Some Integral Mean Estimates for Polynomials with Restricted Zeros (Analysis in Theory and Applications, 2017)
Let $P(z)$ be a polynomial of degree $n$ having all its zeros in $|z|≤ k$. For $k=1$, it is known that for each $r> 0$ and $|α|≥ 1$, $$n(|α|-1)\Big\{\int_{0}^{2π}|P(e^{iθ})|^{r}dθ\Big\}^{\frac{1}{r}}≤ \Big\{\int_{0}^{2π}|1+e^{iθ}|^{r}dθ\Big\}^{\frac{1}{r}}\max_{|z|=1}\big|D_{α}P(z)\big|.$$ In this paper, we shall first consider the case when $k≥ 1$ and present certain generalizations of this inequality. Also for $k≤ 1$, we shall prove an interesting result for Lacunary type of polynomials from which many results can be easily deduced.
Keywords: Polynomial; zeros; polar derivative.