The Schur multiplicative and harmonic convexities of the complete symmetric function

This paper investigates the Schur multiplicative and harmonic convexities of the complete symmetric function \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$F_n(x,r)=\sum _{i_1+i_2+\cdots +i_n=r}x_1^{i_1}x_2^{i_2}\ldots x_n^{i_n}$\end{document} and the function \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\varphi _n(x,r)=\frac{F_n(x,r)}{F_n(x,r-1)}$\end{document} , where i 1 , i 2 , …, i n are nonnegative integers and r ⩾ 1. As applications, some analytic inequalities are established by use of the theory of majorization. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim