Inequalities for sine sums with more variables

Abstract

A result of Vietoris states that if the real numbers $a_1,\ldots ,a_n$ satisfy

$$\begin{aligned} \text{(*) } \qquad a_1\ge \frac{a_2}{2} \ge \cdots \ge \frac{a_n}{n}>0 \quad \text{ and } \quad a_{2k-1}\ge a_{2k} \quad (1\le k\le n/2), \end{aligned}$$

then, for $x_1,\ldots ,x_m>0$ with $x_1+\cdots +x_m <\pi$,

$$\begin{aligned} \begin{aligned} \text{(**) } \qquad \sum _{k=1}^n a_k \frac{\sin (k x_1) \cdots \sin (k x_m)}{k^m}>0. \end{aligned} \end{aligned}$$

We prove that $(**)$ (with “$\ge$” instead of “>”) holds under weaker conditions. It suffices to assume, instead of $(*)$, that

$$\begin{aligned} \sum _{k=1}^N a_k \frac{\sin (kt)}{k}>0 \quad (N=1,\ldots ,n; \, 0<t<\pi ), \end{aligned}$$

and, moreover, $(**)$ is valid for a larger region, namely, $x_1,\ldots ,x_m\in (0,\pi )$.