Convexity theorems for Riesz means (Q1192442)

In a previous note [Mem. Defense Acad. 5, 335-340 (1966; Zbl 0178.058)], M. Riesz's classical convexity theorem was given a new generalization by the author. We refer to its review quotation (there, \(A^ k(x)\) denotes a Riesz \textit{sum} rather than mean). Theorem II of the present paper improves upon the former result in that the review's condition (ii) --- which properly should say \(x^ \beta V(x)\) to be nondecreasing for some \(\beta\geq 0\) --- is replaced by the weaker requirement that there be constants \(h\) and \(H\) such that \(0<h<x'/x<1\) implies \(V(x')/V(x)<H\) [Theorem II covers Theorem I. At Theorems A and I, the exponent of \(W(x)\) in the conclusion lines is misprinted each].