On Two Trigonometric Inequalities of Carslaw and Gasper

Abstract

The inequalities

$$\begin{aligned} A_n(x)=\sum _{k=1}^n \frac{\sin ((2k-1)x)}{2k-1} >0 \quad (n\ge 1; 0<x<\pi ) \end{aligned}$$

and

$$\begin{aligned} B_n(x)=\sum _{k=1}^n\frac{\cos ((2k-1)x)}{2k-1}>0 \quad (n\ge 1; 0\le x<\pi /2) \end{aligned}$$

are due to Carslaw (1917) and Gasper (1977), respectively. We prove the following counterparts:

$$\begin{aligned} A_n(x)+B_n(x)>0 \quad (n\ge 1; 0\le x<3\pi /4) \end{aligned}$$

and

$$\begin{aligned} B_n(x)\ge \frac{64}{75}\cos ^3(x) \quad (n\ge 2; -\pi /2<x<\pi /2). \end{aligned}$$

The constants $3\pi /4$ and 64/75 are best possible.