Some Integral Mean Estimates for Polynomials with Restricted Zeros

Abstract

Let $P(z)$ be a polynomial of degree $n$ having all its zeros in $|z|\leq k$. For $k=1$, it is known that for each $r> 0$ and $|\alpha|\geq 1$, $$n(|\alpha|-1)\Big\{\int_{0}^{2\pi}|P(e^{i\theta})|^{r}d\theta\Big\}^{\frac{1}{r}}\leq \Big\{\int_{0}^{2\pi}|1+e^{i\theta}|^{r}d\theta\Big\}^{\frac{1}{r}}\max_{|z|=1}\big|D_{\alpha}P(z)\big|.$$ In this paper, we shall first consider the case when $k\geq 1$ and present certain generalizations of this inequality. Also for $k\leq 1$, we shall prove an interesting result for Lacunary type of polynomials from which many results can be easily deduced.