- Open Access
Integral inequalities of Hermite-Hadamard type for functions whose third derivatives are convex
© Chun and Qi; licensee Springer. 2013
- Received: 30 November 2012
- Accepted: 26 September 2013
- Published: 7 November 2013
In the paper, the authors establish some new inequalities of Hermite-Hadamard type for functions whose third derivatives are convex.
MSC: 26D15, 26A51, 41A55.
- Hermite-Hadamard’s integral inequality
- convex function
- third derivative
holds. This double inequality is known in the literature as Hermite-Hadamard’s integral inequality for convex functions. The definition of convex functions and Hermite-Hadamard’s integral inequality (1.2) have been generalized, refined, and extended by many mathematicians in a lot of references. Some of them may be recited as follows.
Theorem 1.1 ([, Theorems 2.2 and 2.3])
- (1)If is a convex function on , then(1.3)
- (2)If for is a convex function on , then(1.4)
Theorem 1.2 ([, Theorems 2.2 and 2.3])
Definition 1.1 ()
holds for all and .
Theorem 1.3 ([, Theorem 2])
Definition 1.2 ()
for all and .
Theorem 1.4 ([, Theorem 3.1])
In this paper, we will create some new integral inequalities of Hermite-Hadamard type for functions whose third derivatives are convex.
For establishing some new integral inequalities of Hermite-Hadamard type for functions whose third derivatives are convex, we need an integral identity below.
Lemma 2.1 is thus proved. □
Basing on Lemma 2.1, we now start out to establish some new integral inequalities of Hermite-Hadamard type for functions whose third derivatives are convex.
The proof of Theorem 3.1 is complete. □
The proof of Theorem 3.2 is completed. □
- (1)if , we have
- (2)if , we have
- (3)if , we have
- (4)if , we have
- (5)if , we have
- (6)if , we have
The proof of Theorem 3.3 is complete. □
- (1)if , we have(3.4)
- (2)if , we have
The authors appreciate the editor and anonymous referees for their careful reading, helpful comments on, and valuable suggestions to the original version of this manuscript. This work was partially supported by the NNSF of China under Grant No. 11361038, by the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region under Grant No. NJZY13159, and by the Science Research Funding of the Inner Mongolia University for Nationalities under Grant No. NMD1225, China.
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