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 {article}{amssymb}{empty}{document}Fₙ(x,r)=∑ i₁+i₂+⋯ +iₙ=rx₁i₁x₂i₂… xₙiₙ{document} and the function {article}{amssymb}{empty}{document}φ ₙ(x,r)=Fₙ(x,r)/Fₙ(x,r-1){document} , where i1, i2, …, in are nonnegative integers and r ⩾ 1. As applications, some analytic inequalities are established by use of the theory of majorization.