Integral Inequalities for the Growth and Higher Derivative of Polynomials

Abstract

Let $P(z)$ be a polynomial of degree $n$ which does not vanish in $|z|<1$, it was proved by S. Gulzar [7] that

$$\left|\left|z^{s}P^{(s)}(z)+\beta\frac{n(n-1)...(n-s+1)}{2^{s}}P(z)\right|\right|_{p}\leqslant n(n-1)...(n-s+1)\left|\left|\left(1+\frac{\beta}{2^{s}}\right)z+\frac{\beta}{2^{s}}\right|\right|_{p}\frac{\left|\left|P(z)\right|\right|_{p}}{\left|\left|1+z\right|\right|_{p}}$$

for every $\beta\in\mathbb{C}$ with $|\beta|\leqslant 1$, $1\leqslant s\leqslant n$ and $0\leqslant p<\infty$. In this paper we extend the above result to the growth of polynomials and also generalize the above and other related results in this direction.