1. Introduction

In this paper, by using the some classical integral inequalities, Hölder and Power-Mean integral inequality, we establish some new inequalities for functions whose nth derivatives in absolute value are s-convex functions in the first sense. For some inequalities, generalizations and applications concerning convexity see [1-11]^[1]. Recently, in the literature there are so many papers about n-times differentiable functions on several kinds of convexities and s-convex functions. In references ^{[5, 6, 7, 8]}, readers can find some results about this issue. Many papers have been written by a number of mathematicians concerning inequalities for different classes of convex and s-convex functions in the first sense see for instance the recent papers [12-19]^[12] and the references within these papers. There are quite substantial literatures on such problems. Here we mention the results of [1-19]^[1] and the corresponding references cited therein.

Definition 1.1: A function is said to be convex if the inequality

is valid for all and . If this inequality reverses, then is said to be concave on interval . This definition is well known in the literature.

Definition 1.2: A function is said to be -convex in the first sense if

holds for all , with and for some fixed . It can be easily seen that every 1-convex function is convex.

In ^[21], Dragomir and Fitzpatrick proved a variant of Hadamard’s inequality which holds for -convex mapping in the first sense.

Definition 1.3: The following double inequality is well-known in the literature as Hadamard’s inequality for convex mappings ^{[8, 10]}: Let be a convex mapping defined on the interval in and with . Then the following inequality holds:

Throughout this paper we will use the following notations and conventions. Let , and with and and

be the arithmetic and generalized logarithmic mean for respectively.

We will use the following Lemma ^[20] for we obtain the main results:

Lemma 1.1: Let be -times differentiable mapping on for and , where with , we have the identity

where an empty sum is understood to be nil.

2. Main Results

Proof. If for is -convex function in the first sense on , using Lemma1.1, the Hölder integral inequality and

we have

This completes the proof of theorem.

Corollary 2.1. Under the conditions of Theorem 2.1 for , we obtain the following inequality

which coincide with the Theorem 2.1 in ^[20].

Proposition 2.1. Under the conditions of Corollary 2.1 for , we obtain the following:

Proposition 2.2. For , we obtain the following inequality:

Theorem 2.2. For ; let be n -times differentiable function on and with . If for is -convex function in the first sense on , then the following inequality holds:

Where and , .

This completes the proof of theorem.

Corollary 2.2. Under the conditions of Theorem 2.2 for , we obtain the inequality

Proposition 2.3. Under the conditions of Corollary 2.2 for , we obtain the inequality

Proposition 2.4. For , we obtain the following inequality

Theorem 2.3. For ; let be n -times differentiable function on and with . If and for is -convex function in the first sense on , then the following inequality holds:

Where

Proof: If for is -convex function the first sense on , using Lemma1.1 and the Hölder integral inequality, we have the following inequality:

Corollary 2.3. Under the conditions of Theorem 2.3 for we obtain the inequality

Proposition 2.5. Under the conditions of Corollary 2.3 for we obtain the inequality

Proposition 2.6. For , we obtain the following inequality

Theorem 2.4. For ; let be n -times differentiable function on and with . If and for is -concave function in the first sense on , then the following inequality holds:

Proof: If for is -concave function the first sense on , using Lemma1.1, the Hermite-Hadamard inequality and the Hölder integral inequality, we have the following inequality:

Corollary 2.4. Under the conditions of Theorem 2.4 for , we obtain the inequality

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