Extensions and improvements of Sherman’s and related inequalities for n-convex functions

Abstract This paper gives extensions and improvements of Sherman’s inequality for n-convex functions obtained by using new identities which involve Green’s functions and Fink’s identity. Moreover, extensions and improvements of Majorization inequality as well as Jensen’s inequality are obtained as direct consequences. New inequalities between geometric, logarithmic and arithmetic means are also established.


Introduction
In [1], Sherman proved that the inequality while A T denotes the transpose of A. If is concave, then the reverse inequality in (1) holds. This result generalizes classical Majorization inequality, proved by Hardy et el [2], as well as Jensen's inequality. The purpose of this paper is to extend Sherman's result to the more general class of n-convex functions and to give improvements of Sherman's inequality (1) from which extensions and improvements of Majorization inequality and Jensen's inequality immediately follow. Some related results can be found in [3][4][5][6].
The study of n-convex functions on an interval is the subject of a monograph by Popoviciu [7]. Popoviciu defined n-convexity of a function W OE˛;ˇ ! R in terms of the divided differences (of order n) which are defined recursively as follows: OEx i I D .x i /; i D 0; :::; n OEx 0 ; :::; x n I D OEx 1 ; :::; x n I OEx 0 ; :::; x n 1 I x n x 0 , where x 0 ; x 1 ; :::; x n 2 OE˛;ˇ are mutually different points and the value OEx 0 ; :::; x n I is independent of theirs order. This definition may be extended to include the case in which some or all the points coincide. Assuming that .j 1/ .x/ exists, we define A function W OE˛;ˇ ! R is n-convex (n 0) if for all choices of .n C 1/ distinct points x i 2 OE˛;ˇ; i D 0; :::; n; the inequality OEx 0 ; x 1 ; :::; x n I 0 holds. If this inequality is reversed, then is n-concave. Thus a 1-convex function is nondecreasing and a 2-convex function is convex in the usual sense. An n-convex function need not be n-times differentiable (e.g. W OE0; 1 ! R, .x/ D x 5 3 ). However iff .n/ exists then is n-convex iff .n/ 0 (see [8, p. 16]): In order to develop some inequalities of type (1) for n-convex functions, we use the following Fink's identity [9] .
which holds for every function W OE˛;ˇ ! R such that .n 1/ is absolutely continuous on OE˛;ˇ for some n 1: For n D 1 we take the sum in (4) to be zero.

Some new identities
It is easy to verify that integration by parts yields that for any function 2 C 2 .OE˛;ˇ/ the following identity holds where the function G 1 W OE˛;ˇ OE˛;ˇ ! R is Green's function of the boundary value problem z 00 D 0; z.˛/ D z 0 .ˇ/ D 0 and is defined by u; u Ä s: Green's function G 1 is continuous and convex in u; since it is symmetric, i.e. G 1 .u; s/ D G 1 .s; u/; then also in s.
Here we introduce three new types of Green's functions defined on OE˛;ˇ OE˛;ˇ as follows: All three functions are continuous, symmetric and convex with respect to the both variables u and s. Next we introduce three technical lemmas which give us new identities involving defined Green's functions. . .
from which (11) follows immediately. Similarly, we can prove other two identities.
Lemma 2.2. Let x 2 OE˛;ˇ l ; y 2 OE˛;ˇ m ; a 2 R l and b 2 R m be such that (2) holds for some matrix A 2 M ml .R/ whose entries satisfy the condition P l j D1 a ij D 1 for i D 1; :::; m: Let G k . ; s/; s 2 OE˛;ˇ; k D 1; 2; 3; 4; be defined as in (7)-(10). Then for every 2 C 2 .OE˛;ˇ/; it holds that Proof. Let us consider Green's function G 2 defined by (8). Applying (11) in the difference x j a ij D 0; Moreover, after interchanging the order of summation in (15), we easily get Analogously, we can prove the identities for other three Green's functions.
Lemma 2.3. Let x 2 OE˛;ˇ l ; y 2 OE˛;ˇ m ; a 2 R l and b 2 R m be such that (2) holds for some matrix A 2 M ml .R/ whose entries satisfy the condition P l j D1 a ij D 1 for i D 1; :::; m: Let P .t; s/ and G k . ; s/; s; t 2 OE˛;ˇ; k D 1; 2; 3; 4; be defined as in (5) and (7)-(10), respectively. If a function W OE˛;ˇ ! R is such that .n 1/ is absolutely continuous on OE˛;ˇ for some n 3; then Proof. Applying (4) for 00 , we get 00 .s/ D By an easy calculation, applying (17) in (14), we get After interchanging the order of summation and integration and applying Fubini's theorem we get (16).

