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Advances in Nonlinear Analysis

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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
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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.

Keywords: Majorization; convex function; convex-concave map; HLPK inequality; Sherman inequality; directional derivative; gradient

MSC 2010: 26B25; 26D15; 26D10

1 Introduction

We say that an n-tuple 𝐲=(y1,,yn)Tn is majorized by an n-tuple 𝐱=(x1,,xn)Tn, and write 𝐲𝐱, if

i=1ly[i]i=1lx[i]for l=1,,n,  and  i=1nyi=i=1nxi.

Here x[1]x[n] and y[1]y[n] are the entries of 𝐱 and 𝐲, respectively, arranged in decreasing order [13, p. 8].

It is known that for 𝐱,𝐲n,

𝐲𝐱if and only if𝐲convn𝐱,(1.1)

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

An n×m real matrix 𝐒=(sij) is called column stochastic (resp. row stochastic) if sij0 for i=1,,n, j=1,,m, and all column sums (resp. row sums) of 𝐒 are equal to 1, i.e., i=1nsij=1 for j=1,,m (resp. j=1msij=1 for i=1,,n).

An n×n real matrix 𝐒=(sij) 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 𝔻n.

A doubly stochastic matrix is a convex combination of some permutation matrices, and vice versa [13, Theorem A.2.]. That is, 𝔻n=convn. Therefore, (1.1) takes the following form: for 𝐱,𝐲n,

𝐲𝐱if and only if𝐲=𝐒𝐱

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

A function F:Jn with an interval J is said to be Schur-convex on Jn if for 𝐱,𝐲Jn,

𝐲𝐱impliesF(𝐲)F(𝐱).

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:JR be a real convex function defined on an interval JR.

Then, for x=(x1,x2,,xn)TJn and y=(y1,y2,,yn)TJn,

𝐲𝐱𝑖𝑚𝑝𝑙𝑖𝑒𝑠i=1nf(yi)i=1nf(xi).

Throughout, the symbol ()T denotes the operation of taking the transpose of a matrix. So, 𝐒 is a column stochastic matrix if and only if 𝐒T 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 JR. Let a=(a1,,an)TR+n, b=(b1,,bm)TR+m, x=(x1,,xn)TJn and y=(y1,,ym)TJm.

If

𝐲=𝐒𝐱𝑎𝑛𝑑𝐚=𝐒T𝐛(1.2)

for some n×m row stochastic matrix S=(sij), then

j=1mbjf(yj)i=1naif(xi).(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(𝐲)𝐚,f(𝐱),(1.4)

where , is the standard inner product on n, and f(𝐱)=(f(x1),,f(xn))T and f(𝐲)=(f(y1),,f(yn))T. Further, (1.4) can be restated as

Ψ(𝐛,f(𝐲))Ψ(𝐚,f(𝐱)),(1.5)

where Ψ is the inner product map on n, i.e.,

Ψ(𝐜,𝐳)=𝐜,𝐳for 𝐜,𝐳n.

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

2 Sherman-type inequality for nonlinear maps

In [12], Leon and Proschan gave some interesting inequalities for the Hadamard product map Ψ(𝐱,𝐲)=𝐱𝐲, where 𝐱𝐲=(x1y1,,xnyn)T for 𝐱=(x1,,xn)Tn and 𝐲=(y1,,yn)Tn. They applied a finite reflection group G acting on n with the property that for each gG there exist h,kG such that

𝐱g𝐲=k((h𝐱)𝐲)for 𝐱,𝐲n.(2.1)

Example 2.1.

Let Ψ:n×nn be defined by Ψ(𝐱,𝐲)=𝐱𝐲, the Hadamarad product on n.

If G=n (the permutation group acting on n), then

𝐱p𝐲=p(p-1𝐱𝐲) for 𝐱,𝐲n and pn.

So, (2.1) is met with g=p, h=p-1=pT and k=p.

If G=n (the sign changes group acting on n), then

𝐱c𝐲=(c𝐱)𝐲 for 𝐱,𝐲n and cn.

Therefore, (2.1) holds with g=c, h=c and k=id.

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

We say that a map Ψ:n×nl admits the Leon–Proschan property for the permutation group n if for each gn there exist hn and kl such that

Ψ(𝐱,g𝐲)=kΨ(h𝐱,𝐲)for 𝐱,𝐲n.(2.2)

With k=id, statement (2.2) is called the simplified Leon–Proschan property.

