Nonlinear Sherman-type inequalities

Marek Niezgoda

doi:10.1515/anona-2018-0098

Artikel Open Access

Nonlinear Sherman-type inequalities

Marek Niezgoda

Veröffentlicht/Copyright: 21. September 2018

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Aus der Zeitschrift Advances in Nonlinear Analysis Band 9 Heft 1

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 = ( y 1 , … , y n ) T ∈ ℝ n is majorized by an n-tuple 𝐱 = ( x 1 , … , x n ) T ∈ ℝ 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 ∈ ℝ n ,

(1.1) 𝐲 ≺ 𝐱 if and only if 𝐲 ∈ conv ⁡ ℙ n ⁢ 𝐱 ,

where the symbol 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 = ( 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 ⁡ ℙ n . Therefore, (1.1) takes the following form: for 𝐱 , 𝐲 ∈ ℝ 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 → ℝ with an interval J ⊂ ℝ 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 .

(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 ℝ 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 ℝ n , i.e.,

Ψ ⁢ ( 𝐜 , 𝐳 ) = 〈 𝐜 , 𝐳 〉 for 𝐜 , 𝐳 ∈ ℝ n .

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 Ψ ⁢ ( 𝐱 , 𝐲 ) = 𝐱 ∘ 𝐲 , where 𝐱 ∘ 𝐲 = ( x 1 ⁢ y 1 , … , x n ⁢ y n ) T for 𝐱 = ( x 1 , … , x n ) T ∈ ℝ n and 𝐲 = ( y 1 , … , y n ) T ∈ ℝ n . They applied a finite reflection group G acting on ℝ 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 ∈ ℝ n .

Example 2.1.

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

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

𝐱 ∘ p ⁢ 𝐲 = p ⁢ ( p - 1 ⁢ 𝐱 ∘ 𝐲 ) for 𝐱 , 𝐲 ∈ ℝ n and p ∈ ℙ n .

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

𝐱 ∘ c ⁢ 𝐲 = ( c ⁢ 𝐱 ) ∘ 𝐲 for 𝐱 , 𝐲 ∈ ℝ n and c ∈ 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 = ℙ n of n × n permutation matrices acting on ℝ n .

We say that a map Ψ : ℝ n × ℝ n → ℝ l admits the Leon–Proschan property for the permutation group ℙ n if for each g ∈ ℙ n there exist h ∈ ℙ n and k ∈ ℙ l such that

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

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

Example 2.2.

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

We have

Ψ ⁢ ( 𝐱 , p ⁢ 𝐲 ) = Ψ ⁢ ( p - 1 ⁢ 𝐱 , 𝐲 ) for 𝐱 , 𝐲 ∈ ℝ n and p ∈ ℙ n .

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

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

Ψ ⁢ ( p ⁢ 𝐱 , p ⁢ 𝐲 ) = Ψ ⁢ ( 𝐱 , 𝐲 ) for all 𝐱 , 𝐲 ∈ ℝ n and p ∈ ℙ n .

Then it follows that

Ψ ⁢ ( 𝐱 , p ⁢ 𝐲 ) = Ψ ⁢ ( p - 1 ⁢ 𝐱 , 𝐲 ) for all 𝐱 , 𝐲 ∈ ℝ n and p ∈ 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 ℝ l with l ∈ ℕ .

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

(2.3) f ⁢ ( ( x 1 , … , x n ) T ) = ( f ⁢ ( x 1 ) , … , f ⁢ ( x n ) ) T for x 1 , … , x n ∈ ℝ

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

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

In other words,

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

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

𝒜 Ψ = { 𝐱 ∈ ℝ n : for 𝐲 , 𝐳 ∈ ℝ n inequality 𝐲 ≤ 𝐳 implies 〈 𝐱 , 𝐲 〉 ≤ 〈 𝐱 , 𝐳 〉 } = ℝ + 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:

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

Ψ ⁢ ( 𝐱 , g ⁢ 𝐲 ) = Ψ ⁢ ( h ⁢ 𝐱 , 𝐲 ) for 𝐱 , 𝐲 ∈ ℝ n .
For each 𝐱 ∈ ℝ n the one-variable map Ψ ⁢ ( 𝐱 , ⋅ ) is convex (with respect to ≤ ) on ℝ n , i.e., for 𝐲 1 , … , 𝐲 m ∈ ℝ 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 ) .
For each 𝐲 ∈ ℝ n the one-variable map Ψ ⁢ ( ⋅ , 𝐲 ) is concave (with respect to ≤ ) on ℝ n , i.e., for 𝐱 1 , … , 𝐱 m ∈ ℝ 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 ∈ ℝ 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 ℝ 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 ℝ 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, ℙ 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

for each g ∈ ℙ n ,

(2.10) Ψ ⁢ ( 𝐱 , g ⁢ 𝐲 ) = Ψ ⁢ ( θ ⁢ ( g ) ⁢ 𝐱 , 𝐲 ) for 𝐱 , 𝐲 ∈ ℝ 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 ⁡ ℙ n , we obtain that 𝐒 = ∑ i = 1 m t i ⁢ g i for some g 1 , … , g m ∈ ℙ 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

