Hermite-Hadamard inequality for functions whose derivatives absolute values are preinvex

In this article, we extend some estimates of the right-hand side of a Hermite-Hadamard-type inequality for preinvex functions. Then, a generalization to functions of several variables on invex subsets of is introduced.

Let be an interval on the real line ℝ, let be a convex function and let , . We consider the well-known Hadamard’s inequality

Both inequalities hold in the reversed direction if f is concave. We note that Hadamard’s inequality may be regarded as a refinement of the concept of convexity and it follows easily from Jensen’s inequality. Hadamard’s inequality for convex functions has received renewed attention in recent years and a remarkable variety of refinements and generalizations have been found (see, for example, [1-4]).

The classical Hermite-Hadamard inequality provides estimates of the mean value of a continuous convex function .

Dragomir and Agarwal [5] used the formula,

to prove the following results.

Theorem 1.1Assumewithandis a differentiable function on. Ifis convex onthen the following inequality holds true

Theorem 1.2Assumewithandis a differentiable function on. Assumewith. Ifis convex onthen the following inequality holds true

Ion [6] presented some estimates of the right-hand side of a Hermite-Hadamard-type inequality in which some quasi-convex functions are involved.

In recent years, several extensions and generalizations have been considered for classical convexity. A significant generalization of convex functions is that of invex functions introduced by Hanson [7]. Weir and Mond [8] introduced the concept of preinvex functions and applied it to the establishment of the sufficient optimality conditions and duality in nonlinear programming. Aslam Noor [9,10] introduced the Hermite-Hadamard inequality for preinvex and log-preinvex functions.

In this article, we generalize the results in [6] for functions whose first derivatives absolute values are preinvex. Also some results for functions whose second derivatives absolute values are preinvex will be given. Now, we recall some notions in invexity analysis which will be used throughout the article (see [11,12] and references therein).

Definition 1.1 A set is said to be invex with respect to the map , if for every and ,

It is obvious that every convex set is invex with respect to the map , but there exist invex sets which are not convex (see [11]). Let be an invex set with respect to . For every the η-path joining the points x and is defined as follows

Definition 1.2 Let be an invex set with respect to . Then, the function is said to be preinvex with respect to η, if for every and ,

Every convex function is a preinvex with respect to the map but the converse does not holds. For properties and applications of preinvex functions, see [12,13] and references therein.

The organization of the article is as follows: In Section 2, some generalizations of Hermite-Hadamard-type inequality for first-order differentiable functions are given. Section 3 is devoted to a generalization to several variable preinvex functions. Hermite-Hadamard-type inequality for second-order differentiable functions are studied in Section 23.

In this section, we introduce some generalizations of Hermite-Hadamard-type inequality for functions whose first derivatives absolute values are preinvex. We begin with the following lemma which is a generalization of Lemma 2.1 in [5] to invex setting.

Lemma 2.1Letbe an open invex subset with respect toandwith. Suppose thatis a differentiable function. Ifis integrable on theθ-paththen, the following equality holds

(7)

Proof Suppose that . Since A is an invex set with respect to θ, for every we have . Integrating by parts implies that

(8)

which completes the proof. □

Theorem 2.1Letbe an open invex subset with respect to. Suppose thatis a differentiable function. Ifis preinvex onAthen, for everywiththe following inequality holds

(9)

Proof Suppose that . Since A is an invex set with respect to θ, for every we have . By preinvexity of and Lemma 2.1 we get

(10)

where,

 □

Now, we give an example of an invex set with respect to an θ which is satisfies the conditions of Theorem 2.1.

Example 2.1 Suppose that and the function is defined by

Clearly K is an open invex set with respect to θ. Suppose that and , hence, . Now,

and

where .

Another similar result is embodied in the following theorem.

Theorem 2.2Letbe an open invex subset with respect to. Suppose thatis a differentiable function. Assume thatwith. Ifis preinvex onAthen, for everywiththe following inequality holds

(11)

Proof Suppose that . By assumption, Hölder’s inequality and the proof of Theorem 4.1 we have

(12)

where . □

Note that if and for every then, we can deduce Theorems 1.1 and 1.2, from Theorems 2.1 and 2.2, respectively.

The aim of this section is to extend the Proposition 1 in [6] and Theorem 2.2 to functions of several variables defined on invex subsets of .

The mapping is said to be satisfies the condition C if for every and ,

Note that, in Example 2.1, θ satisfies the condition C.

For every and every from condition C we have

see [12] for details.

Proposition 3.1Letbe an invex set with respect toandis a function. Suppose thatηsatisfies conditionConS. Then, for everythe functionfis preinvex with respect toηonη-pathif and only if the functiondefined by

is convex on.

Proof Suppose that φ is convex on and , . Fix . Since η satisfies condition C, by (13) we have

Hence, f is preinvex with respect to η on η-path .

Conversely, let and the function f be preinvex with respect to η on η-path . Suppose that . Then, for every we have

Therefore, φ is quasi-convex on . □

The following theorem is a generalization of Proposition 1 in [6].

Theorem 3.1Letbe an open invex set with respect to. Assume thatηsatisfies conditionC. Suppose that for everythe functionis preinvex with respect toηonη-path. Then, for everywiththe following inequality holds,

(16)

Proof Let and with . Since f is preinvex with respect to η on η-path by Proposition 3.1 the function defined by

is convex on . Now, we define the function as follows

Obviously for every we have

hence, . Applying Theorem 1.1 to the function ϕ implies that

and we deduce that (16) holds. □

In this section, we introduce some generalizations of Hermite-Hadamard-type inequality for functions whose second derivatives absolute values are preinvex. We begin with the following lemma (see Lemma 1 in [14] and Lemma 4 in [15]).

Lemma 4.1Letbe an open invex subset with respect toandwith. Suppose thatis a differentiable function. Ifis integrable on theθ-path, then, the following equality holds

(17)

Proof Suppose that . Since A is an invex set with respect to θ, for every we have . Integrating by parts implies that

(18)

which completes the proof. □

Theorem 4.1Letbe an open invex subset with respect toandfor all. Suppose thatis a twice differentiable function onA. Ifis preinvex onAandis integrable on theθ-path, then, the following inequality holds

(19)

Proof Suppose that . Since A is an invex set with respect to θ, for every we have . By preinvexity of and Lemma 4.1 we get

(20)

which completes the proof. □

The corresponding version for powers of the absolute value of the second derivative is incorporated in the following theorem.

Theorem 4.2Letbe an open invex subset with respect toandfor all. Suppose thatis a twice differentiable function onAandis preinvex onA, for. Ifis integrable on theθ-path, then, the following inequality holds

(21)

Proof By preinvexity of , Lemma 4.1 and using the well-known Hölder integral inequality, we get

which completes the proof. □

A more general inequality is given using Lemma 4.1, as follows:

Theorem 4.3Letbe an open invex subset with respect toandfor all. Suppose thatis a twice differentiable function onAandis preinvex onA, for. Ifis integrable on theθ-path, then, the following inequality holds

(22)

Proof By preinvexity of , Lemma 4.1 and using the well-known weighted power mean inequality, we get

which completes the proof. □

The authors declare that they have no competing interests.

All authors conceived of the study, and participated in the design and coordination. All authors read and approved the final manuscript.

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