Marek Niezgoda

Nonlinear Sherman-type inequalities

Open Access
De Gruyter | Published online: September 21, 2018

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 = ( y 1 , , y n ) T n is majorized by an n-tuple 𝐱 = ( x 1 , , x n ) T n , and write 𝐲 𝐱 , if

i = 1 l y [ i ] i = 1 l x [ i ] for  l = 1 , , n ,    and    i = 1 n y i = i = 1 n x i .

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 , 𝐲 n ,

(1.1) 𝐲 𝐱 if and only if 𝐲 conv n 𝐱 ,

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 = ( s i j ) is called column stochastic (resp. row stochastic) if s i j 0 for i = 1 , , n , j = 1 , , m , and all column sums (resp. row sums) of 𝐒 are equal to 1, i.e., i = 1 n s i j = 1 for j = 1 , , m (resp. j = 1 m s i j = 1 for i = 1 , , n ).

An n × n real matrix 𝐒 = ( s i j ) 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 = conv n . Therefore, (1.1) takes the following form: for 𝐱 , 𝐲 n ,

𝐲 𝐱 if and only if 𝐲 = 𝐒𝐱

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

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

𝐲 𝐱 implies F ( 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 : J R be a real convex function defined on an interval J R .

Then, for x = ( x 1 , x 2 , , x n ) T J n and y = ( y 1 , y 2 , , y n ) T J n ,

𝐲 𝐱 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 i = 1 n f ( y i ) i = 1 n f ( x i ) .

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 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 R . Let a = ( a 1 , , a n ) T R + n , b = ( b 1 , , b m ) T R + m , x = ( x 1 , , x n ) T J n and y = ( y 1 , , y m ) T J m .

If

(1.2) 𝐲 = 𝐒𝐱 𝑎𝑛𝑑 𝐚 = 𝐒 T 𝐛

for some n × m row stochastic matrix S = ( s i j ) , then

(1.3) j = 1 m b j f ( y j ) i = 1 n a i f ( x i ) .

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

(1.4) 𝐛 , f ( 𝐲 ) 𝐚 , f ( 𝐱 ) ,

where , is the standard inner product on n , and f ( 𝐱 ) = ( f ( x 1 ) , , f ( x n ) ) T and f ( 𝐲 ) = ( f ( y 1 ) , , f ( y n ) ) T . Further, (1.4) can be restated as

(1.5) Ψ ( 𝐛 , f ( 𝐲 ) ) Ψ ( 𝐚 , f ( 𝐱 ) ) ,

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

Ψ ( 𝐜 , 𝐳 ) = 𝐜 , 𝐳 for  𝐜 , 𝐳 n .

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 ) = 𝐱 𝐲 , where 𝐱 𝐲 = ( x 1 y 1 , , x n y n ) T for 𝐱 = ( x 1 , , x n ) T n and 𝐲 = ( y 1 , , y n ) T n . They applied a finite reflection group G acting on n with the property that for each g G there exist h , k G such that

(2.1) 𝐱 g 𝐲 = k ( ( h 𝐱 ) 𝐲 ) for  𝐱 , 𝐲 n .

Example 2.1.

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

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

𝐱 p 𝐲 = p ( p - 1 𝐱 𝐲 )  for  𝐱 , 𝐲 n  and  p n .

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

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

𝐱 c 𝐲 = ( c 𝐱 ) 𝐲  for  𝐱 , 𝐲 n  and  c n .

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 × n l admits the Leon–Proschan property for the permutation group n if for each g n there exist h n and k l such that

(2.2) Ψ ( 𝐱 , g 𝐲 ) = k Ψ ( h 𝐱 , 𝐲 ) for  𝐱 , 𝐲 n .

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

Example 2.2.

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

We have

Ψ ( 𝐱 , p 𝐲 ) = Ψ ( p - 1 𝐱 , 𝐲 )  for  𝐱 , 𝐲 n  and  p n .

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

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

Ψ ( p 𝐱 , p 𝐲 ) = Ψ ( 𝐱 , 𝐲 ) for all  𝐱 , 𝐲 n  and  p n .

Then it follows that

Ψ ( 𝐱 , p 𝐲 ) = Ψ ( p - 1 𝐱 , 𝐲 ) for all  𝐱 , 𝐲 n  and  p n ,

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

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

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

(2.3) f ( ( x 1 , , x n ) T ) = ( f ( x 1 ) , , f ( x n ) ) T for  x 1 , , x n

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

𝒜 Ψ = { 𝐱 n :  for  𝐲 , 𝐳 n  inequality  𝐲 𝐳  implies  Ψ ( 𝐱 , 𝐲 ) Ψ ( 𝐱 , 𝐳 ) } .

In other words,

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

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

𝒜 Ψ = { 𝐱 n :  for  𝐲 , 𝐳 n  inequality  𝐲 𝐳  implies  𝐱 , 𝐲 𝐱 , 𝐳 } = + n .

