Norm inequalities and characterizations of inner product spaces

Abstract

Let (X,) be a real normed space and let θ : (0,∞) → (0,∞) be an increasing function such that t → t/θ(t) is non-decreasing on (0,∞) . For such function, we introduce the notion of θ-angular distance aθ [x,y], where x,y ϵ X \{0}, and showthat X is an inner product space if and only if aθ [x,y] ≤ 2 x-y/θ x+θ y for each x,y ϵ X \{0}. Then, in order to generalize the Dunkl-Williams constant of X [10], we introduce a new geometric constant CF (X) for X wrt F , where F : (0,∞)×(0,∞)→(0,∞) is a given function, and obtain some characterizations of inner product spaces related to the constant CF (X) . Our results generalize and extend various known results in the literature. © ELEMEN , Zagreb.