Example 2.2.

Take Ψ:n×nl to be given by Ψ(𝐱,𝐲)=Φ(𝐱+𝐲), where Φ:nl is a permutation-invariant function.

We have

Ψ(𝐱,p𝐲)=Ψ(p-1𝐱,𝐲) for 𝐱,𝐲n and pn.

Therefore, (2.2) holds with g=p, h=p-1=pT and k=id.

More generally, let Ψ:n×nl be a permutation-invariant function in the sense

Ψ(p𝐱,p𝐲)=Ψ(𝐱,𝐲)for all 𝐱,𝐲n and pn.

Then it follows that

Ψ(𝐱,p𝐲)=Ψ(p-1𝐱,𝐲)for all 𝐱,𝐲n and pn,

which is the simplified Leon–Proschan property with g=p and h=p-1=pT.

Throughout, stands for the componentwise order on l with l.

For a given function f:, we extend f to n by

f((x1,,xn)T)=(f(x1),,f(xn))Tfor x1,,xn(2.3)

In the sequel, for a map Ψ:n×nl, we consider the set

𝒜Ψ={𝐱n: for 𝐲,𝐳n inequality 𝐲𝐳 implies Ψ(𝐱,𝐲)Ψ(𝐱,𝐳)}.

In other words,

𝒜Ψ={𝐱n: the one-variable map Ψ(𝐱,) is nondecreasing on n}.(2.4)

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

𝒜Ψ={𝐱n: for 𝐲,𝐳n inequality 𝐲𝐳 implies 𝐱,𝐲𝐱,𝐳}=+n.

Theorem 2.3.

Let Ψ:Rn×RnRl be a map. Let f:RR be a convex function. Assume the following conditions:

  • (i)

    The map Ψ admits the simplified Leon–Proschan property, that is, for each gn there exists hn such that

    Ψ(𝐱,g𝐲)=Ψ(h𝐱,𝐲)for 𝐱,𝐲n.

  • (ii)

    For each 𝐱n the one-variable map Ψ(𝐱,) is convex (with respect to ) on n , i.e., for 𝐲1,,𝐲mn, t1,,tm0, i=1mti=1,

    Ψ(𝐱,i=1mti𝐲i)i=1mtiΨ(𝐱,𝐲i).

  • (iii)

    For each 𝐲n the one-variable map Ψ(,𝐲) is concave (with respect to ) on n , i.e., for 𝐱1,,𝐱mn, t1,,tm0, i=1mti=1,

    Ψ(i=1mti𝐱i,𝐲)i=1mtiΨ(𝐱i,𝐲).

Fix any x,yRn and a,bRn with bAΨ. If

𝐲=i=1mtigi𝐱𝑎𝑛𝑑𝐚=i=1mtihi𝐛(2.5)

for some giPn and hiPn such that Ψ(hix,y)=Ψ(x,giy), i=1,,m, then the following Sherman-type inequality holds:

Ψ(𝐛,f(𝐲))Ψ(𝐚,f(𝐱)).(2.6)

Proof.

For any 𝐳n we get

Ψ(𝐛,i=1mtigi𝐳)i=1mtiΨ(𝐛,gi𝐳)=i=1mtiΨ(hi𝐛,𝐳)Ψ(i=1mtihi𝐛,𝐳)=Ψ(𝐚,𝐳).(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(𝐱), from (2.7) we have

Ψ(𝐛,i=1mtigif(𝐱))Ψ(𝐚,f(𝐱)).(2.8)

Because the extension (2.3) of f is convex on n (with respect to ), we find that

f(i=1mtigi𝐱)i=1mtif(gi𝐱).

Hence, by the monotonicity of Ψ(𝐛,) on n (with respect to ) (see (2.4)), we obtain

Ψ(𝐛,f(i=1mtigi𝐱))Ψ(𝐛,i=1mtif(gi𝐱)).

It is not hard to check that

f(gi𝐱)=gif(𝐱),i=1,,m.

Therefore, the last inequality becomes

Ψ(𝐛,f(i=1mtigi𝐱))Ψ(𝐛,i=1mtigif(𝐱)).(2.9)

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

Ψ(𝐛,f(𝐲))=Ψ(𝐛,f(i=1mtigi𝐱))Ψ(𝐛,i=1mtigif(𝐱))Ψ(𝐚,f(𝐱)).

