Further improvements of Askey-Steinig’s inequalities for finite sums involving sine and cosine

In 1974, Askey and Steinig proved that for all n ≥ 0 n\geq 0 and x ∈ ( 0 , 2 π ) x\in (0,2\pi ) the trigonometric sums sin ⁡ ( x / 4 ) 1 + sin ⁡ ( 5 x / 4 ) 2 + ⋯ + sin ⁡ ( ( 4 n + 1 ) x / 4 ) n + 1 \begin{equation*} \frac {\sin (x/4)}{1}+\frac {\sin (5x/4)}{2}+\cdots + \frac {\sin ((4n+1)x/4)}{n+1} \end{equation*} and cos ⁡ ( x / 4 ) 1 + cos ⁡ ( 5 x / 4 ) 2 + ⋯ + cos ⁡ ( ( 4 n + 1 ) x / 4 ) n + 1 \begin{equation*} \frac {\cos (x/4)}{1}+\frac {\cos (5x/4)}{2}+\cdots + \frac {\cos ((4n+1)x/4)}{n+1} \end{equation*} are positive. Recently, the Askey-Steinig inequalities were improved by the present authors. In this paper, we further improve these inequalities and provide new sharp upper and lower bounds for the two sums given above.