Some Results for $(s,m)$-convex Function in the Second Sense

Yıl 2022, Cilt: 4 Sayı: 1, 1 - 8, 30.04.2022

Meltem Sertbaş İlknur Mihyaz

https://doi.org/10.47087/mjm.1037337

Öz

Convex functions, like differentiable functions, have a important role in many fields of pure and applied mathematics. It connects concepts from topology, algebra, geometry and analysis, and is an important tool in optimization, mathematical programming and game theory. Also, inequalities for convex function has receives special attention by many researchers because the theory of convex functions has applications in different field of science like biology, economy and optimization.
In this paper, it is given some properties for an (s,m)-convex function defined on [0,d], d>0 in the first sense and the second sense with m\in (0,1). Also, some integral inequalities are examined for any non positive (s,m)-convex function in the second sense with any measure space.

Anahtar Kelimeler

Integral inequalities;, Measure space., (s;m)-convex function

Kaynakça

• Bai, R. F., Qi, F., Xi ,B.Y.: Hermite–Hadamard type inequalities for the m-and (\alpha,m)-logarithmically convex functions. Filomat 27(1), 1-7 (2013)
• Bakula, M.K., Özdemir, M.E., Pecaric ,J.: Hadamard type inequalities form-convex and (\alpha,m)-convexfunctions.J. Inequal. Pure Appl. Math. 9(4), Article ID 96 (2008)
• Bakula,M.K.,Pecaric ,J.,Ribicic, M.:Companion inequalities to Jensen’s inequality for m-convex and (alpha,m)-convex functions. J. Inequal. Pure Appl. Math. 7(5), Article ID 194 (2006)
• Kunt,M., Işcan,I.: Hermite–Hadamard type inequalities for harmonically (\alpha,m)-convex functions by using fractional integrals. Konuralp J. Math. 5(1), 201-213 (2017)
• Latif, M.A., Dragomir, S.S., Momoniat,E.: On Hermite–Hadamard type integral inequalities for n-times differentiable m- and (\alpha, m)-logarithmically convex functions. Filomat 30(11), 3101-3114 (2016)
• Özdemir, M.E., Set,E.,Sarıkaya, M.Z.: Some new Hadamard type inequalities for co-ordinatedm-convexand (alpha, m)-convex functions. Hacet. J. Math. Stat. 40(2), 219-229 (2011)
• M. E. Özdemir, H. Kavurmacı and E. Set, Ostrowski’s Type Inequalities for (\alpha,m)−Convex Functions, Kyungpook Math. J. 50(2010), 371-378
• Set,E., Özdemir,M.E., Sarıkaya,M.Z., Karakoç,F, Hermite–Hadamard type inequalitiesfor (\alpha,m)-convex functions via fractional integrals. Moroccan J. Pure Appl. Anal. 3(1), 15-21 (2017)
• Set,E., Sardari ,M.,Özdemir, M.E.,Rooin, J.: On generalization sof the Hadamard inequality for (\alpha,m)-convex functions. Kyungpook Math. J. 52, 307-317 (2012)
• S. Özcan, Hermite–Hadamard type inequalities for m-convex and (\alpha, m)-convex functions, J. Inequal. Appl. 175, 1-10, 2020.
• Klaricic ́ Bakula, M., Pecaric , J., Ribic ́ic ́, M.: Hadamard type inequalities for m-convex and (s;m)- convex functions. J. Inequal. Pure Appl. Math. 5(4), Art 96 (2008)
• Bracamonte, M., Gime ́nez, J., Vivas-Cortez, M.J.: Hermite-Hadamard-Fejer type inequalities for strongly (s, m)-convex functions with modulus $c$ in second sense. Appl. Math. Inf. Sci. 10(6), 2045-2053 (2016)
• Yang, Z., Li, Y., Du, T.: A generalization of Simpson type inequality via differentiable functions using (s, m)-convex functions. Italian J. Pure Appl. Math. 35, 327-338 (2015)
• V. G. Miheşan, A generalization of the convexity, Seminar on Functional Equations, Approx. and Convex., Cluj-Napoca (Romania) (1993)
• Eftekhari, N.: Some remarks on $(s; m)$-convexity in the second sense. J. Math. Inequal. 8 (2), 489-495 (2014)
• W. W. Breckner, Stetigkeitsaussagen fu ̈r eine Klasse verallgemeinerter konvexer funktionen in topologischen linearen Rau-men, Publ. Inst. Math. (Beograd) (N.S.), 23 (1978), 13-20. 1, 1.1, 1
• Dragomir, S.S., Fitzpatrick,S.: The Hadamard’s inequality for s-convex functions in the second sense. Demonstr. Math. 32, 4, 687-696 (1999)
• T. Lian , W. Tang, R. Zhou, Fractional Hermite–Hadamard inequalities for (s, m)-convex or s-concave functions, J Inequal Appl., 2018:240 (2018)
• Vivas-Cortez M., Fejér type inequalities for (s, m)- convex functions in second sense. Appl Math Inform Sci 10, 1–8 (2016) .
• Vivas-Cortez M., Ostrowski and Jensen-type inequalities via (s, m)-convex functions in the second sense, Bol. Soc. Mat. Mex. (2020) 26:287–302.
• Yeh, J., Real Analysis, World Scientific Publishing Co. Pte. Ltd., USA, 2006
• Cerone, P., Dragomir, S.S.: Ostrowski type inequalities for functions whose derivatives satisfy certain convexity assumptions. Demonstratio Math. 37(2), 299–308 (2004)
• Alomari, M., Darus, M., Dragomir, S.S., Cerone, P.: Ostrowski type inequalities for functions whose derivatives are s-convex in the second sense. Appl. Math. Lett. 23, 1071–1076 (2010)
• A.M. Bruckner, E. Ostrow, Some function classes related to the class of convex functions, Pacific J. Math., 12 (1962), 1203-1215.
• Z. Pavic, M. A. Ardıç, \textit{The most important inequalities of m-convex functions, } Turk J Math., \textbf{41}, (2017) 625-635.