Remarks on the monotonicity and convexity of Jensen's function

Abstract

. Let x 1 , x 2 ,... , x n be nonnegative real numbers. The Jensen function of { x i } i = 1 is de fi ned as J s ( x ) = ( ∑ in = 1 x is ) 1 / s , also known as the L p -norm. It is well-known that J s ( x ) is decreasing on s ∈ ( 0 , + ∞ ) . Moreover, Beckenbach [Amer. Math. Monthly, 53 (1946), 501– 505] proved further that J s ( x ) is a convex function on s ∈ ( 0 , + ∞ ) . The goal of this note is two-fold. We fi rst 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 J s ( x ) by using the H¨older 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-P´oya inequality is also included. (2010):