{y}}{2}\right)+\Phi\left(\frac{\mathbf{y}+\mathbf{z}}{2}\right)+\Phi\left(\frac{\mathbf{z}+\mathbf{x}}{2}\right)\right]+\frac{1}{6}\omega\left(\frac{\left\|\mathbf{x}-\mathbf{y}\right\|}{2}\right)\\\
+\frac{1}{6}\omega\left(\frac{\left\|2\mathbf{z}-\mathbf{x}-\mathbf{y}\right\|}{6}\right)+\frac{1}{3}\omega\left(\frac{\left\|2\mathbf{y}-\mathbf{x}-\mathbf{z}\right\|}{6}\right)+\frac{1}{3}\omega\left(\frac{\left\|\mathbf{z}-\mathbf{y}\right\|}{2}\right).$

$(b)$ If
$\mathbf{x}\geq\mathbf{y}\geq(\mathbf{x}+\mathbf{y}+\mathbf{z})/3\geq\,\mathbf{z},$
then

$\frac{\Phi(\mathbf{x})+\Phi(\mathbf{y})+\Phi(\mathbf{z})}{3}+\Phi\left(\frac{\mathbf{x}+\mathbf{y}+\mathbf{z}}{3}\right)\\\
\geq\frac{2}{3}\,\left[\Phi\left(\frac{\mathbf{x}+\mathbf{y}}{2}\right)+\Phi\left(\frac{\mathbf{y}+\mathbf{z}}{2}\right)+\Phi\left(\frac{\mathbf{z}+\mathbf{x}}{2}\right)\right]+\frac{1}{3}\omega\left(\frac{\left\|\mathbf{x}-\mathbf{y}\right\|}{2}\right)\\\
+\frac{1}{6}\omega\left(\frac{\left\|2\mathbf{x}-\mathbf{y}-\mathbf{z}\right\|}{6}\right)+\frac{1}{3}\omega\left(\frac{\left\|2\mathbf{y}-\mathbf{x}-\mathbf{z}\right\|}{6}\right)+\frac{1}{6}\omega\left(\frac{\left\|\mathbf{z}-\mathbf{y}\right\|}{2}\right).$

###### Proof.

$(a)$ In this case

$(\mathbf{x}+\mathbf{y})/2\geq(\mathbf{x}+\mathbf{z})/2\geq(\mathbf{y}+\mathbf{z})/2\text{
and }\frac{\mathbf{x}+\mathbf{z}}{2}\geq\mathbf{y.}$

so the conclusion follows from Theorem 4 applied to the families of points

$\mathbf{x}_{1}=\mathbf{x}_{2}=(\mathbf{x}+\mathbf{y})/2\geq\mathbf{x}_{3}=\mathbf{x}_{4}=(\mathbf{x}+\mathbf{z})/2\geq\mathbf{x}_{5}=\mathbf{x}_{6}=(\mathbf{y}+\mathbf{z})/2$

and

$\mathbf{y}_{1}=\mathbf{x}\geq\mathbf{y}_{2}=\mathbf{y}_{3}=\mathbf{y}_{4}=(\mathbf{x}+\mathbf{y}+\mathbf{z})/3\geq\mathbf{y}_{5}=\mathbf{y}\geq\mathbf{y}_{6}=\mathbf{z,}$

by noticing that

$\displaystyle(\mathbf{x}+\mathbf{y})/2$ $\displaystyle\leq\mathbf{x}$
$\displaystyle(\mathbf{x}+\mathbf{y})/2+(\mathbf{x}+\mathbf{y})/2$
$\displaystyle\leq\mathbf{x}+(\mathbf{x}+\mathbf{y}+\mathbf{z})/3$
$\displaystyle(\mathbf{x}+\mathbf{y})/2+(\mathbf{x}+\mathbf{y})/2+(\mathbf{x}+\mathbf{z})/2$
$\displaystyle\leq\mathbf{x}+(\mathbf{x}+\mathbf{y}+\mathbf{z})/3+(\mathbf{x}+\mathbf{y}+\mathbf{z})/3$

