SOME INEQUALITIES FOR B -1 -CONVEX FUNCTIONS VIA FRACTIONAL INTEGRAL OPERATOR

Year 2019, Volume: 9 Issue: 3, 620 - 625, 01.09.2019

I. Yesilce

Abstract

In this paper, B-1-convexity which is an abstract convexity type is studied. In addition, some new Hermite-Hadamard type inequalities for B-1-convex functions involving Riemann-Liouville type integral operators that are more general from classic integral operators are proven.

Keywords

Abstract convexity , B-1-convexity , fractional integral , Hermite-Hadamard inequalities.

References

• [1] G. Adilov, (2011), Increasing Co-radiant Functions and Hermite-Hadamard Type Inequalities, Mathematical Inequalities and Applications, 14 (1), pp. 45-60.
• [2] G. Adilov, S. Kemali, (2009), Abstract convexity and Hermite-Hadamard Type Inequalities, Journal of Inequalities and Applications, 2009, 13 pages.
• [3] G. Adilov, S. Kemali, (2007), Hermite-Hadamard-Type Inequalities For Increasing Positively Homogeneous Functions, Journal of Inequalities and Applications, 2007, 10 pages.
• [4] G. Adilov and A. Rubinov, (2006), B-convex Sets and Functions, Numerical Functional Analysis and Optimization, 27 (3-4), pp. 237-257.
• [5] G. Adilov and G. Tinaztepe, (2009), The Sharpening of Some Inequalities via Abstract Convexity, Mathematical Inequalities and Applications, 12 (1), pp. 33-51.
• [6] G. Adilov and I. Yesilce, (2017), B −1−convex Functions, Journal of Convex Analysis, 24 (2), pp. 505-517.
• [7] G. Adilov and I. Yesilce, (2012), B −1−convex Sets and B −1−measurable Maps, Numerical Functional Analysis and Optimization, 33 (2), pp. 131-141.
• [8] G. Adilov and I. Yesilce, (2012), On Generalization of the Concept of Convexity, Hacettepe Journal of Mathematics and Statistics, 41 (5), pp. 723-730.
• [9] W. Briec, Q.B. Liang, (2011), On Some Semilattice Structures for Production Technologies, Eur. J. Oper. Res., 215, pp. 740749.
• [10] Z. Dahmani, (2010), On Minkowski and Hermite-Hadamard Integral Inequalities via Fractional Integration, Ann. Funct. Anal, 1 (1), pp. 51-58.
• [11] S.S. Dragomir, C.E.M. Pearce, (2000), Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University.
• [12] J. Hadamard, (1893), Etude sur les proprietes des fonctions entieres et en particulier d’une fonction consideree par Riemann, Journal des Mathematiques Pures et Appliquees, 58, pp. 171-215.
• [13] Ch. Hermite, (1883), Sur deux limites d’une integrale define, Mathesis, 3, pp. 82.
• [14] S. Kemali, I. Yesilce, G. Adilov, (2015), B-convexity, B −1 -convexity, and Their Comparison, Numerical Functional Analysis and Optimization, 36 (2), pp. 133-146.
• [15] A. A. Kilbas, O. I. Marichev, S. G. Samko, (1993), Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach, Switzerland.
• [16] C.P. Niculescu, L.-E. Persson, (2003), Old and New on the Hermite-Hadamard Inequality, Real Analysis Exchange, 29 (2), pp. 663-685.
• [17] M.Z. Sarikaya, E. Set, H. Yaldiz, N. Basak, (2013), HermiteHadamards Inequalities for Fractional Integrals and Related Fractional Inequalities, Mathematical and Computer Modelling, 57 (9), pp. 2403-2407.
• [18] G. Tinaztepe, I. Yesilce and G. Adilov, (2014), Separation of B −1−convex Sets by B −1−measurable Maps, Journal of Convex Analysis, 21 (2), pp. 571-580.
• [19] I. Yesilce, (2018), Inequalities for B-convex Functions via Generalized Fractional Integral, Journal of Inequalities and Applications (submitted).
• [20] I. Yesilce, G. Adilov, (2017), Hermite-Hadamard Inequalities for B-convex and B −1 -convex Functions, International Journal of Nonlinear Analysis and Applications, 8 (1), pp. 225-233.