Better Approaches for n -Times Differentiable Convex Functions

: In this work, by using an integral identity together with the Hölder–˙I¸scan inequality we establish several new inequalities for n -times differentiable convex and concave mappings. Furthermore, various applications for some special means as arithmetic, geometric, and logarithmic are given.


Introduction
Definition 1. A function f : I ⊆ R → R is said to be convex if the inequality is valid for all x, y ∈ I and t ∈ [0, 1]. If this inequality reverses, then f is said to be concave on interval I = ∅.
The above is a well known definition in the literature. Convexity theory has been appeared as a powerful technique to study a wide class of unrelated problems in pure and applied sciences (see, for example [1][2][3][4]). Recently, in the literature many researchers contributed their research on n-times differentiable functions on several kinds of convexities (see, for example [1][2][3][5][6][7][8]) and the references within these papers.
The classical Hermite-Hadamard inequality provides estimates of the mean value of a continuous convex or concave function.
Definition 2. f : I ⊆ R → R be a concave function on the interval I of real numbers and a, b ∈ I with a < b. The inequality f (a) is known in the literature as a Hermite-Hadamard's inequality for convex functions. Both inequalities holds if f is concave.
A refinement of Hölder integral inequality better approach than Hölder integral inequality can be given as follows: Theorem 1 (Hölder-İşcan Integral Inequality [13]). Let p > 1 and 1 In this paper, by using the Hölder-İşcan integral inequality (better approach than Hölder integral inequality) and together with an integral identity, we present a rather generalization of Hadamard type inequalities for functions whose derivative is absolute value at the certain power are convex and concave.
Let 0 < a < b, and throughout this paper we will use for the arithmetic, geometric, generalized logarithmic mean, respectively.

Main Results
We will use the following Lemma to obtain our main results.

Lemma 1 ([8]
). Let f : I ⊆ R → R be n-times differentiable mapping on I • for n ∈ N and f (n) ∈ L [a, b], where a, b ∈ I • with a < b, we have the identity where an empty sum is understood to be nil.
Theorem 2. For n ∈ N; let f : I ⊆ (0, ∞) → R be n-times differentiable function on I • and a, b ∈ I • with a < b. If f (n) ∈ L [a, b] and f (n) q for q > 1 is convex on interval [a, b], then the following inequality holds Proof. If f (n) q for q > 1 is convex on interval [a, b], then by using Lemma 1, the Hölder-İşcan integral inequality and from the following inequality This completes the proof of Theorem 2.

Corollary 1.
Under the conditions of Theorem 2 for n = 1 we have the following inequality: Proof. Under the assumption of the Proposition 1, let is convex on (0, ∞) and the result follows directly from Theorem 2.
Example 1. If we take m = 2, n = 1, p = q = 2 in the inequality (4), then we have the following inequality: Proposition 2. Let a, b ∈ (0, ∞) with a < b, q > 1 and n ∈ N, then we have Proof. Under the assumption of the Proposition 2, let f (x) = ln x, x ∈ (0, ∞). Then is convex on (0, ∞) and the result follows directly from Theorem 2.
Example 2. If we take n = 1, p = q = 2 in the inequality (4), then we have the following inequality: Proof. Under the assumption of the Proposition 3, let f (x) = q m+q x m q +1 , x ∈ (0, ∞). Then is convex on (0, ∞) and the result follows directly from Corollary 1.

Example 3.
If we take m = 2, p = q = 2 in the inequality (5), then we have the following inequality: Theorem 3. For n ∈ N; let f : I ⊆ (0, ∞) → R be n-times differentiable function on I • and a, b ∈ I • with , then the following inequality holds Proof. Since f (n) q for q > 1 is convex on interval [a, b], by using Lemma 1 and the Hölder-İşcan integral inequality, we obtain the following inequality: This completes the proof of Theorem 3, after a little simplifications.
The inequality (6) gives better results than the inequality (7). Indeed, using the inequality which shows that the inequality (6) gives better results than the inequality (7).

Corollary 2.
Under the conditions of Theorem 3 for n = 1, we have the following inequality: Proposition 4. Let a, b ∈ (0, ∞) with a < b, q > 1 and m, n ∈ N with m ≥ n, then we have Proof. Under the assumption of the Proposition 4, let f (x) = x m , x ∈ (0, ∞). Then is convex on (0, ∞) and the result follows directly from Theorem 3, respectively.
Proposition 5. Let a, b ∈ (0, ∞) with a < b, q > 1 and n ∈ N, then we have Proof. Under the assumption of the Proposition 5, let f (x) = ln x, x ∈ (0, ∞). Then is convex on (0, ∞) and the result follows directly from Theorem 3.
Proof. The result follows directly from Corollary 2 for the function This completes the proof of Proposition.

Corollary 3.
For m = 1 from Proposition 6, we obtain the following inequality: Theorem 4. For n ∈ N; let f : I ⊂ (0, ∞) → R be n-times differentiable function on I • and a, b ∈ I • with a < b. If f (n) ∈ L [a, b] and f (n) q for q > 1 is concave on [a, b] , then the following inequality holds

Proof.
Since the function f (n) q for q > 1 is concave on interval [a, b], with respect to Hermite-Hadamard integral inequality, we get and thus we have Proposition 8. Let a, b ∈ (0, ∞) with a < b, q > 1 and n ∈ N, then we have Proof. Under the assumption of the Proposition 8, let f (x) = ln x, x ∈ (0, ∞). Then is convex on (0, ∞) and the result follows directly from the Theorem 4.

Conclusions
By using an integral identity together with the Hölder-İşcan integral inequality (which is a better approach than Hölder integral inequality), we obtain several new inequalities for n-times differentiable convex and concave mappings. We would like to emphasize that some new integral inequalities can be obtained by using a similar method to different types of convex functions.