On Hadamard-Type Inequalities Involving Several Kinds of Convexity
© Erhan Set et al. 2010
Received: 14 May 2010
Accepted: 23 August 2010
Published: 26 August 2010
where is a convex function on the interval of real numbers and with This inequality is one of the most useful inequalities in mathematical analysis. For new proofs, note worthy extension, generalizations, and numerous applications on this inequality; see ([1–6]) where further references are given.
(see , Page 1). Geometrically, this means that if , and are three distinct points on the graph of with between and , then is on or below chord
It is said to be log-concave if the inequality in (1.3) is reversed.
In , Toader defined -convexity as follows.
In , Miheşan defined -convexity as in the following:
Denote by the class of all -convex functions on for which . It can be easily seen that for -convexity reduces to -convexity and for , -convexity reduces to the concept of usual convexity defined on , .
In , Gill et al. established the following results.
For some recent results related to the Hadamard's inequalities involving two -convex functions, see  and the references cited therein. The main purpose of this paper is to establish the general version of inequalities (1.7) and new Hadamard-type inequalities involving two -convex functions, two -convex functions, or two -convex functions using elementary analysis.
2. Main Results
We start with the following theorem.
the theorem is proved.
Combining (2.16), we get the desired inequalities (2.10). The proof is complete.
Combining (2.21), we get the required inequalities (2.17). The proof is complete.
Rewriting (2.27) and (2.28), we get the required inequality in (2.22). The proof is complete.
Rewriting (2.34) and (2.35), we get the required inequality in (2.29). The proof is complete.
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