This completes the proof. ∎

By 𝕄n we denote the space of all n×n real matrices. Clearly, n𝔻n𝕄n.

Corollary 2.4.

Let f:RR be a convex function and let Ψ:Rn×RnRl be a map satisfying assumptions (ii) and (iii) of Theorem 2.3. Additionally, let θ:MnMn be a linear map such that

  • (i’)

    for each gn,

    Ψ(𝐱,g𝐲)=Ψ(θ(g)𝐱,𝐲)for 𝐱,𝐲n.(2.10)

Fix any x,yRn and a,bRn with bAΨ. If

𝐲=𝐒𝐱𝑎𝑛𝑑𝐚=θ(𝐒)𝐛(2.11)

for some SDn, then inequality (2.6) holds.

Proof.

Since 𝐒convn, we obtain that 𝐒=i=1mtigi for some g1,,gmn and t1,,tm0 with t1++tm=1. Then (2.11) implies (2.5). So, it is enough to apply Theorem 2.3. ∎

Corollary 2.5.

Let f:RR be a convex function and let Ψ:Rn×RnRl 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 gn

    Ψ(g𝐱,g𝐲)=Ψ(𝐱,𝐲)for 𝐱,𝐲n.

Fix any x,yRn and a,bRn with bAΨ. If

𝐲=𝐒𝐱𝑎𝑛𝑑𝐚=𝐒T𝐛(2.12)

for some SDn, then inequality (2.6) holds.

Proof.

It follows from (i’) that

Ψ(𝐱,g𝐲)=Ψ(g-1𝐱,𝐲)for 𝐱,𝐲n and gn.

However, for gn one has g-1=gT. So, (2.10) is met with θ(g)=g-1=gT for gn. Therefore, the usage of Corollary 2.4 with θ=()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 ψ:n the directional derivative 𝐲ψ(𝐱) of ψ at the point 𝐱 in the direction 𝐲 is given by

𝐲ψ(𝐱)=limt0ψ(𝐱+t𝐲)-ψ(𝐱)t

(provided the limit there exists).

It is readily seen that if ψ is permutation-invariant, i.e., ψ(p𝐱)=ψ(𝐱) for 𝐱n and pn, then

p𝐲ψ(p𝐱)=𝐲ψ(𝐱)for 𝐱,𝐲n and pn.(3.1)

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

By taking Ψ(𝐱,𝐲)=𝐲ψ(𝐱) with 𝐱,𝐲n, we find that

𝒜Ψ={𝐱n: the one-variable map Ψ(𝐱,)=ψ(𝐱) is nondecreasing on n}.

Theorem 3.1.

Let ψ:RnR be a Schur-convex function. Assume that for any x,yRn there exists the directional derivative yψ(x) of ψ at the point x in the direction y. Let f:RR be convex. Assume that assumptions (ii) and (iii) of Theorem 2.3 are satisfied for Ψ(x,y)=yψ(x) with x,yRn.

Fix any x,yRn and a,bRn with bAΨ. If

𝐲=𝐒𝐱𝑎𝑛𝑑𝐚=𝐒T𝐛(3.2)

for some SDn, then the following Sherman-type inequality holds:

f(𝐲)ψ(𝐛)f(𝐱)ψ(𝐚).(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 Ψ(𝐱,𝐲)=𝐲ψ(𝐱) for 𝐱,𝐲n. 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 𝐲ψ(𝐱), viewed as a function of a direction 𝐲, is linear and continuous on n. Then there exists the gradient ψ(𝐱) of ψ at the point 𝐱 such that

𝐲ψ(𝐱)=ψ(𝐱),𝐲for 𝐱,𝐲n,

where , is the standard inner product on n.

For the map Ψ(𝐱,𝐲)=ψ(𝐱),𝐲 with 𝐱,𝐲n, we find that

𝒜Ψ={𝐱n: the one-variable map ψ(𝐱), is nonnegative on n}={𝐱n:ψ(𝐱)+n}.

Thus the condition 𝐛𝒜Ψ means that ψ(𝐛)+n.

Corollary 3.2.

Let ψ:RnR be a Gâteaux differentiable Schur-convex function. Let f:RR be convex. Assume that the assumption (iii) of Theorem 2.3 is satisfied for Ψ(x,y)=ψ(x),y with x,yRn.

