FEJ&#201;R TYPE INEQUALITIES FOR HARMONICALLY-CONVEX FUNCTIONS WITH APPLICATIONS

Muhammad Amer Latif; Sever Silvestru Dragomir; Ebrahim Momoniat

doi:10.11948/2017050

2017 Volume 7 Issue 3

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2017 > 7(3): 795-813

Next Article Previous Article

Muhammad Amer Latif, Sever Silvestru Dragomir, Ebrahim Momoniat. FEJÉR TYPE INEQUALITIES FOR HARMONICALLY-CONVEX FUNCTIONS WITH APPLICATIONS[J]. Journal of Applied Analysis & Computation, 2017, 7(3): 795-813. doi: 10.11948/2017050

Citation:

Muhammad Amer Latif, Sever Silvestru Dragomir, Ebrahim Momoniat. FEJÉR TYPE INEQUALITIES FOR HARMONICALLY-CONVEX FUNCTIONS WITH APPLICATIONS[J]. Journal of Applied Analysis & Computation, 2017, 7(3): 795-813. doi: 10.11948/2017050

FEJÉR TYPE INEQUALITIES FOR HARMONICALLY-CONVEX FUNCTIONS WITH APPLICATIONS

1 School of Computer Science and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa;
2 School of Engineering and Science, Victoria University, PO Box 14428 Melbourne City, MC 8001, Australia

Abstract

Abstract

In this paper, a new weighted identity involving harmonically symmetric functions and differentiable functions is established. By using the notion of harmonic symmetricity, harmonic convexity and some auxiliary results, some new Fejér type integral inequalities are presented. Applications to special means of positive real numbers are given as well.
- Hermite-Hadamard's inequality /
- Fejér's inequality /
- convex function /
- harmonically-convex function /
- Hölder's inequality /
- power mean inequality
MSC: 26D15;26A51;26E60;41A55

FullText(HTML)

