SUB- AND SUPERADDITIVE PROPERTIES OF FEJÉR'S SINE POLYNOMIAL

Abstract

Let $ S_n(x)=\sum_{k=1}^{n}({\sin(kx)})/{k} $ be Fejér's sine polynomial. We prove the following statements.

1. The inequality $ (S_n(x+y))^{\alpha} (x+y)^{\beta} \leq (S_n(x))^{\alpha}x^{\beta}+ (S_n(y))^{\alpha}y^{\beta} {(n\in \mathbb{N}; \; \alpha, \beta \in \mathbb{R})}$ holds for all $x,y \in (0,\pi)$ with $x+y<\pi$ if and only if $\alpha\geq 0$ and $\alpha+\beta \leq 1$.

2. The converse of the above inequality is valid for all $x,y \in (0,\pi)$ with $x+y<\pi$ if and only if $\alpha\leq 0$ and $\alpha+\beta \geq 1$.

3. For all $n\in\mathbb{N}$ and $x,y \in [0,\pi]$ we have $ 0\leq S_n(x)+S_n(y)- S_n(x+y)\leq \frac{3}{2}\sqrt{3}.$ Both bounds are best possible.