Jensen-Type Inequalities for ( h , g ; m )-Convex Functions

: Jensen-type inequalities for the recently introduced new class of ( h , g ; m ) -convex functions are obtained, and certain special results are indicated. These results generalize and extend corresponding inequalities for the classes of convex functions that already exist in the literature. Schur-type inequalities are given.


Introduction
A convex function is one whose epigraph is a convex set, or, as in the basic definition: A function f : I ⊆ R → R is said to be convex function if (1) holds for all points x and y in I and all λ ∈ [0, 1].
It is called strictly convex if the inequality (1) holds strictly whenever x and y are distinct points and λ ∈ (0, 1). If − f is convex (respectively, strictly convex) then we say that f is concave (respectively, strictly concave). If f is both convex and concave, then f is said to be affine.
The following lemma is equivalent to the definition of a convex function.

Lemma 1 ([1], p. 2)
. Let x 1 , x 2 , x 3 ∈ I be such that x 1 < x 2 < x 3 . The function f : I → R is convex if and only if the following inequality holds By mathematical induction, we can extend the inequality (1) to the convex combinations of finitely many points in I and next to random variables associated to arbitrary probability spaces. These extensions are known as the discrete Jensen inequality and the integral Jensen inequality, respectively. Theorem 1 (The discrete Jensen inequality). A real-valued function f defined on an interval I is convex if and only if for all x 1 , . . . , x n in I and all scalars λ 1 , . . . , λ n in [0 The above inequality is strict if f is strictly convex, all the points x i are distinct and all scalars λ i are positive.
Proving Jensen's inequality in a more general setting is the main motivation for this paper. We will use the recently introduced new class of convexity: ). Let h be a nonnegative function on J ⊆ R, (0, 1) ⊆ J, h ≡ 0 and g be a positive function on I ⊆ R. Let m ∈ (0, 1]. A function f : I → R is said to be (h, g; m)-convex function if it is nonnegative and if holds for all x, y ∈ I and all λ ∈ (0, 1). If (2) holds in the reversed sense, then f is said to be (h, g; m)-concave function.
This class generalizes quite a number of different convexities which exist in the literature. For different choices of functions h, g and parameter m in (2), an (h, g; m)-convex function becomes P-function [3], h-convex function [4], m-convex function [5], (h − m)convex function [6], (s, m)-Godunova-Levin function of the second kind [7], exponentially convex function [8], exponentially s-convex in the second sense [9], and so on. For example, setting h(λ) = λ s , s ∈ (0, 1], g(x) = e −αx , α ∈ R, the (h, g; m)-convexity reduces to exponentially (s, m)-convexity in the second sense from [10]: More on properties of (h, g; m)-convex functions can be found in [2]. Here are a few of them: Let g 1 , g 2 be positive functions on I ⊆ R and let m 1 , If the functions f 1 g 1 and f 2 g 2 are monotonic in the same sense, i.e., where m = max{m 1 , m 2 }, then f 1 f 2 is an (ch, g 1 g 2 ; m)-convex function.
In recent work, we investigated the Hermite-Hadamard inequality for (h, g; m)-convex functions [2], its weighted version-the Féjer inequality [11] and Lah-Ribarič inequality from which the inequalities of Giaccardi, Popoviciu and Petrović for (h, g; m)-convex functions are obtained [12]. Here, we will obtain Schur-type inequalities in Section 2 and Jensen-type inequalities in Section 3 for (h, g; m)-convex functions, which will generalize and extend corresponding inequalities for the classes of convex functions that already exist in the literature. For this, we need super(sub)multiplicative functions: for all x, y ∈ J. If inequality (4) is reversed, then h is said to be a submultiplicative function. If the equality holds in (4), then h is said to be a multiplicative function.

Schur Type Inequalities
We start with a result related to the definition of (h, g; m)-convex functions.
Proposition 5. Let f be a nonnegative (h, g; m)-convex function on I ⊆ R, where h is a nonnegative supermultiplicative function on J ⊆ R, (0, 1) ⊆ J, h ≡ 0, g is a positive function on I and m ∈ (0, 1]. Then, for x 1 , If f is an (h, g; m)-concave function where h is a submultiplicative function, then inequality (5) is reversed.
Proof. Let f be an (h, g; m)-convex function and x 1 , x 2 , x 3 ∈ I. From the assumptions, we have Since h is a supermultiplicative function and Hence, (5) is proven. Analogously follows reversed inequality (5) if f is an (h, g; m)-concave function where h is a submultiplicative function.

Recall the Schur inequality:
If x, y, z are positive numbers and if λ is real, then with equality if and only if x = y = z.
This inequality follows from (5) for f (x) = x λ , λ ∈ R , h(x) = 1 x , g ≡ 1 and m = 1. A related inequality was proved in [13] by Mitrinović and Pečarić: where f is a Godunova-Levin function that is an (h, g; m) ≡ (x −1 , 1; 1)-convex function: Next inequality is of Schur type for (x −k , g; m)-convex (and concave) functions, obtained for h(x) = 1 x k , k ∈ R: Corollary 1. Let f be a positive (x −k , g; m)-convex function on I ⊆ R, where k ∈ R, g is a positive function on I and m ∈ (0, 1]. Then, for x 1 , x 2 , x 3 ∈ I, x 1 < x 2 < x 3 , the following inequality holds If the function f is a positive (x −k , g; m)-concave function, then inequality (7) is reversed.
As an example of a special case, if we set h(x) = x s , s ∈ (0, 1], g(x) = e −αx , α ∈ R, then we obtain following Schur type inequality for convexity (3), i.e., exponentially (s, m)-convex functions in the second sense.
Corollary 2. Let f be an exponentially (s, m)-convex function in the second sense on I ⊆ R, where s, m ∈ (0, 1]. Then, for x 1 , x 2 , x 3 ∈ I, x 1 < x 2 < x 3 the following inequality holds If the function f is an exponentially (s, m)-concave function in the second sense, then inequality (8) is reversed. Remark 1. Using special functions for h and/or g, as well as choosing a fixed parameter for m, Schur-type inequalities for different types of convexity can be derived. For instance, setting g ≡ 1 and m = 1 in (5) and (7), we obtain results for h-convex functions given in [4].

Jensen-Type Inequalities for (h, g; m)-Convex Functions
We continue with Jensen-type inequalities for (h, g; m)-convex functions, where h is supermultiplicative function. In the following, for n ∈ N, let We will set empty products equal to 1, for example G n n+1 = n ∏ j=n+1 g(X j ) ≡ 1.
Notice that P 1 = p 1 , X 1 = x 1 and G n n = g(X n ). The following recursive formulas hold Theorem 2 (The Jensen inequality for an (h, g; m)-convex function). Let p 1 , . . . , p n be positive real numbers. Let f be a nonnegative (h, g; m)-convex function on [0, ∞) such that I ⊆ [0, ∞), where h is a nonnegative supermultiplicative function on J ⊆ R, (0, 1) ⊆ J, h ≡ 0, g is a positive function on [0, ∞) and m ∈ (0, 1]. Then, for x 1 , . . . , x n ∈ I the following inequality holds If f is an (h, g; m)-concave function where h is a submultiplicative function, then inequality (14) is reversed.
Proof. We will prove the theorem by the mathematical induction.

Remark 2.
As before, if we use special h, g and m in (14), then we obtain Jensen-type inequalities for different types of convexity. Hence, Theorem 2 is a generalization of Jensen's inequality for h-convex functions given in [4].