Theorem 2.3.

Let Ψ : R n × R n R l be a map. Let f : R R be a convex function. Assume the following conditions:

  1. (i)

    The map Ψ admits the simplified Leon–Proschan property, that is, for each g n there exists h n such that

    Ψ ( 𝐱 , g 𝐲 ) = Ψ ( h 𝐱 , 𝐲 ) for  𝐱 , 𝐲 n .

  2. (ii)

    For each 𝐱 n the one-variable map Ψ ( 𝐱 , ) is convex (with respect to ) on n , i.e., for 𝐲 1 , , 𝐲 m n , t 1 , , t m 0 , i = 1 m t i = 1 ,

    Ψ ( 𝐱 , i = 1 m t i 𝐲 i ) i = 1 m t i Ψ ( 𝐱 , 𝐲 i ) .

  3. (iii)

    For each 𝐲 n the one-variable map Ψ ( , 𝐲 ) is concave (with respect to ) on n , i.e., for 𝐱 1 , , 𝐱 m n , t 1 , , t m 0 , i = 1 m t i = 1 ,

    Ψ ( i = 1 m t i 𝐱 i , 𝐲 ) i = 1 m t i Ψ ( 𝐱 i , 𝐲 ) .

Fix any x , y R n and a , b R n with b A Ψ . If

(2.5) 𝐲 = i = 1 m t i g i 𝐱 𝑎𝑛𝑑 𝐚 = i = 1 m t i h i 𝐛

for some g i P n and h i P n such that Ψ ( h i x , y ) = Ψ ( x , g i y ) , i = 1 , , m , then the following Sherman-type inequality holds:

(2.6) Ψ ( 𝐛 , f ( 𝐲 ) ) Ψ ( 𝐚 , f ( 𝐱 ) ) .

Proof.

For any 𝐳 n we get

(2.7) Ψ ( 𝐛 , i = 1 m t i g i 𝐳 ) i = 1 m t i Ψ ( 𝐛 , g i 𝐳 ) = i = 1 m t i Ψ ( h i 𝐛 , 𝐳 ) Ψ ( i = 1 m t i h i 𝐛 , 𝐳 ) = Ψ ( 𝐚 , 𝐳 ) .

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

(2.8) Ψ ( 𝐛 , i = 1 m t i g i f ( 𝐱 ) ) Ψ ( 𝐚 , f ( 𝐱 ) ) .

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

f ( i = 1 m t i g i 𝐱 ) i = 1 m t i f ( g i 𝐱 ) .

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

Ψ ( 𝐛 , f ( i = 1 m t i g i 𝐱 ) ) Ψ ( 𝐛 , i = 1 m t i f ( g i 𝐱 ) ) .

It is not hard to check that

f ( g i 𝐱 ) = g i f ( 𝐱 ) , i = 1 , , m .

Therefore, the last inequality becomes

(2.9) Ψ ( 𝐛 , f ( i = 1 m t i g i 𝐱 ) ) Ψ ( 𝐛 , i = 1 m t i g i f ( 𝐱 ) ) .

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 = 1 m t i g i 𝐱 ) ) Ψ ( 𝐛 , i = 1 m t i g i f ( 𝐱 ) ) Ψ ( 𝐚 , f ( 𝐱 ) ) .

This completes the proof. ∎

By 𝕄 n we denote the space of all n × n real matrices. Clearly, n 𝔻 n 𝕄 n .

Corollary 2.4.

Let f : R R be a convex function and let Ψ : R n × R n R l be a map satisfying assumptions (ii) and (iii) of Theorem 2.3. Additionally, let θ : M n M n be a linear map such that

  1. (i’)

    for each g n ,

    (2.10) Ψ ( 𝐱 , g 𝐲 ) = Ψ ( θ ( g ) 𝐱 , 𝐲 ) for  𝐱 , 𝐲 n .

Fix any x , y R n and a , b R n with b A Ψ . If

(2.11) 𝐲 = 𝐒𝐱 𝑎𝑛𝑑 𝐚 = θ ( 𝐒 ) 𝐛

for some S D n , then inequality (2.6) holds.

Proof.

Since 𝐒 conv n , we obtain that 𝐒 = i = 1 m t i g i for some g 1 , , g m n and t 1 , , t m 0 with t 1 + + t m = 1 . Then (2.11) implies (2.5). So, it is enough to apply Theorem 2.3. ∎

Corollary 2.5.

Let f : R R be a convex function and let Ψ : R n × R n R l be a map satisfying assumptions (ii) and (iii) of Theorem 2.3. Additionally, assume that

  1. (i’)

    a map Ψ is permutation-invariant in the sense that for each g n

    Ψ ( g 𝐱 , g 𝐲 ) = Ψ ( 𝐱 , 𝐲 ) for  𝐱 , 𝐲 n .