and

$(\mathbf{x}+\mathbf{y})/2+(\mathbf{x}+\mathbf{y})/2+(\mathbf{x}+\mathbf{z})/2+(\mathbf{x}+\mathbf{z})/2\\\
\leq\mathbf{x}+(\mathbf{x}+\mathbf{y}+\mathbf{z})/3+(\mathbf{x}+\mathbf{y}+\mathbf{z})/3+(\mathbf{x}+\mathbf{y}+\mathbf{z})/3\\\
(\mathbf{x}+\mathbf{y})/2+(\mathbf{x}+\mathbf{y})/2+(\mathbf{x}+\mathbf{z})/2+(\mathbf{x}+\mathbf{z})/2+(\mathbf{y}+\mathbf{z})/2\\\
\leq\mathbf{x}+(\mathbf{x}+\mathbf{y}+\mathbf{z})/3+(\mathbf{x}+\mathbf{y}+\mathbf{z})/3+(\mathbf{x}+\mathbf{y}+\mathbf{z})/3+\mathbf{y}\\\
(\mathbf{x}+\mathbf{y})/2+(\mathbf{x}+\mathbf{y})/2+(\mathbf{x}+\mathbf{z})/2+(\mathbf{x}+\mathbf{z})/2+(\mathbf{y}+\mathbf{z})/2+(\mathbf{y}+\mathbf{z})/2\\\
=\mathbf{x}+(\mathbf{x}+\mathbf{y}+\mathbf{z})/3+(\mathbf{x}+\mathbf{y}+\mathbf{z})/3+(\mathbf{x}+\mathbf{y}+\mathbf{z})/3+\mathbf{y}+\mathbf{z}.$

$(b)$ The proof is similar, by considering the families

$\displaystyle\mathbf{x}_{1}$
$\displaystyle=\mathbf{x}_{2}=(\mathbf{x}+\mathbf{y})/2\geq\mathbf{x}_{3}=\mathbf{x}_{4}=(\mathbf{x}+\mathbf{z})/2\geq\mathbf{x}_{5}=\mathbf{x}_{6}=(\mathbf{y}+\mathbf{z})/2$
$\displaystyle\mathbf{y}_{1}$
$\displaystyle=\mathbf{x},\quad\mathbf{y}_{2}=\mathbf{y},\quad\mathbf{y}_{3}=\mathbf{y}_{4}=\mathbf{y}_{5}=(\mathbf{x}+\mathbf{y}+\mathbf{z})/3,\quad\mathbf{y}_{6}=\mathbf{z.}$

∎

In the case where $E=\mathbb{R}$ and $C$ is an interval of $\mathbb{R},$ we
have
$\left[\mathbf{z},\mathbf{x}\right]=\left[\mathbf{z},\mathbf{y}\right]\cup\left[\mathbf{y},\mathbf{x}\right],$
so $(\mathbf{x}+\mathbf{y}+\mathbf{z})/3$ lies automatically in one of the
intervals $\left[\mathbf{z},\mathbf{y}\right]$ and
$\left[\mathbf{y},\mathbf{x}\right].$ This allows us to recover the
aforementioned result of Popoviciu.

##  6\. Appendix: Examples of differentiable functions which are isotone
and/or admit isotone differentials

######  Example 1.

Let $I$ be one of the intervals $(-\infty,0],$ $[0,\infty)$ or
$(-\infty,\infty)).$ The perspective function associated to a convex function
$f:I\rightarrow\mathbb{R}$ is the convex function

$\tilde{f}:I\times(0,\infty),\text{\quad}\tilde{f}(x,y)=yf(x/y).$

See _[19]_, Section $3.5$. Assuming $f$ of class $C^{2},$ then

$\frac{\partial\tilde{f}}{\partial
x}=f^{\prime}\left(\frac{x}{y}\right),\quad\frac{\partial\tilde{f}}{\partial
y}=-\frac{x}{y}f^{\prime}\left(\frac{x}{y}\right)+f\left(\frac{x}{y}\right)\text{\quad
and }\frac{\partial^{2}\tilde{f}}{\partial x\partial
y}=-\frac{x}{y^{2}}f^{\prime\prime}\left(\frac{x}{y}\right).$

As a consequence, if $I=(-\infty,0)$, then $d\tilde{f}$ is isotone; if in
addition $f$ is nonnegative and increasing, then $\tilde{f}$ itself is
isotone.

######  Example 2.

Let $p\in(1,\infty)$. The function

$\Phi:L^{p}\left(\mathbb{R}\right)\rightarrow\mathbb{R},\text{\quad}\Phi(f)=\left\|f\right\|_{p}^{p}=\int_{\mathbb{R}}\left|f\right|^{p}\mathrm{d}t$

is convex and differentiable, its differential being defined by the formula

$d\Phi(f)(h)=p\int_{\mathbb{R}}h\left|f\right|^{p-1}\operatorname*{sgn}f\mathrm{d}t\text{\quad
for all }f,h\in L^{p}\left(\mathbb{R}\right).$

See _[19]_, Proposition $3.7.8$, p. $151$. Clearly, $\Phi$ and its
differential are isotone on the positive cone of
$L^{p}\left(\mathbb{R}\right).$ A variant of this example within the framework
of Schatten classes is provided by Theorem $16$ in _[13]_.

