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.
KeywordsHermite-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. □
3 Hermite-Hadamard type inequalities for convex functions
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.
- Dragomir SS: Two mappings on connection to Hadamard’s inequality. J. Math. Anal. Appl. 1992, 167(1):49–56. Available online at 10.1016/0022-247X(92)90233-4MathSciNetView ArticleGoogle Scholar
- Kirmaci US: Inequalities for differentiable mappings and applications to special means of real numbers to midpoint formula. Appl. Math. Comput. 2004, 147(1):137–146. Available online at 10.1016/S0096-3003(02)00657-4MathSciNetView ArticleGoogle Scholar
- Dragomir SS, Pečarić J, Persson L-E: Some inequalities of Hadamard type. Soochow J. Math. 1995, 21(3):335–341.MathSciNetGoogle Scholar
- Alomari M, Hussain S: Two inequalities of Simpson type for quasi-convex functions and applications. Appl. Math. E-Notes 2011, 11: 110–117.MathSciNetGoogle Scholar
- Hudzik H, Maligranda L: Some remarks on s -convex functions. Aequ. Math. 1994, 48(1):100–111. Available online at 10.1007/BF01837981MathSciNetView ArticleGoogle Scholar
- Chun L, Qi F: Integral inequalities of Hermite-Hadamard type for functions whose 3rd derivatives are s -convex. Appl. Math. 2012, 3(11):1680–1685. Available online at 10.4236/am.2012.311232View ArticleGoogle Scholar
- Bai R-F, Qi F, Xi B-Y: Hermite-Hadamard type inequalities for the m - and -logarithmically convex functions. Filomat 2013, 27(1):1–7. Available online at 10.2298/FIL1301001BMathSciNetView ArticleGoogle Scholar
- Bai S-P, Qi F:Some inequalities for --convex functions on the co-ordinates. Glob. J. Math. Anal. 2013, 1(1):22–28.MathSciNetGoogle Scholar
- Bakula MK, Özdemir ME, Pečarić J: Hadamard type inequalities for m -convex and ( α , m ) -convex functions. J. Inequal. Pure Appl. Math. 2008., 9(4): Article ID 96. Available online at http://www.emis.de/journals/JIPAM/article1032.htmlGoogle Scholar
- Kavurmaci, H, Avci, M, Özdemir, ME: New inequalities of Hermite-Hadamard type for convex functions with applications. Available online at arXiv:1006.1593. e-printatarXiv.org
- Qi F, Wei Z-L, Yang Q: Generalizations and refinements of Hermite-Hadamard’s inequality. Rocky Mt. J. Math. 2005, 35(1):235–251. Available online at 10.1216/rmjm/1181069779MathSciNetView ArticleGoogle Scholar
- Shuang Y, Yin H-P, Qi F: Hermite-Hadamard type integral inequalities for geometric-arithmetically s -convex functions. Analysis 2013, 33(2):197–208. Available online at 10.1524/anly.2013.1192MathSciNetView ArticleGoogle Scholar
- Xi B-Y, Qi F: Some Hermite-Hadamard type inequalities for differentiable convex functions and applications. Hacet. J. Math. Stat. 2013, 42(3):243–257.MathSciNetGoogle Scholar
- Xi B-Y, Qi F: Hermite-Hadamard type inequalities for functions whose derivatives are of convexities. Nonlinear Funct. Anal. Appl. 2013, 18(2):163–176.Google Scholar
- Xi B-Y, Qi F: Some inequalities of Hermite-Hadamard type for h -convex functions. Adv. Inequal. Appl. 2013, 2(1):1–15.MathSciNetGoogle Scholar
- Xi B-Y, Wang Y, Qi F:Some integral inequalities of Hermite-Hadamard type for extended -convex functions. Transylv. J. Math. Mech. 2013, 5(1):69–84.MathSciNetGoogle Scholar
- Zhang T-Y, Ji A-P, Qi F: Integral inequalities of Hermite-Hadamard type for harmonically quasi-convex functions. Proc. Jangjeon Math. Soc. 2013, 16(3):399–407.MathSciNetGoogle Scholar
- Zhang T-Y, Ji A-P, Qi F: On integral inequalities of Hermite-Hadamard type for s -geometrically convex functions. Abstr. Appl. Anal. 2013., 2013: Article ID 560586. Available online at 10.1155/2012/560586Google Scholar
- Zhang T-Y, Ji A-P, Qi F: Some inequalities of Hermite-Hadamard type for GA-convex functions with applications to means. Matematiche 2013, 68(1):229–239. Available online at 10.4418/2013.68.1.17MathSciNetGoogle Scholar
- Dragomir SS, Pearce CEM RGMIA Monographs. In Selected Topics on Hermite-Hadamard Type Inequalities and Applications. Victoria University, Melbourne; 2000. Available online at http://rgmia.org/monographs/hermite_hadamard.htmlGoogle Scholar
- Niculescu CP, Persson L-E: Convex functions and their applications. In CMS Books in Mathematics. Springer, Berlin; 2005.Google Scholar
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