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

Ψ ⁢ ( g ⁢ 𝐱 , g ⁢ 𝐲 ) = Ψ ⁢ ( 𝐱 , 𝐲 ) for 𝐱 , 𝐲 ∈ ℝ 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 ∈ ℝ n and g ∈ ℙ n .

However, for g ∈ ℙ n one has g - 1 = g T . So, (2.10) is met with θ ⁢ ( g ) = g - 1 = g T for g ∈ ℙ 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 : ℝ n → ℝ 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 ∈ ℝ n and p ∈ ℙ n , then

(3.1) ∇ p ⁢ 𝐲 ⁡ ψ ⁢ ( p ⁢ 𝐱 ) = ∇ 𝐲 ⁡ ψ ⁢ ( 𝐱 ) for 𝐱 , 𝐲 ∈ ℝ n and p ∈ ℙ n .

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

By taking Ψ ⁢ ( 𝐱 , 𝐲 ) = ∇ 𝐲 ⁡ ψ ⁢ ( 𝐱 ) with 𝐱 , 𝐲 ∈ ℝ n , we find that

𝒜 Ψ = { 𝐱 ∈ ℝ n : the one-variable map Ψ ⁢ ( 𝐱 , ⋅ ) = ∇ ⋅ ⁡ ψ ⁢ ( 𝐱 ) is nondecreasing on ℝ 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 ∈ 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 ∇ 𝐲 ⁡ ψ ⁢ ( 𝐱 ) , viewed as a function of a direction 𝐲 , is linear and continuous on ℝ 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 ∈ ℝ n ,

where 〈 ⋅ , ⋅ 〉 is the standard inner product on ℝ n .

For the map Ψ ⁢ ( 𝐱 , 𝐲 ) = 〈 ∇ ⁡ ψ ⁢ ( 𝐱 ) , 𝐲 〉 with 𝐱 , 𝐲 ∈ ℝ n , we find that

𝒜 Ψ = { 𝐱 ∈ ℝ n : the one-variable map 〈 ∇ ⁡ ψ ⁢ ( 𝐱 ) , ⋅ 〉 is nonnegative on ℝ n }

= { 𝐱 ∈ ℝ n : ∇ ⁡ ψ ⁢ ( 𝐱 ) ∈ ℝ + n } .

Thus the condition 𝐛 ∈ 𝒜 Ψ means that ∇ ⁡ ψ ⁢ ( 𝐛 ) ∈ ℝ + 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 ∈ ℝ 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 ∈ ℝ n .

In this case, condition ∇ ⁡ ψ ⁢ ( 𝐛 ) ∈ ℝ + n means that 𝐛 ∈ 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 ⁢ ( 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 ) ∈ ℝ 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 ∈ ℝ n that

𝒜 Ψ = { 𝐱 ∈ ℝ n : the one-variable map Ψ ⁢ ( 𝐱 , ⋅ ) = ∇ 𝐱 ⁡ ψ ⁢ ( ⋅ ) is nondecreasing on ℝ 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 ∈ ℝ 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 ∈ ℝ n we have

𝒜 Ψ = { 𝐱 ∈ ℝ n : the one-variable map Ψ ⁢ ( 𝐱 , ⋅ ) = 〈 ∇ ⁡ ψ ⁢ ( ⋅ ) , 𝐱 〉 is nondecreasing on ℝ n } .

Condition 𝐛 ∈ 𝒜 Ψ means that the one-variable map Ψ ⁢ ( 𝐛 , ⋅ ) = 〈 ∇ ⁡ ψ ⁢ ( ⋅ ) , 𝐛 〉 is nondecreasing on ℝ n , that is, for 𝐲 , 𝐳 ∈ ℝ 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 ∈ ℝ n ,

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

It is readily seen that

(3.12) ∇ ⁡ ψ ⁢ ( 𝐱 ) = ( exp ⁡ 〈 𝐱 , 𝐞 〉 ) ⁢ 𝐞 for 𝐱 ∈ ℝ 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

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Artikel in diesem Heft

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

Schlagwörter für diesen Artikel

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

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