Hermite-Hadamard Type Inequalities for the Functions Whose Absolute Values of First Derivatives are $p$-Convex

Year 2021, Volume: 4 Issue: 2, 88 - 99, 01.06.2021

Sevda Sezer

https://doi.org/10.33401/fujma.881979

Cited By: 3

https://izlik.org/JA69MD83NG

Abstract

In this paper, we extend some estimates of a Hermite-Hadamard type inequality for functions whose absolute values of the first derivatives are $p$-convex. By means of the obtained inequalities, some bound functions involving beta functions and hypergeometric functions are derived as applications. Also, we suggest an upper bound for error in numerical integration of $p$-convex functions via composite trapezoid rule.

Keywords

Hermite-Hadamard inequality , $p$-convex function , Convexity

References

• [1] G. Adilov, I. Yesilce, On generalizations of the concept of convexity, Hacet. J. Math. Stat., 41(5) (2012), 723-730.
• [2] G. Adilov, I. Yesilce, B^-11-convex functions, J. Convex Anal., 24(2) (2017), 505-517.
• [3] G. R. Adilov, S. Kemali, Abstract convexity, and Hermite-Hadamard type inequalities, J. Inequal. Appl., 2009 (2009), Article ID 943534, 13 pages, DOI:10.1155/2009/943534.
• [4] W. Briec, C. Horvath, B-convexity, Optimization, 53(2) (2004), 103-127.
• [5] S. I. Butt, A. Kashuri, M. Tariq, J. Nasir, A. Aslam, W. Gao, n–polynomial exponential type p–convex function with some related inequalities and their applications, Heliyon, 6(11) (2020), e05420.
• [6] Z. B. Fang, R. Shi, On the (p,h)-convex function and some integral inequalities, J. Inequal. Appl., 45 (2014), 1-16.
• [7] S. Kemali, G. Tınaztepe, G. Adilov, New type inequalities for B^-11-convex functions involving Hadamard fractional integral, Ser. Math. Inform., 33(5) (2018), 697-704.
• [8] S. Kemali, I. Yesilce, G. Adilov, B-convexity, B􀀀1-convexity, and their comparison, Numer. Funct. Anal. Optim., 36(2) (2015), 133-146.
• [9] W. Orlicz, A note on modular spaces I, Bull. Acad. Polon. Soi., Ser. Math. Astronom Phys., 9 (1961), 157-162.
• [10] G. Tinaztepe, I. Yesilce, G. Adilov, Separation of B^-1-convex sets by B^-1-measurable maps, J. Convex Anal., 21(2) (2014), 571-580.
• [11] I. Yesilce, G. Adilov, Some operations on B^-1-convex sets, J. Math. Sci. Adv. Appl., 39(1) (2016), 99-104.
• [12] I. Yesilce, Inequalities for B-convex functions via generalized fractional integrals, J. Inequal. Appl., 194 (2019), DOI 10.1186/s13660-019-2150-3.
• [13] S. S. Dragomir, J. Pecaric, L. E. Persson, Some inequalities of Hadamard type, Soochow J. Math., 21(3) (1995), 335-341.
• [14] S. S. Dragomir, C. Pearce, Selected topics on Hermite-Hadamard inequalities and applications, Math. Prep. Archive, 3 (2003), 463-817.
• [15] S. S. Dragomir, R. P. Agarwal, N. S. Barnett, Inequalities for Beta and Gamma functions via some classical and new integral inequalities, J. Inequal. Appl., 5 (2000), 103–165. [16] ˙I. ˙Is¸can, Ostrowski type inequalities for p-convex functions, NTMSCI, 4(3) (2016), 140-150.
• [17] A. Bayoumi, A. Fathy Ahmed, p-convex functions in discrete sets, Int. J. Eng. Appl. Sci., 4(10) (2017), 63-66.
• [18] N. T. Peck, Banach-Mazur distances and projections on p-convex spaces, Math. Z., 177(1) (1981), 131-142.
• [19] J. Bastero, J. Bernues, A. Pena, The theorems of Caratheodory and Gluskin for 0 < p < 1, Proc. Amer. Math. Soc., 123(1) (1995), 141-144.
• [20] J. Bernu´es, A. Pena, On the shape of p-convex hulls, 0 < p < 1, Acta Math. Hungar., 74(4) (1997), 345-353.
• [21] J. Kim, V. Yaskin, A. Zvavitch, The geometry of p-convex intersection bodies, Adv. Math., 226(6) (2011), 5320-5337.
• [22] S. Sezer, Z. Eken, G. Tinaztepe, G. Adilov p-convex functions and some of their properties, Numer. Funct. Anal. Optim., 42(4) (2021), 443-459.
• [23] Z. Eken, S. Kemali, G. Tınaztepe, G. Adilov, The Hermite-Hadamard inequalities for p-convex functions, Hacet. J. Math. Stat., (in press).