A mean value inequality for the cyclic functions

Abstract

Let ${\phi }_n(x)=\sum _{\nu =0}^{\mathrm{\infty }}\frac{x^{n\nu }}{(n\nu )!},n\in \mathbb{N},$ be the cyclic function of order n and let $\begin{array}{rl}{L}_r(x,y)& ={\left(\frac{x^r-y^r}{r(x-y)}\right)}^{1/(r-1)}(r\ne 0,1),\\ L_0(x,y)& =\frac{x-y}{\log x-\log y},L_1(x,y)=\frac{1}{e}{\left(\frac{x^x}{y^y}\right)}^{1/(x-y)}\end{array}$ be Stolarsky's one-parameter mean value family. We prove that for each n the inequality ${\phi }_n(L_{\alpha }(x,y))<\frac{1}{y-x}{\int }_x^y{\phi }_n(t)\mathrm{d}t(\alpha \in \mathbb{R})$ holds for all $x,y\in \mathbb{R}$ with 0<x<y if and only if $\alpha \le n+1$.

Keywords:

• Cyclic function
• Stolarsky's mean value family
• inequality

2020 Mathematics Subject Classifications:

• 26D07
• 33E12

Acknowledgments

I thank Professor S. Yakubovich for inspiring comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).