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REFINEMENTS OF HERMITE-HADAMARD TYPE INEQUALITIES FOR CONVEX FUNCTIONS VIA FRACTIONAL INTEGRALS†
REFINEMENTS OF HERMITE-HADAMARD TYPE INEQUALITIES FOR CONVEX FUNCTIONS VIA FRACTIONAL INTEGRALS†
Journal of Applied Mathematics & Informatics. 2015. Jan, 33(1_2): 119-125
Copyright © 2015, Korean Society of Computational and Applied Mathematics
  • Received : June 24, 2014
  • Accepted : September 25, 2014
  • Published : January 30, 2015
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RUIYIN XIANG

Abstract
In this note, two new mappings associated with convexity are propoesd, by which we obtain some new Hermite-Hadamard type inequalities for convex functions via Riemann-Liouville fractional integrals. We conclude that the results obtained in this work are the refinements of the earlier results. AMS Mathematics Subject Classification : 26D15, 26A51.
Keywords
1. Introduction
If f : I R is a convex function on the interval I , then for any a, b I with a b we have the following double inequality
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This remarkable result is well known in the literature as the Hermite-Hadamard inequality.
Since then, some refinements of the Hermite-Hadamard inequality for convex functions have been extensively obtained by a number of authors (e.g., [1] , [2] , [3] , [4] , [5] , [6] , [7] , [8] , [9] and [10] ).
In [4] , S. S. Dragomir proposed the following Hermite-Hadamard type in-equalities which refine the first inequality of (1).
Theorem 1.1 ( [4] ). Let f is convex on [ a, b ]. Then H is convex, increasing on [0, 1], and for all t ∈ [0, 1], we have
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where
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An analogous result for convex functions which refines the second inequality of (1) is obtained by G. S. Yang and M. C. Hong in [13] as follows.
Theorem 1.2 ( [13] ). Let f is convex on [ a, b ]. Then P is convex, increasing on [0, 1], and for all t ∈ [0, 1], we have
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where
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G. S. Yang and K. L. Tseng in [12] established some generalizations of (2) and (3) based on the following results.
Theorem 1.3 ( [12] ). Let f : [ a, b ] → R be a convex function , 0 < α < 1,
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and let h be defined by h ( t ) = (1 − β ) f ( A βt ) + βf ( A + (1 − β ) t ), t ∈ [0, u 0 ]. Then h is convex, increasing on [0, u 0 ] and for all t ∈ [0, u 0 ],
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It is remarkable that M. Z. Sarikaya et al. [11] proved the following interesting inequalities of Hermite-Hadamard type involving Riemann-Liouville fractional integrals.
Theorem 1.4 ( [11] ). Let f : [ a, b ] → R be a positive function with a < b and f L 1 [ a, b ]. If f is a convex function on [ a, b ], then the following inequalities for fractional integrals hold:
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with α > 0.
We remark that the symbols
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denote the left-sided and right-sided Riemann-Liouville fractional integrals of the order α ≥ 0 with a ≥ 0 which are defined by
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and
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respectively. Here, Г( α ) is the Gamma function defined by
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In this paper, we establish some new Hermite-Hadamard type inequalities for convex functions via Riemann-Liouville fractional integrals which refine the inequalities of (4).
2. Main results
Lemma 2.1. Let f : [ a, b ] → R be a convex function and h be defined by
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Then h ( t ) is convex, increasing on [0, b a ] and for all t ∈ [0, b a ],
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Proof . We can obtain the result by taking
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in Theorem 1.3. □
Theorem 2.2. Let f : [ a, b ] → R be a positive function with a < b and f L 1 [ a, b ]. If f is a convex function on [ a, b ], then WH is convex and monotonically increasing on [0, 1] and
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with α > 0, where
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Proof . Firstly, let t 1 , t 2 , β ∈ [0, 1], then
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Since f is convex, we get
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So
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from which we get W H is convex on [0, 1]. Next, by elementary calculus, we have
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It follows from Lemma 2.1 that
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is increasing on [0, b a ]. Since
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is nonnegative, hence W H ( t ) is increasing on [0, 1]. Finally, from
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and
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we have completed the proof. □
Similarly, we have the following theorem:
Theorem 2.3. Let f be defined as in Theorem 2.2, then W P is convex and monotonically increasing on [0, 1] and
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with α > 0, where
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Proof . We note that if f is convex and g is linear, then the composition f g is convex. Also we note that a positive constant multiple of a convex function and a sum of two convex functions are convex, hence
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are convex, from which we get that W P ( t ) is convex. Next, by elementary calculus, we have
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It follows from Lemma 2.1 that
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and k ( t ) = b a − (1 − t ) x are increasing on [0, b a ] and [0, 1], respectively. Hence
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is increasing on [0, 1]. Since
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is nonnegative, it follows that W P is monotonically increasing on [0, 1]. Finally, from
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and
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we get the desired result. □
Corollary 2.4. With assumptions in Theorem 2.2, if α = 1, we get
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where H ( t ) is defined as Theorem 1.1, which is just the result in Theorem 1.1.
Corollary 2.5. With assumptions in Theorem 2.3, if α = 1, we get
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where P ( t ) is defined as Theorem 1.2, which is just the result in Theorem 1.2.
3. Conclusion
In this note, we obtain some new Hermite-Hadamard type inequalities for convex functions via Riemann-Liouville fractional integrals. We conclude that the results obtained in this work are the refinements of the earlier results. An interesting topic is whether we can use the methods in this paper to establish the Hermite-Hadamard inequalities for convex functions on the co-ordinates via Riemann-Liouville fractional integrals.
Acknowledgements
This work is supported by Youth Project of Chongqing Three Gorges University of China (No.13QN11).
BIO
Ruiyin Xiang received M.Sc. from Chongqing University. Since 1995 he has been teaching at Chongqing Three Gorges University. His research interests include integral inequalities on several kinds of convex functions and applications.
School of Mathematics and Statistics, Chongqing Three-Gorges University, Wanzhou, Chongqing, 404000, P.R.China.
e-mail: xiangry@163.com
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