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

Mahir Kadakal*, Huriye Kadakal, İmdat İşcan

Department of Mathematics, Faculty of Sciences and Arts, Giresun University-Giresun-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

Citation: 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, 5 (2). 63-68. January 2017. doi:10.12691/tjant-5-2-4

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]. 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] and the references within these papers. There are quite substantial literatures on such problems. Here we mention the results of [1-19] 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

References

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[2]   | S.-P. Bai, S.-H. Wang And F. Qi, “Some Hermite-Hadamard type inequalities for n-time differentiable (a,m) -convex functions”, Jour. of Ineq. and Appl., 2012, 2012: 267.
[3]   | P. Cerone, S. S. Dragomir and J. Roumeliotis, “Some Ostrowski type inequalities for n-time differentiable mappings and applications”, Demonstratio Math., 32 (4) (1999), 697-712.
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[5]   | S. S. Dragomir and C. E. M. Pearce, “Selected Topics on Hermite-Hadamard Inequalities and Applications”, RGMIA Monographs, Victoria University, 2000.
[6]   | D.-Y. Hwang, “Some Inequalities for n-time Differentiable Mappings and Applications”, Kyung. Math. Jour., 43 (2003), 335-343.
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[16]   | A. Saglam, M. Z. Sarıkaya and H. Yıldırım, “Some new inequalities of Hermite-Hadamard’s type”, Kyung. Math. Jour., 50 (2010), 399-410.
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[19]   | B.-Y. Xi and F. Qi, “Some integral inequalities of Hermite-Hadamard type for convex functions with applications to means”, J. Funct. Spaces Appl., (2012).
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