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  • SUB- AND SUPERADDITIVE PROPERTIES OF FEJÉR'S SINE POLYNOMIAL

  • HORST ALZER, STAMATIS KOUMANDOS
  • Journal:
    Bulletin of the London Mathematical Society / Volume 38 / Issue 2 / April 2006
    Published online by Cambridge University Press:
    16 March 2006, pp. 261-268
    Print publication:
    April 2006

  • 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.