Fix any x , y R n and a , b R n with b A Ψ . If

(2.12) 𝐲 = 𝐒𝐱 𝑎𝑛𝑑 𝐚 = 𝐒 T 𝐛

for some S D n , then inequality (2.6) holds.

Proof.

It follows from (i’) that

Ψ ( 𝐱 , g 𝐲 ) = Ψ ( g - 1 𝐱 , 𝐲 ) for  𝐱 , 𝐲 n  and  g n .

However, for g n one has g - 1 = g T . So, (2.10) is met with θ ( g ) = g - 1 = g T for g n . 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 𝐲 ψ ( 𝐱 ) of ψ at the point 𝐱 in the direction 𝐲 is given by

𝐲 ψ ( 𝐱 ) = lim t 0 ψ ( 𝐱 + t 𝐲 ) - ψ ( 𝐱 ) t

(provided the limit there exists).

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

(3.1) p 𝐲 ψ ( p 𝐱 ) = 𝐲 ψ ( 𝐱 ) for  𝐱 , 𝐲 n  and  p n .

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

By taking Ψ ( 𝐱 , 𝐲 ) = 𝐲 ψ ( 𝐱 ) with 𝐱 , 𝐲 n , we find that

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

Theorem 3.1.

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

Fix any x , y R n and a , b R n with b A Ψ . If

(3.2) 𝐲 = 𝐒𝐱 𝑎𝑛𝑑 𝐚 = 𝐒 T 𝐛

for some S D n , then the following Sherman-type inequality holds:

(3.3) f ( 𝐲 ) ψ ( 𝐛 ) f ( 𝐱 ) ψ ( 𝐚 ) .

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 ) = 𝐲 ψ ( 𝐱 ) for 𝐱 , 𝐲 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 𝐲 ψ ( 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

𝐲 ψ ( 𝐱 ) = ψ ( 𝐱 ) , 𝐲 for  𝐱 , 𝐲 n ,

where , is the standard inner product on n .

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

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

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

Corollary 3.2.

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

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

𝐲 = 𝐒𝐱 𝑎𝑛𝑑 𝐚 = 𝐒 T 𝐛

for some S D n , then the following Sherman-type inequality holds:

(3.4) ψ ( 𝐛 ) , f ( 𝐲 ) ψ ( 𝐚 ) , f ( 𝐱 ) .

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

(3.5) ψ ( 𝐱 ) = 𝐱 2 = 𝐱 , 𝐱 for  𝐱 n ,

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

(3.6) ψ ( 𝐱 ) = 2 𝐱 for  𝐱 n .

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 ) = 𝐱 ψ ( 𝐲 ) with 𝐱 , 𝐲 n that

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

Theorem 3.3.

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

Fix any x , y R n and a , b R n with b A Ψ . If

(3.7) 𝐲 = 𝐒𝐱 𝑎𝑛𝑑 𝐚 = 𝐒 T 𝐛

for some S D n , then the following Sherman-type inequality holds:

(3.8) 𝐛 ψ ( f ( 𝐲 ) ) 𝐚 ψ ( f ( 𝐱 ) ) .

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 ) = 𝐱 ψ ( 𝐲 ) , 𝐱 , 𝐲 n . 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 , 𝐲 n we have

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

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

(3.9) 𝐲 𝐳 implies ψ ( 𝐲 ) , 𝐛 ψ ( 𝐳 ) , 𝐛 .

Corollary 3.4.

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

Fix any x , y R n and a , b R n with b A Ψ . If

𝐲 = 𝐒𝐱 𝑎𝑛𝑑 𝐚 = 𝐒 T 𝐛

for some S D n , then the following Sherman-type inequality holds:

(3.10) ψ ( f ( 𝐲 ) ) , 𝐛 ψ ( f ( 𝐱 ) ) , 𝐚 .

Proof.

It is sufficient to apply Theorem 3.3. ∎

To illustrate the last result, choose

(3.11) ψ ( 𝐱 ) = exp 𝐱 , 𝐞 for  𝐱 n ,

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

It is readily seen that

(3.12) ψ ( 𝐱 ) = ( exp 𝐱 , 𝐞 ) 𝐞 for  𝐱 n .

Here

Ψ ( 𝐱 , 𝐲 ) = ψ ( 𝐲 ) , 𝐱 = ( exp 𝐲 , 𝐞 ) 𝐞 , 𝐱 = 𝐞 , 𝐱 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 𝒜 Ψ means that (3.9) holds in the form

𝐲 𝐳 implies 𝐞 , 𝐛 exp 𝐲 , 𝐞 𝐞 , 𝐛 exp 𝐳 , 𝐞 ,

which is true whenever 𝐞 , 𝐛 0 .

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

𝐞 , 𝐛 exp f ( 𝐲 ) , 𝐞 𝐞 , 𝐚 exp f ( 𝐱 ) , 𝐞 .

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Received: 2018-04-23
Revised: 2018-06-28
Accepted: 2018-07-09
Published Online: 2018-09-21

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