######  Example 3.

The negative entropy function,
$E\left(\mathbf{x}\right)=\sum_{k=1}^{N}x_{k}\log x_{k},$ is
$C^{\infty}$-differentiable on

$\mathbb{R}_{++}^{N}=\\{\mathbf{x}=\left(x_{1},...,x_{N}\right)\in\mathbb{R}^{N}:x_{1},\ldots,x_{N}>0\\}$

and strongly convex on any compact subset $K$ of $\mathbb{R}_{++}^{N}$. The
differential of $\Phi$ is the map
$d\Phi:\mathbb{R}_{++}^{N}\rightarrow\left(\mathbb{R}^{N}\right)^{\ast}$ given
by the formula

$d\Phi(\mathbf{x})\mathbf{v}=\sum\nolimits_{k=1}^{N}(1+\log
x_{k})v_{k},\text{\quad}\mathbf{x}\in\mathbb{R}_{++}^{N}~{}\text{and
}\mathbf{v}\in\mathbb{R}^{N},$

so that $\mathbf{x}\leq\mathbf{y}$ in $\mathbb{R}_{++}^{N}$ implies
$d\Phi(\mathbf{x})\leq d\Phi(\mathbf{y}).$

######  Example 4.

The log-sum-exp function is defined on $\mathbb{R}^{N}$ by the formula

$\operatorname*{LSE}(\mathbf{x})=\log(\sum\nolimits_{k=1}^{N}e^{x_{k}}),\text{\quad}\mathbf{x}\in\mathbb{R}^{N}.$

This function is infinitely differentiable, isotone and convex, but it is not
strongly convex. See _[19]_, Example $3.8.9$, pp. _157-158_. A simple argument
showing that the differential of $\operatorname*{LSE}$ is not isotone is given
in the comments after Lemma _6_. The log-sum-exp function is the Legendre-
Fenchel conjugate of the restriction of the negative entropy function $E$ to
the simplex $\Delta=\left\\{x:\mathbf{x}\in\mathbb{R}^{N},\text{
}\sum_{k=1}^{N}x_{k}=1\right\\}.$ See _[7]_, p. $93$. Since $E$ is strongly
convex, it follows from Lemma _3_ that the log-sum-exp function is strongly
smooth.

######  Example 5.

$($Trace functions of matrices$)$ Denote by
$\operatorname*{Sym}(N,\mathbb{R)}$ the ordered Banach space of all $N\times
N$-dimensional symmetric matrices with real coefficients endowed with the
Frobenius norm and the _Löwner ordering_ ,

$A\leq B\text{ if and only if }\langle
A\mathbf{x},\mathbf{x}\rangle\leq\langle B\mathbf{x},\mathbf{x}\rangle\text{
for all }\mathbf{x}\in\mathbb{R}^{N}.$

If $f:\mathbb{R\rightarrow R}$ is a continuously differentiable $($strongly$)$
convex function, then the formula

$\Phi(A)=\operatorname*{trace}(f(A))$

defines a differentiable $($strongly$)$ convex function on
$\operatorname*{Sym}(n,\mathbb{R})$. Since
$d\Phi(A)X=\operatorname*{trace}\left(f^{\prime}(A)X\right)$ and $f^{\prime}$
is isotone, it follows that $d\Phi$ is isotone too. According to Weyl’s
monotonicity principle $($see _[19]_, Corollary $4.4.3$, p. $203),$__ the
function $\Phi$ is isotone if $f$ itself is isotone.

Two particular cases are of a special interest:

$(a)$ The operator analogue of the negative entropy function presented in
Example $4$ is the negative von Neumann entropy, defined on the compact convex
set
$C=\left\\{A\in\operatorname*{Sym}\nolimits^{++}(N,\mathbb{R}):\operatorname*{trace}(A)=1\right\\}$
via the formula

$S(A)=\operatorname*{trace}\left(A\log
A\right)=\sum\nolimits_{k=1}^{N}\lambda_{k}(A)\log\lambda_{k}(A),$

where $\lambda_{1}(A),...,\lambda_{N}(A)$ are the eigenvalues of $A$ counted
with their multiplicity. According to the preceding discussion, this function
is convex and differentiable and its differential is isotone. One can prove
$($using Lemma _3_ $)$ that the negative von Neumann entropy is $1/2$-strongly
convex and its Legendre-Fenchel conjugate, the convex function
$\log(\operatorname*{trace}(e^{A}))),$ is $2$-smooth. See _[13]_, Theorem
$16).$

$(b)$ The function $\operatorname*{trace}(e^{A})$ is $\log$-convex and
continuously differentiable, with isotone differential. However, the
differential of the convex function $\log(\operatorname*{trace}(e^{A}))$ is
not anymore isotone. See Example $5,$ which discusses the case of diagonal
matrices.

Acknowledgement. The author would like to thank Ştefan Cobzaş, Sorin G. Gal
and Flavia-Corina Mitroi-Symeonidis for useful conversations on the subject of
this paper and to the reviewer for many valuable and constructive comments
that have improved the final version of the paper.

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