Sherman's type inequalities
We begin this section with the following result which concerns Sherman's type inequalities for real, not necessary nonnegative entries of vectors a, b and matrix A.
Theorem 3.1. Let x 2 OE˛;ˇ l ; y 2 OE˛;ˇ m ; a 2 R l and b 2 R m be such that (2) holds for some matrix A 2 M ml .R/ whose entries satisfy the condition P l j D1 a ij D 1 for i D 1; :::; m: Let G k . ; s/; s 2 OE˛;ˇ; k D 1; 2; 3; 4; be defined as in (7)-(10). Then the following statements are equivalent: .i i/ For every k D 1; 2; 3; 4 and s 2 OE˛;ˇ; it holds that Furthermore, the statements (i) and (ii) are also equivalent if one changes the sing of inequality in both (18) and (19).
Proof. (i))(ii) Let (i) hold. Since G k . ; s/; s 2 OE˛;ˇ; k D 1; 2; 3; 4; is continuous and convex on OE˛;ˇ; then also a j G k .x j ; s/: (ii))(i) Let (ii) hold. Let us consider the function G 2 . ; s/; s 2 OE˛;ˇ; defined by (8). For every function 2 then also (18) holds. Note that it is not necessary to demand the existence of the second derivative of the function : The differentiability condition can be directly eliminated by using the fact that a continuous convex function is possible to approximate uniformly by convex polynomials (see [8, p. 172]). The same conclusion we have for other three Green's functions. The last part statement of theorem can be proved analogously.
Next, we develop Sherman's type inequalities for n-convex functions.
Theorem 3.2. Let x 2 OE˛;ˇ l ; y 2 OE˛;ˇ m ; a 2 R l and b 2 R m be such that (2) holds for some matrix A 2 M ml .R/ whose entries satisfy the condition P l j D1 a ij D 1 for i D 1; :::; m: Let P .t; s/ and G k . ; s/; s; t 2 OE˛;ˇ; k D 1; 2; 3; 4; be defined as in (5) and (7)-(10), respectively. Let W OE˛;ˇ ! R be n-convex function such that .n 1/ is absolutely continuous on OE˛;ˇ for some n 3: If the reverse of (20) holds, then the reverse of (21) holds.
Proof. Under the assumptions of theorem, (16) holds: Since .n 1/ is absolutely continuous on OE˛;ˇ, then .n/ exists almost everywhere (see [10]). By assumption, is n-convex on OE˛;ˇ, therefore .n/ 0 on OE˛;ˇ. Using this fact and the assumption (20) in combination with (16) When we take in account Sherman's condition of nonnegativity of vectors a, b and matrix A, we obtain the following extensions.
.i i/ If n is odd, then for t Ä s; the inequalities (20) and (21) hold, while for s Ä t; the reverse inequalities in (20) and (21) hold.
Proof. (i)-(ii) Since G k . ; s/; s 2 OE˛;ˇ; is convex on OE˛;ˇ; by Sherman's theorem If n is even, then while for odd n; the reversed inequality in (24) holds while the inequality (25) remains the same. Now, applying Theorem 3.2, we conclude (i) and (ii).
(iii) If (21) holds, the right hand side can be written in the form where F k is defined as in (22). If F k is convex, then by Sherman's theorem we have i.e. the right hand side of (21) is nonnegative and (23) holds. i.e. the right hand side of (21) is nonnegative what we need to prove.

Related results
Motivated by (21), we define .s t / n 3 P .t; s/dsˇq dt The constant on the right hand side is sharp for 1 < p Ä 1 and the best possible for p D 1: Proof. Applying the well-known Hölder inequality to (16), we have Considering the functions B k we obtain the following estimations for the given remainders R k .

Applications
In this section, considering the particular cases of the previous results, we show some consequences. As applications, we obtain extensions and improvements of Majorization and discrete Jensen's inequality as well as inequalities between geometric, logarithmic and arithmetic means. As a direct consequence of Theorem 3.3 we get the following corollary.
i.e. we get improvements when 0 .ˇ/ 0 .˛/: If we denote A m D P m i D1 a i and put y 1 D y 2 D ::: D y m D 1 A m P m iD1 a i x i ; we get the following Jensen's type inequality i.e. if in addition 0 .ˇ/ 0 .˛/; then we get double inequality which improves Jensen's inequality. Especially, choosing Green's function G 2 ; defined by (8) and setting m D 2; A 2 D 2; x 1 D˛; In the sequel, applying the double inequality (35) to some concrete functions, we derive some new inequalities. i.e. we get the inequalities between the geometric mean G D p xy; the logarithmic mean L D y x ln y ln x and the arithmetic mean A D xCy 2 ; in form G Ä L .ln y ln x/ 2 4 C G Ä A: The constant on the right hand side is sharp for 1 < p Ä 1 and the best possible for p D 1: Remark 5.6. Choosing Green's function G 2 ; defined by (8), and setting l D m D 2; a i D b i D 1; i D 1; 2; x 1 D˛; x 2 Dˇ; y 1 D y 2 D˛C2 with t Ä s; from (36)