Around Jensen's inequality for strongly convex functions

作者：Moradi Hamid Reza^*; Omidvar Mohsen Erfanian; Khan Muhammad Adil; Nikodem Kazimierz

来源：Aequationes Mathematicae, 2018, 92(1): 25-37.

DOI：10.1007/s00010-017-0496-5

• Strongly convex functions
• Operator inequality
• Jensen's inequality

In this paper we use basic properties of strongly convex functions to obtain new inequalities including Jensen type and Jensen-Mercer type inequalities. Applications for special means are pointed out as well. We also give a Jensen's operator inequality for strongly convex functions. As a corollary, we improve the Holder-McCarthy inequality under suitable conditions. More precisely we show that if Sp (A) subset of (1, infinity), then
< Ax, x >(r) <= < A(r)x, x > - r(2) - r/2 (< A(2)x, x > - < Ax, x >(2)), r >= 2
and if Sp (A) subset of (0, 1), then
< A(r)x, x > <= < Ax, x >(r) + r - r(2)/2 (< Ax, x >(2) - < A(2)x, x >), 0 < r < 1
for each positive operator A and x is an element of H with parallel to x parallel to = 1.

• 出版日期2018-2