Hermite-Hadamard type inequalities for n-times differentiable and preinvex functions
© Wang and Qi; licensee Springer. 2014
Received: 29 August 2013
Accepted: 8 January 2014
Published: 30 January 2014
In the paper, by creating an integral identity involving an n-times differentiable function, the authors establish some new Hermite-Hadamard type inequalities for preinvex functions and generalize some known results.
MSC:26D15, 26A51, 26B12, 41A55, 49J52.
Throughout this paper, let and ℕ denote the set of all positive integers.
Let us recall some definitions of various convex functions.
holds for all and . If the inequality (1) reverses, then f is said to be concave on I.
Definition 2 
A set is said to be invex with respect to the map , if for every and .
It is obvious that every convex set is invex with respect to the map , but there exist invex sets which are not convex. See , for example.
Definition 3 
Definition 4 
Let be an invex set with respect to . A function is said to be preinvex with respect to η, if for every and .
If f is concave on , then the inequality (3) is reversed.
The inequality (3) has been generalized by many mathematicians. Some of them may be recited as follows.
Theorem 1 [, Theorem 2.2]
Theorem 2 [, Theorem 1]
Theorem 3 [, Theorem 2.3]
Theorem 4 [, Theorem 2.1]
Theorem 5 [, Theorem 4.1]
Theorem 6 [, Theorem 4.3]
Recently, some related inequalities for preinvex functions were also obtained in [7, 8]. Some integral inequalities of Hermite-Hadamard type for other kinds of convex functions were also established in [9–16] and references cited therein.
In this paper, by creating an integral identity involving an n-times differentiable function, the authors will establish some new Hermite-Hadamard type inequalities for preinvex functions and generalize some of the above mentioned results.
2 A lemma
In order to obtain our main results, we need the following lemma.
where the above summation is zero for .
Hence, the identity (1) holds for .
When and , suppose that the identity (1) is valid.
Therefore, when , the identity (1) holds. By induction, the proof of Lemma 1 is complete. □
which may be found in .
3 Hermite-Hadamard type inequalities for preinvex functions
Now we start out to establish some new Hermite-Hadamard type inequalities for n-times differentiable and preinvex functions.
Theorem 7 is thus proved. □
- 1.if , then
if and , then the inequality (8) is valid.
Theorem 8 is thus proved. □
The proof of Theorem 9 is complete. □
The proof of Theorem 10 is complete. □
The authors appreciate anonymous referees for their valuable comments on and careful corrections to 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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