Integral inequalities under beta function and preinvex type functions
© Ahmad. 2016
Received: 12 December 2015
Accepted: 15 April 2016
Published: 26 April 2016
In the present paper, the notion of P-preinvex function is introduced and new integral inequalities for this kind of function along with beta function are establised. The work extends the results appeared in the literature.
KeywordsEuler beta function Integral inequality Holder’s inequality P-preinvex function
Mathematics Subject Classification33B15 26B25 26D15 26D10
Convexity plays an important role in economics, management science, engineering, finanace and optimization theory. Many interesting generalizations and extensions of classical convexity have been used in optimization and mathematical inequalities. Hanson (1981) introduced the concept of invexity. These functions were named invex by Craven (1981) and \(\eta\)-convex by Kaul and Kaur (1980). Weir and Mond (1988) introduced the concept of preinvex function. Later, Mohan and Neogy (1995) presented few properties of preinvex functions. Some refinements of the mathematical inequalities on convex and generalized convex functions have been investigated in Barani et al. (2012), Chalco-Cano et al. (2012), Dragomir (2001), Dragomir and Agarwal (1998), Fok and Vong (2015), Matloka (2014), Muddassar and Bhatti (2013) and Pachpatte (2004).
Let S be a nonempty subset of \(R^n\) and let \(\eta : S \times S \rightarrow R^n.\)
It is obvious that every convex set is invex with respect to \(\eta (u, v) = u - v,\) but there exist invex sets which are not convex (see Mohan and Neogy 1995).
Recently, Liu (2014) obtained several integral inequalities for the left hand side of (1) under the following P-convexity:
The main purpose of this paper is to introduce the class of P-preinvex function and derive new inequalities for the left hand side of (1) under these assumptions. The presented results generalize the results of Liu (2014) and references cited therein.
New integral inequalities
Note that every P-convex function (Liu 2014) is a P-preinvex function with respect to \(\eta (u,v) = u - v\) for any \(t \in [0, 1].\)
The following definition will be used in the sequel.
Intergal inequalities involving prequasi-invex
I state the following theorems as the proof follow on the same lines of the theorems of “New integral inequalities” section.
In this paper, I have introduced the P-preinvex function and used it along with beta function to establish the new integral type inequalities. I also stated the other integral type inequalities under prequasi-invex function. The presented results may be futher generalized under weaker convexity assumptions.
This research is supported by King Fahd University of Petroleum and Minerals, Saudi Arabia under the Internal Project No. IN131038. The author is thankful to the anonymous referees for their valuable suggestions, which have substantially improved the presentation of the paper.
The author declares that he has no competing interests.
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