**Turkish Journal of Analysis and Number Theory**

##
Some New Integral Inequalities for *n*-times Differentiable *s*- Convex functions in the first sense

**Mahir Kadakal**^{1,}, **Huriye Kadakal**^{2}, **İmdat İşcan**^{1}

^{1}Department of Mathematics, Faculty of Sciences and Arts, Giresun University-Giresun-TÜRKİYE

^{2}Institute of Science, Ordu University-Ordu-TÜRKİYE

### Abstract

In this work, by using an integral identity together with both the Hölder and the Power-Mean integral inequality we establish several new inequalities for *n*-time differentiable -convex functions in the first sense.

**Keywords:** convex function, - convex function in the first sense, hölder integral inequality and power-mean integral inequality

Received January 03, 2017; Revised February 04, 2017; Accepted February 13, 2017

**Copyright**© 2017 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- Mahir Kadakal, Huriye Kadakal, İmdat İşcan. Some New Integral Inequalities for
*n*-times Differentiable*s*- Convex functions in the first sense.*Turkish Journal of Analysis and Number Theory*. Vol. 5, No. 2, 2017, pp 63-68. http://pubs.sciepub.com/tjant/5/2/4

- Kadakal, Mahir, Huriye Kadakal, and İmdat İşcan. "Some New Integral Inequalities for
*n*-times Differentiable*s*- Convex functions in the first sense."*Turkish Journal of Analysis and Number Theory*5.2 (2017): 63-68.

- Kadakal, M. , Kadakal, H. , & İşcan, İ. (2017). Some New Integral Inequalities for
*n*-times Differentiable*s*- Convex functions in the first sense.*Turkish Journal of Analysis and Number Theory*,*5*(2), 63-68.

- Kadakal, Mahir, Huriye Kadakal, and İmdat İşcan. "Some New Integral Inequalities for
*n*-times Differentiable*s*- Convex functions in the first sense."*Turkish Journal of Analysis and Number Theory*5, no. 2 (2017): 63-68.

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### 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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