Remarks on the monotonicity and convexity of Jensen's function

Let x1, x2, ⋯, xnbe nonnegative real numbers. The Jensen function of {xi}ni=1is defined as Js(x) = (Σni=1xsi)1/s, also known as the Lp-norm. It is well-known that Js(x) is decreasing on s ∈ (0, + ∞). Moreover, Beckenbach [Amer. Math. Monthly, 53 (1946), 501. 505] proved further that Js(x) is a convex function on s ∈ (0, + ∞). The goal of this note is two-fold. We first revisit the skillful treatment of the proof of Beckenbach, and then we simplify the proof slightly. Additionally, we give a new proof of the convexity of Js(x) by using the Hölder inequality, our proof is more succinct and short. On the other hand, we investigate a Jensen-type inequality that arised from Fourier analysis by Stein and Weiss. As a byproduct, the Hardy-Littlewood-Pöya inequality is also included. © 2021 Element D.O.O.. All rights reserved.