Abstract
In this paper, we introduce the notion of strongly \({\varphi_{h}}\) -convex functions with respect to c > 0 and present some properties and representation of such functions. We obtain a characterization of inner product spaces involving the notion of strongly \({\varphi_{h}}\) -convex functions. Finally, a version of Hermite–Hadamard-type inequalities for strongly \({\varphi_{h}}\) -convex functions is established.
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Angulo H., Gimenez J., Moros A.M., Nikodem K.: On strongly h-convex functions. Ann. Funct. Anal. 2(2), 85–91 (2011)
Bakula, M.K.; Pečarić, J.: Note on some Hadamard-type inequalities. J. Inequal. Pure Appl. Math. 5(3) article 74 (2004)
Chu Y.M., Wang G.D., Zhang X.H.: Schur convexity and Hadmard’s inequality. Math. Inequal. Appl. 4, 725–731 (2010)
Cristescu, G.: Hadamard type inequalities for \({\varphi}\) -convex functions. Annals of the University of Oradea, Fascicle of Management and Technological Engineering, CD-Rom Edition, III(XIII) (2004)
Cristescu, G.; Lupşa, L.: Non-Connected Convexities and Applications. Kluwer Academic Publishers, Dordrecht / Boston / London (2002)
Dragomir S.S., Agarwal R.P.: Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula. Appl. Math. Lett 11(5), 91–95 (1998)
Dragomir, S.S.; Pearce, C. E. M.: Selected Topics on Hermite–Hadamard Inequalities and Applications. RGMIA Monographs, Victoria University (2000)
Dragomir S.S., Fitzpatrik S.: The Hadamard’s inequality for s-convex functions in the second sense. Demonstr Math. 32(4), 687–696 (1999)
Dragomir, S.S.; Pečarić, J.; Persson, L.E.: Some inequalities of Hadamard type. Soochow J. Math. 21, 335–241 (1995)
Guessab, A.; Schmeisser, G.: Sharp integral inequalities of the Hermite–Hadamard type. J. Approx. Theory 115(2), 260–288 (2002)
Merentes, N.; Nikodem, K.: Remarks on strongly convex functions. Aequationes Math. 80(1–2), 193–199 (2010)
Nikodem, K.; Pales, Zs.: Characterizations of inner product spaces be strongly convex functions. Banach J. Math. Anal. 5(1), 83–87 (2011)
Pečarić, J.E.; Proschan, F.; Tong, Y.L.: Convex Functions, Partial Orderings and Statistical Applications. Academic Press, Boston (1992)
Polyak B.T.: Existence theorems and convergence of minimizing sequences in extremum problems with restictions. Soviet Math. Dokl. 7, 72–75 (1966)
Sarikaya, M.Z.: On Hermite Hadamard-type inequalities for strongly \({\varphi }\) -convex functions. Southeast Asian Bull. Math., accepted (2013a)
Sarikaya, M.Z.: On Hermite Hadamard-type inequalities for \({\varphi_{h}}\) -convex functions. Submitted (2013)
Sarikaya, M.Z.; Saglam, A.; Yıldırım, H.: On some Hadamard-type inequalities for h-convex functions. J. Math. Inequal. 2(3), 335–341 (2008)
Sarikaya, M.Z.; Set, E.; Ozdemir, M.E.: On some new inequalities of Hadamard type involving h-convex functions. Acta Mathematica Universitatis Comenianae, vol. LXXIX, 2, pp. 265–272 (2010)
Set, E.; Özdemir, M.E.; Dragomir, S.S.: On the Hermite–Hadamard inequality and other integral inequalities involving two functions. J. Inequal. Appl. Article ID 148102 (2010)
Set, E.; Özdemir, M.E.; Dragomir, S.S.: On Hadamard-Type inequalities involving several kinds of convexity. J. Inequal. Appl. Article ID 286845 (2010)
Varošanec, S.: On h-convexity. J. Math. Anal. Appl. 326, 303–311 (2007)
Youness, E.A.: E-Convex Sets, E-convex functions and E-convex programming. J. Optim. Theory Appl. 102(2), 439–450 (1999)
Zhang, X.M.; Chu, Y.M.: Convexity of the integral arithmetic mean of a convex function. Rocky Mt. J. Math. 3, 1061–1068 (2010)
Zhang, X.M.; Chu, Y.M.; Zhang, X.H.: The Hermite–Hadamard type inequality of GA-convex functions and its applications. J. Inequal. Appl., Art. ID 507560 (2010)
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Sarikaya, M.Z. On strongly \({\varphi_{h}}\) -convex functions in inner product spaces. Arab. J. Math. 2, 295–302 (2013). https://doi.org/10.1007/s40065-013-0069-y
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DOI: https://doi.org/10.1007/s40065-013-0069-y