Some weighted integral inequalities for differentiable preinvex and prequasiinvex functions with applications
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RESEARCH
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Some weighted integral inequalities for differentiable preinvex and prequasiinvex functions with applications Muhammad Amer Latif1* and Sever Silvestru Dragomir1,2 * Correspondence: [email protected] 1 School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa Full list of author information is available at the end of the article
Abstract In this paper, we present weighted integral inequalities of Hermite-Hadamard type for differentiable preinvex and prequasiinvex functions. Our results, on the one hand, give a weighted generalization of recent results for preinvex functions and, on the other hand, extend several results connected with the Hermite-Hadamard type integral inequalities. Applications of the obtained results are provided as well. MSC: 26D15; 26D20; 26D07 Keywords: Hermite-Hadamard’s inequality; invex set; preinvex function; prequasiinvex; Hölder’s integral inequality; power-mean inequality
1 Introduction Let f : I ⊆ R → R be a convex mapping and a, b ∈ I with a < b. Then
a+b f
≤ b–a
a
b
f (x) dx ≤
f (a) + f (b) .
(.)
Both the inequalities in (.) hold in reversed direction if f is concave. Inequalities (.) are famous in mathematical literature due to their rich geometrical significance and applications and are known as the Hermite-Hadamard inequalities (see []). For several results which generalize, improve and extend inequalities (.), we refer the interested reader to [–]. In [], Dragomir and Agarwal obtained the following inequalities for differentiable functions which estimate the difference between the middle and the rightmost terms in (.). Theorem [] Let f : I ⊆ R → R be a differentiable mapping on I ◦ , where a, b ∈ I with a < b, and f ∈ L([a, b]). If |f | is a convex function on [a, b], the following inequality holds: b b – a f (a) + f (b) f (a) + f (b) . – f (x) dx ≤ b–a a
(.)
Theorem [] Let f : I ⊆ R → R be a differentiable mapping on I ◦ , where a, b ∈ I with p a < b, and f ∈ L([a, b]). If |f | p– is a convex function on [a, b], the following inequality ©2013 Latif and Dragomir; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Latif and Dragomir Journal of Inequalities and Applications 2013, 2013:575 http://www.journalofinequalitiesandapplications.com/content/2013/1/575
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holds: b p f (a) + f (b) p b – a p– – f (x) dx ≤ + f (b) p– , f (a) b–a a (p + ) p where p > and
p
+
q
(.)
= .
In [], Pearce and Pečarić gave an improvement and simplification of the constant in Theorem and consolidated these results with Theorem . The following is the main result from []. Theorem [] Let f : I ⊆
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