Basic Inequalities for (m,M)-Ψ-Convex Functions when Ψ=-ln


Download PDF

Authors: S. S. DRAGOMIR AND I. GOMM

DOI: 10.46793/KgJMat2002.313D

Abstract:

In this paper we establish some basic inequalities for (m,  M  )-Ψ-convex functions when Ψ = ln. Applications for power functions and weighted arithmetic mean and geometric mean are also provided.



Keywords:

Convex functions, special convexity, weighted arithmetic and geometric means, logarithmic function.



References:

[1]   S. S. Dragomir, On a reverse of Jessen’s inequality for isotonic linear functionals, Journal of Inequalities in Pure and Applied Mathematics 2(3) (2001), Article ID 36.

[2]   S. S. Dragomir, Bounds for the normalized Jensen functional, Bull. Aust. Math. Soc. 74(3) (2006), 417–478.

[3]   S. S. Dragomir, A survey on Jessen’s type inequalities for positive functionals, in: P. M. Pardalos et al. (Eds.), Nonlinear Analysis, Springer Optimization and Its Applications 68, In Honor of Themistocles M. Rassias on the Occasion of his 60th Birthday, Springer Science+Business Media, LLC 2012, DOI 10.1007/978-1-4614-3498-6_12.

[4]   S. S. Dragomir, A note on Young’s inequality, Research Group in Mathematical Inequalities and Applications 18 (2015), Article ID 126, [http://rgmia.org/papers/v18/v18a126.pdf].

[5]   S. S. Dragomir, A note on new refinements and reverses of Young’s inequality, Research Group in Mathematical Inequalities and Applications 18 (2015), Article ID 131, [http://rgmia.org/papers/v18/v18a131.pdf].

[6]   S. S. Dragomir, Additive inequalities for weighted harmonic and arithmetic operator means, Research Group in Mathematical Inequalities and Applications 19 (2016), Article ID 6, [http://rgmia.org/papers/v19/v19a06.pdf].

[7]   S. S. Dragomir and N. M. Ionescu, On some inequalities for convex-dominated functions, L’Anal. Num. Théor. L’Approx. 19(1) (1990), 21–27.

[8]   S. Furuichi, Refined Young inequalities with Specht’s ratio, J. Egyptian Math. Soc. 20 (2012), 46–49.

[9]   S. Furuichi, On refined Young inequalities and reverse inequalities, J. Math. Inequal. 5 (2011), 21–31.

[10]   S. Furuichi and N. Minculete, Alternative reverse inequalities for Young’s inequality, J. Math. Inequal. 5(4) (2011), 595–600.

[11]   F. Kittaneh and Y. Manasrah, Improved Young and Heinz inequalities for matrix, J. Math. Anal. Appl. 361 (2010), 262–269.

[12]   F. Kittaneh and Y. Manasrah, Reverse Young and Heinz inequalities for matrices, Linear Multilinear Algebra 59 (2011), 1031–1037.

[13]   W. Liao, J. Wu and J. Zhao, New versions of reverse Young and Heinz mean inequalities with the Kantorovich constant, Taiwanese J. Math. 19(2) (2015), 467–479.

[14]   M. Tominaga, Specht’s ratio in the Young inequality, Sci. Math. Jpn. 55 (2002), 583–588.

[15]   G. Zuo, G. Shi and M. Fujii, Refined Young inequality with Kantorovich constant, J. Math. Inequal. 5 (2011), 551–556.