References

[1]	G. D. Anderson, M.K. Vamanamurthy and M. Vuorinen, Generalized convexity and inequalities, J. Math. Anal. Appl., 2007, 335, 1294-1308. Google Scholar
[2]	N. S. Barnett, P. Cerone and S. S. Dragomir, Some new inequalities for Hermite-Hadamard divergence in information theory, Stochastic analysis and applications. Vol. 3, 7-19, Nova Sci. Publ., Hauppauge, NY, 2003. Google Scholar
[3]	P. S. Bullen, Handbook of Means and Their Inequalities. Mathematics and its Applications, Dordrecht/Boston/London, 560, Kluwer Academic Publishers, 2003. Google Scholar
[4]	P. Cerone and S. S. Dragomir, Mathematical Inequalities. A Perspective, CRC Press, Boca Raton, FL, 2011. x+391 pp. ISBN:978-1-4398-4896-8 Google Scholar
[5]	F. Chen and S. Wu, Fejér and Hermite-Hadamard type inequalities for harmonically convex functions, J. Appl. Math., 2014, Article ID 386806, 6 pages. Google Scholar
[6]	F. Chen and S. Wu, Hermite-Hadamard type inequalities for harmonically sconvex functions, Sci. World J., 2014(7), Article ID 279158. Google Scholar
[7]	S. S. Dragomir, Operator Inequalities of Ostrowski and Trapezoidal Typ, Springer Briefs in Mathematics. Springer, New York, 2012. x+112 pp. ISBN:978-1-4614-1778-1 Google Scholar
[8]	S. S. Dragomir and R. P. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 1998, 11(5), 91-95. Google Scholar
[9]	S. S. Dragomir and C. E. M. Pearce, Selected topics on Hermite-Hadamard type inequalities and applications, RGMIA Monographs:Victoria University, 2000. Google Scholar
[10]	L. Féjer, Über die Fourierreihen, Ⅱ, Math. Naturwiss, Anz. Ungar. Akad. Wiss., 1906, 24, 369-390. (In Hungarian). Google Scholar
[11]	V. N. Huy and N. T. Chung, Some generalizations of the Fejér and HermiteHadamard inequalities in Hölder spaces, J. Appl. Math. Inform., 2011, 29(3-4), 859-868. Google Scholar
[12]	J. Hua, B. -Y. Xi and F. Qi, Hemite-Hadamard type inequalities for geometrically-arithmetically s-convex functions, Commun. Korean Math. Soc., 2014, 29(1), 51-63. Google Scholar
[13]	J. Hua, B. -Y. Xi and F. Qi, Inequalities of Hermite-Hadamard type involving an s-convex function with applications, Appl. Math. Comput., 2014, 246, 752-760. Google Scholar
[14]	İ. İşcan, _Hemite-Hadamard type inequalities for GA-s-convex functions, Le Matematiche, LXIX 2014, (Ⅱ), 129-146. Google Scholar
[15]	İ. İşcan, _Hermite-Hadamard type inequalities for harmonically convex functions, Hacet. J. Math. Stat., 2014, 43(6), 935-942. Google Scholar
[16]	İ. İşcan, _Hermite-Hadamard and Simpson-like type inequalities for differentiable harmonically convex functions, J. Math., 2014, Article ID 346305, 10 pages. Google Scholar
[17]	İ. İşcan and S. Wu, _Hermite-Hadamard type inequalities for harmonically convex functions via fractional integrals, Appl. Math. Comput., 2014, 238, 237-244. Google Scholar
[18]	A. P. Ji, T. Y. Zhang and F. Qi, Integral Inequalities of Hermite-Hadamard Type for (α,m)-GA-Convex Functions, J. Funct. Space. Appl., 2013, Article ID 823856, 8 pages. Google Scholar
[19]	M. A. Latif, New Hermite-Hadamard type integral inequalities for GA-convex functions with applications, Analysis, 2014, 34(4), 379-389. Google Scholar
[20]	M. V. Mihai, M. A. Noor, K. I. Noor and M. U. Awan, Some integral inequalities for harmonic h-convex functions involving hypergeometric functions, Appl. Math. Comput., 2015, 252, 257-262. Google Scholar
[21]	M. A. Noor, K. I. Noor and M. U. Awana, Integral inequalities for coordinated harmonically convex functions, Complex Var. Elliptic Eqn., 2014, 60(6), 776-786. Google Scholar
[22]	M. A. Noor, K. I. Noor, M. U. Awan and S. Costache, Some integral inequalities for harmonically h-convex functions, U.P.B Sci. Bull. Serai A., 2015, 77(1), 5-16. Google Scholar
[23]	J. E. Pečarić, F. Proschan and Y. L. Tong, Convex Functions, Partial Orderings and Statistical Applications (Mathematics in Science and Engineering), Boston/San Diego/New York/London/Sydney/Tokyo/Toronto, 187, Academic Press Inc., 1992. Google Scholar
[24]	M. Z. Sarikaya, On new Hermite Hadamard Fejér type integral inequalities, Stud. Univ. Babeş-Bolyai Math., 2012, 57(3), 377-386. Google Scholar
[25]	Y. Shuang, H. P. Yin and F. Qi, Hermite-Hadamard type integral inequalities for geometric-arithmetically s-convex functions, Analysis, 2013, 33, 1001-1010. Google Scholar
[26]	B. -Y. Xi and F. Qi, Hemite-Hadamard type inequalities for geometrically rconvex functions, Studia Sci. Math. Hungar., 2014, 51(4), 530-546. Google Scholar
[27]	T. Y. Zhang, A. P. Ji and F. Qi, Some inequalities of Hermite-Hadamard type for GA-Convex functions with applications to means, Le Matematiche, 2013, 48(1), 229-239. Google Scholar
[28]	T. Y. Zhang, A. P. Ji and F. Qi, Integral inequalities of Hermite-Hadamard type for harmonically quasi-convex functions, Proceedings of Proc. Jangjeon Math. Soc., 2013, 16(3), 399-407. Google Scholar