Refinements of Hermite-Hadamard Inequalities for Functions When a Power of the Absolute Value of the Second Derivative Is P-Convex
Abstract
We extend some estimates of the right-hand side of Hermite-Hadamard-type inequalities for functions whose second derivatives absolute values are P-convex. Applications to some special means are considered.
1. Introduction
It is well known that the Hermite-Hadamard inequality plays an important role in nonlinear analysis. Over the last decade, this classical inequality has been improved and generalized in a number of ways; there have been a large number of research papers written on this subject, (see, [2–13]) and the references therein. In [13] Dragomir and Agarwal established the following results connected with the right-hand side of (1.1) as well as applied them for some elementary inequalities for real numbers and numerical integration.
Theorem 1.1. Assume that a, b ∈ ℝ with a < b and f : [a, b] → ℝ is a differentiable function on (a, b). If |f′| is convex on [a, b], then the following inequality holds:
Theorem 1.2. Assume that a, b ∈ ℝ with a < b and f : [a, b] → ℝ is a differentiable function on (a, b). Assume p ∈ ℝ with p > 1. If |f′|p/(p−1) is convex on [a, b], then the following inequality holds:
In [1] Pearce and Pečarić proved the following theorem.
Theorem 1.3. Let f : I → ℝ be a differentiable function on I∘, a, b ∈ I∘ with a < b. If is convex on [a, b], for q ≥ 1, then the following inequality holds:
The generalizations of the Theorems 1.1 and 1.2 are introduced by Ion in [14] for quasiconvex functions and are given in [6] to differentiable P-convex functions. Then, Alomari et al. in [2] improved the results in [14] and Theorem 1.3, for twice differentiable quasiconvex functions.
On the other hand, Dragomir et al. in [11] defined the following class of functions.
Definition 1.4. Let I⊆ℝ be an interval. The function f : I → ℝ is said to be P-convex (or belong to the class P(I)) if it is nonnegative and, for all x, y ∈ I and λ ∈ [0,1], satisfies the inequality
Note that P(I) contain all nonnegative convex and quasiconvex functions. Since then numerous articles have appeared in the literature reflecting further applications in this category, see [3, 6, 12, 15, 16] and references therein. Ozdemir and Yildiz in [15] proved the following results.
Theorem 1.5. Let f : I → ℝ be a twice differentiable function on I∘ and a, b ∈ I∘ with a < b. If |f′′| is P-convex, 0 ≤ λ ≤ 1, then the following inequality holds:
Corollary 1.6. If in Theorem 1.5 one chooses λ = 1, one obtains
Theorem 1.7. Let f : I → ℝ be a twice differentiable function on I∘ and a, b ∈ I∘ with a < b. If |f′′|q is P-convex, 0 ≤ λ ≤ 1 and q ≥ 1, then the following inequality holds:
Corollary 1.8. If in Theorem 1.7 one chooses λ = 1, one obtains
The main purpose of this paper is to establish the refinements of results in [15]. Applications for special means are considered.
2. Main Results
In order to prove our main theorems, we need the following Lemma in [5] throughout this paper.
Lemma 2.1. Suppose that f : I → ℝ is a twice differentiable function on I∘, the interior of I. Assume that a, b ∈ I∘, with a < b and f′′, is integrable on [a, b]. Then, the following equality holds:
In the following theorem, we will propose some new upper bound for the right-hand side of (1.1) for P-convex functions, which is better than the inequality had done in [15].
Theorem 2.2. Let f : I → ℝ be a twice differentiable function on I∘ such that |f′′| is a P-convex function on I. Suppose that a, b ∈ I∘ with a < b and f′′ ∈ L1[a, b]. Then, the following inequality holds:
Proof. Since |f′′| is a P-convex function, by using Lemma 2.1 we get
An immediate consequence of Theorem 2.2 is as follows.
Corollary 2.3. Let f be as in Theorem 2.2, if in addition
(i) f′′((a + b)/2) = 0, then one has
(ii) f′′(a) = f′′(b) = 0, then one has
The corresponding version for powers of the absolute value of the second derivative is incorporated in the following theorem.
Theorem 2.4. Let f : I → ℝ be a differentiable function on I∘. Assume that p ∈ ℝ, p > 1 such that is a P-convex function on I. Suppose that a, b ∈ I∘ with a < b and f′′ ∈ L1[a, b]. Then, the following inequality holds:
Proof. By assumption, Lemma 2.1 and Hölder′s inequality, we have
The following corollary is an immediate consequence of Theorem 2.4.
Corollary 2.5. Let f be as in Theorem 2.4, if in addition
(i) f′′((a + b)/2) = 0, then one has
(ii) f′′(a) = f′′(b) = 0, then one has
Another similar result may be extended in the following theorem.
Theorem 2.6. Let f : I → ℝ be a differentiable function on I∘. Assume that q ≥ 1 such that is a P-convex function on I. Suppose that a, b ∈ I∘ with a < b and f′′ ∈ L1[a, b]. Then, the following inequality holds:
Proof. Suppose that a, b ∈ I∘. From Lemma 2.1 and using well-known power mean inequality, we get
3. Applications to Special Means
Now, we consider the applications of our theorems to the special means. We consider the means for arbitrary real numbers α, β (α ≠ β). We take the following
(1) Arithmetic mean:
(2) Logarithmic mean:
(3) Generalized log-mean:
Now, using the results of Section 2, we give some applications for special means of real numbers.
Proposition 3.1. Let a, b ∈ ℝ, a < b, and n ∈ ℕ, n ≥ 2. Then, one has
Proof. The assertion follows from Theorem 2.2 applied to the P-convex function f(x) = xn, x ∈ ℝ.
Proposition 3.2. Let a, b ∈ ℝ, a < b, and 0 ∉ [0,1]. Then, for all p > 1 one has
Proof. The assertion follows from Theorem 2.4 applied to the P-convex function f(x) = 1/x, x ∈ [a, b].
Proposition 3.3. Let a, b ∈ ℝ, a < b, and n ∈ ℕ, n ≥ 2. Then, for all q ≥ 1 one has
Proof. The assertion follows from Theorem 2.6 applied to the P-convex function f(x) = xn, x ∈ ℝ.
