Integral inequalities of Hermite-Hadamard type for functions whose derivatives are α-preinvex
© Wang et al.; licensee Springer. 2014
Received: 16 October 2013
Accepted: 13 February 2014
Published: 25 February 2014
In the article, the authors introduce a new notion, ‘α-preinvex function’, establish an integral identity for the newly introduced function, and find some Hermite-Hadamard type integral inequalities for a function of which the power of the absolute value of the first derivative is α-preinvex.
MSC: 26D15, 26A51, 26B12, 41A55.
Let us recall some definitions of various convex functions.
holds for all and .
Definition 2 ()
is valid for all and , then we say that is an m-convex function on .
Definition 3 ()
is valid for all and , then we say that is an -convex function on .
Definition 5 ()
Definition 6 ()
Let us reformulate some inequalities of Hermite-Hadamard type for the above mentioned convex functions.
Theorem 1 ([, Theorem 2.2])
Theorem 3 ([, Theorem 3.1])
Theorem 4 ([, Theorem 2.1])
In this article, we will introduce a new notion ‘α-preinvex function’, establish an integral identity for such a kind of functions, and find some Hermite-Hadamard type integral inequalities for a function that the power of the absolute value of its first derivative is α-preinvex.
2 A new definition and a lemma
The so-called ‘α-preinvex function’ may be introduced as follows.
Remark 1 If and is an α-preinvex function, then is a preinvex function.
For establishing our new integral inequalities of Hermite-Hadamard type for α-preinvex functions, we need the following integral identity.
The proof of Lemma 1 is completed. □
3 Some new integral inequalities of Hermite-Hadamard type
We are now in a position to establish some Hermite-Hadamard type integral inequalities for a function that the power of the absolute value of its first derivative is α-preinvex.
The proof of Theorem 5 is completed. □
The proof of Theorem 6 is complete. □
Substituting (14) and (15) into (13) results in (12). The proof of Theorem 7 is complete. □
The proof of Theorem 8 is complete. □
The proof of Theorem 9 is complete. □
The authors appreciate the anonymous referees for their helpful corrections to and valuable comments on the original version of this paper. This work was partially supported by the NNSF under Grant No. 11361038 of China and by the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region under Grant No. NJZY13159, China.
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