Sharp inequalities for sine polynomials

Abstract:

Let $\displaystyle F_n(x)=\sum_{k=1}^n \frac{\sin(kx)}{k}$ and $\displaystyle C_n(x)=\sum_{k=1}^n\frac{\sin((2k-1)x)}{2k-1}$.
The classical inequalities $$ 0＜F_n(x)＜\int_0^{\pi}\frac{\sin(t)}{t}dt=1.85193\dots \quad\mbox{and} \quad 0＜ C_n(x)\leq 1 $$ are valid for all $n\geq 1$ and $x\in (0,\pi)$. All constant bounds are sharp. We present the following refinements of the lower bound for $F_n(x)$ and the upper bound for $C_n(x)$.
• Let $\mu\geq 1$. The inequality $\displaystyle \frac{\sin(x)}{\mu-\cos(x)}＜F_n(x) $ holds for all odd $n\geq 1$ and $x\in (0,\pi)$ if and only if $\mu\geq 2$.
  
• For all $n\geq 2$ and $x\in [0,\pi]$, we have $\displaystyle C_n(x) \leq 1-\lambda \sin(x) $ with the best possible constant factor $\lambda=\sqrt[3]{9}-2$.
Moreover, we offer a companion to the inequality $C_n(x)＞0$.
• Let $n\geq 1$. The inequality $\displaystyle 0 \leq \sum_{k=1}^n \left( \delta(n) -(k-1)k\right) \sin((2k-1)x) $ holds for all $x\in [0,\pi]$ if and only if $\delta(n)\geq (n^2-1)/2$.
This extends a result of Dimitrov and Merlo, who proved the inequality for the special case $\delta(n)=n(n+1)$. The following inequality for the Chebyshev polynomials of the second kind plays a key role in our proof of (iii).

• Let $m\geq 0$. For all $t\in\mathbb{R}$, we have $\displaystyle \left( m^2(1-t^2)-1 \right) U^2_m(t) +(m+1) U_{2m}(t)\leq m(m+1). $ The upper bound is sharp.