Around Jensen’s inequality for strongly convex functions

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,...

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Bibliographic Details
Published in:Aequationes mathematicae Vol. 92; no. 1; pp. 25 - 37
Main Authors: Moradi, Hamid Reza, Omidvar, Mohsen Erfanian, Adil Khan, Muhammad, Nikodem, Kazimierz
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.02.2018
Subjects:
Analysis
Combinatorics
Mathematics
Mathematics and Statistics
Operator inequality
Secondary 46L05
Strongly convex functions
Jensen’s inequality
26B25
Primary 47A63
ISSN:0001-9054, 1420-8903
Online Access:Get full text
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Summary: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 Hölder-McCarthy inequality under suitable conditions. More precisely we show that if S p A ⊂ 1 , ∞ , then A x , x r ≤ A r x , x - r 2 - r 2 A 2 x , x - A x , x 2 , r ≥ 2 and if S p A ⊂ 0 , 1 , then A r x , x ≤ A x , x r + r - r 2 2 A x , x 2 - A 2 x , x , 0 < r < 1 for each positive operator A and x ∈ H with x = 1 .
ISSN:0001-9054
1420-8903
DOI:10.1007/s00010-017-0496-5