Around Jensen’s inequality for strongly convex functions - Moradi, Hamid Reza; Omidvar, Mohsen Erfanian; Adil Khan, Muhammad; Nikodem, Kazimierz (Aequationes mathematicae, 2018)
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 .