On Hermite–Hadamard-Type Inequalities for Functions Satisfying Second-Order Differential Inequalities

: Hermite–Hadamard inequality is a double inequality that provides an upper and lower bounds of the mean (integral) of a convex function over a certain interval. Moreover, the convexity of a function can be characterized by each of the two sides of this inequality. On the other hand, it is well known that a twice differentiable function is convex, if and only if it admits a nonnegative second-order derivative. In this paper, we obtain a characterization of a class of twice differentiable functions (including the class of convex functions) satisfying second-order differential inequalities. Some special cases are also discussed


Introduction
Inequalities involving convex functions are very useful in many branches of mathematics.The Hermite-Hadamard inequality is the one of the most important inequality for convex functions.This inequality provides an upper and lower bounds of the mean of a convex function over a certain interval.It is mostly used in mathematics to study the properties of convex functions and their applications in optimization and approximation theory, see, e.g., [1][2][3].
A real-valued function f defined in an interval I is convex if: for every ι ∈ [0, 1] and y, z ∈ I.If f is twice differentiable, then f is convex, if and only if its second derivative is nonnegative.The Hermite-Hadamard inequality can be stated as follows: Let f be a real-valued convex function in an interval I.Then, for all x, y ∈ I with x < y, we have: Many generalizations and extensions of (1) can be found in the literature.For instance, Dragomir and Agarwal [2] studied the following class of functions: They proved that, if f ∈ F , then: Some improvements and extensions of the above result have been obtained by some authors, see, e.g., [4][5][6][7].Other extensions of (1) to various classes of functions have been obtained: s-convex functions [8][9][10][11], log-convex functions [12][13][14], h-convex functions [15,16], and m-convex functions [17][18][19][20].For other classes of functions, we refer to [21][22][23][24] and the references therein.Some extensions of Hermite-Hadamard inequality to a higher dimension can be found in [25][26][27][28][29].
It is interesting to notice that each of the two sides of (1) provides a characterization of convex functions.Namely, if f is a real valued continuous function in an interval I, then the following statements are equivalent: (i) f is convex; (ii) For all x, y ∈ I with x < y: (iii) For all x, y ∈ I with x < y: The proof of the implication (ii) =⇒ (i) can be found in ( [30], p. 98).For the proof of the implication (iii) =⇒ (i), we refer to (Problem Q, [31], p. 15).On the other hand, one can check easily that (iii) is equivalent to: for all x, y ∈ I with x < y, where: Observe that H is the unique (nonnegative) solution to the boundary value problem: From the above remarks, we deduce that, if f is twice differentiable in I, then f ≥ 0 (i.e., f is convex), if and only if (3) holds for all x, y ∈ I with x < y.Thus, (3) provides a characterization of twice differentiable functions in I, having a nonnegative second derivative.
Motivated by the above discussion, our aim in this paper is to obtain a characterization of the class of twice continuously differentiable functions f in I, satisfying second-order differential inequalities of the form: where α is twice continuously differentiable in I and β, γ are continuous in I.We shall assume that for all x, y ∈ I with x < y, there exists a unique nonnegative solution H to the boundary value problem: The rest of the paper is organized as follows.Section 2 is devoted to the main results and their proofs.Namely, we establish a characterization of the class of functions f satisfying differential inequalities of the form (4).In Section 3, we discuss some special cases of (4).

Main Results
For any interval J of R, by C n (J), where n ≥ 0 is a natural number, we mean the space of n-continuously differentiable functions in J.
Let I be an open interval of R. Let α ∈ C 1 (I) and β, γ ∈ C(I).Throughout this section, it is assumed that for all x, y ∈ I with x < y, there exists a unique nonnegative solution We are concerned with the class of functions f ∈ C 2 (I) satisfying the second-order differential inequality: Our main result, which is stated below, provides a characterization of this class of functions.
Replacing f by − f and γ by −γ in Theorem 1, we obtain the following result.
Theorem 2. Let α ∈ C 1 (I), β, γ ∈ C(I) and f ∈ C 2 (I).The following statements are equivalent: (ii) For all x, y ∈ I with x < y, it holds that: From Theorem 1, we deduce the following result.
Similarly, from Theorem 2, we deduce the following result.
then for all x, y ∈ I with x < y, we have: where H 1 (resp.H 2 ) is the unique nonnegative solution to (13) (resp.(14)) and H is defined by (15).
Corollary 3. Let f ∈ C 2 (I) be a convex function.Then, for all x, y ∈ I with x < y, we have: Proof.Taking: in Corollary 1, we obtain: Then, by (12), we obtain (18).
Similarly, from Corollary 2, we deduce the following result.
Corollary 4. Let f ∈ C 2 (I) be a concave function.Then, for all x, y ∈ I with x < y, we have

Applications
In this section, some special cases of Theorems 1 and 2 are discussed.Namely, we provide characterizations of various classes of functions satisfying differential inequalities of type (6).We first consider the classes of functions: and where ∈ R is a constant.Observe that for = 0, F + 0 reduces to the class of twice continuously differentiable convex functions, while F − 0 reduces to the class of twice continuously differentiable concave functions.We recall that in [29], Niculescu and Persson proved that, if f ∈ F + , then for all x, y ∈ I with x < y, it holds that: Furthermore, if f ∈ F − , then for all x, y ∈ I with x < y, it holds that: In this section, we show that (21) (resp.( 22)) provides a characterization of the class of functions F + (resp.F − ).We next consider the classes of functions and where λ > 0. Observe that when λ = 0, G + 0 reduces to the class of twice continuously differentiable convex functions, while G − 0 reduces to the class of twice continuously differentiable concave functions.

Characterizations of the Classes of Functions F ±
Let I be an open interval of R. Let ∈ R. The following result provides a characterization of the class of functions F + defined by (19).Proof.Observe that: Hence, by Theorem 1, f ∈ F + , if and only if, for all x, y ∈ I with x < y, it holds that: where is the unique (nonnegative) solution to the boundary value problem: On the other hand, for all x, y ∈ I with x < y, we have: which shows that ( 25) is equivalent to (21).
Similarly, using Theorem 2 (or replacing f by − f and by − in Corollary 5), we obtain the following characterization of the class of functions F − defined by (20).
The following result provides a characterization of the class of functions G + λ defined by (23).(α(z)H (z)) + β(z)H(z) = −1, x < z < y, H(x) = H(y) = 0 admits a unique nonnegative solution H.We show that the considered differential inequality is equivalent to: We also discussed some special cases of α, β and γ, and provided some characterizations in those cases.
In this work, only second-order differential inequalities are investigated.It would be interesting to show whether it is possible to obtain a characterization of functions f satisfying higher-order differential inequalities.For instance, the class of functions f satisfying f ≥ 0 in I deserves to be studied.

Corollary 6 .Lemma 1 .,
Let f ∈ C 2 (I).The following statements are equivalent: (i) f ∈ F − ; (ii) For all x, y ∈ I with x < y, (22) holds.3.2.Characterizations of the Classes of Functions G ± λLet I be an open interval of R and λ > 0. We first need the following lemma.Its proof is elementary; we omit the details.For all x, y ∈ I with x < y, the following boundary value problem:H (z) − λH(z) = −1, x < z < y, H(x) = H(y) = 0admits a unique nonnegative solution given by: x ≤ z ≤ y.