Fix any x,yRn and a,bRn with ψ(b)R+n. If

𝐲=𝐒𝐱𝑎𝑛𝑑𝐚=𝐒T𝐛

for some SDn, then the following Sherman-type inequality holds:

ψ(𝐛),f(𝐲)ψ(𝐚),f(𝐱).(3.4)

Proof.

Condition (ii) of Theorem 2.3 holds by the linearity of Ψ(𝐱,𝐲)=ψ(𝐱),𝐲 with respect to 𝐲. Therefore, it is sufficient to apply Theorem 3.1. ∎

By putting

ψ(𝐱)=𝐱2=𝐱,𝐱for 𝐱n,(3.5)

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

ψ(𝐱)=2𝐱for 𝐱n.(3.6)

In this case, condition ψ(𝐛)+n means that 𝐛+n.

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

𝐛,f(𝐲)𝐚,f(𝐱),

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=(1,,1)n with a doubly stochastic 𝐒, we can get Theorem A.

It is not hard to verify for the map Ψ(𝐱,𝐲)=𝐱ψ(𝐲) with 𝐱,𝐲n that

𝒜Ψ={𝐱n: the one-variable map Ψ(𝐱,)=𝐱ψ() is nondecreasing on n}.

Theorem 3.3.

Let ψ:RnR be a Schur-convex function. Assume that for any x,yRn there exists the directional derivative xψ(y) of ψ at the point x in the direction y. Let f:RR be convex. Assume that assumptions (ii) and (iii) of Theorem 2.3 are satisfied for Ψ(x,y)=xψ(y) with x,yRn.

Fix any x,yRn and a,bRn with bAΨ. If

𝐲=𝐒𝐱𝑎𝑛𝑑𝐚=𝐒T𝐛(3.7)

for some SDn, then the following Sherman-type inequality holds:

𝐛ψ(f(𝐲))𝐚ψ(f(𝐱)).(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 Ψ(𝐱,𝐲)=𝐱ψ(𝐲), 𝐱,𝐲n. Furthermore, (3.7) gives (2.12), which implies (2.6). Finally, we get (3.8), as claimed. ∎

For the map Ψ(𝐱,𝐲)=ψ(𝐲),𝐱 with 𝐱,𝐲n we have

𝒜Ψ={𝐱n: the one-variable map Ψ(𝐱,)=ψ(),𝐱 is nondecreasing on n}.

Condition 𝐛𝒜Ψ means that the one-variable map Ψ(𝐛,)=ψ(),𝐛 is nondecreasing on n, that is, for 𝐲,𝐳n,

𝐲𝐳impliesψ(𝐲),𝐛ψ(𝐳),𝐛.(3.9)

Corollary 3.4.

Let ψ:RnR be a Gâteaux differentiable Schur-convex function. Let f:RR be convex. Assume that assumptions (ii) and (iii) of Theorem 2.3 are satisfied for Ψ(x,y)=ψ(y),x with x,yRn.

Fix any x,yRn and a,bRn with bAΨ. If

𝐲=𝐒𝐱𝑎𝑛𝑑𝐚=𝐒T𝐛

for some SDn, then the following Sherman-type inequality holds:

ψ(f(𝐲)),𝐛ψ(f(𝐱)),𝐚.(3.10)

Proof.

It is sufficient to apply Theorem 3.3. ∎

To illustrate the last result, choose

ψ(𝐱)=exp𝐱,𝐞for 𝐱n,(3.11)

where 𝐞=(1,,1)n. This is a Schur-convex function.

It is readily seen that

ψ(𝐱)=(exp𝐱,𝐞)𝐞for 𝐱n.(3.12)

Here

Ψ(𝐱,𝐲)=ψ(𝐲),𝐱=(exp𝐲,𝐞)𝐞,𝐱=𝐞,𝐱exp𝐲,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 𝐛𝒜Ψ means that (3.9) holds in the form

𝐲𝐳implies𝐞,𝐛exp𝐲,𝐞𝐞,𝐛exp𝐳,𝐞,

which is true whenever 𝐞,𝐛0.

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

𝐞,𝐛expf(𝐲),𝐞𝐞,𝐚expf(𝐱),𝐞.

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About the article

Received: 2018-04-23

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, DOI: https://doi.org/10.1515/anona-2018-0098.

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© 2020 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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