REFINEMENT OF TURÁN-TYPE INEQUALITY FOR A POLYNOMIAL

Let $p(z)$ be a polynomial of degree $n$. Then the polar derivative with respect to real or complex number $\alpha$ is defined by $D_\alpha p(z)=n p(z)+(\alpha-z) p^{\prime}(z)$. Govil and Mctume [14] proved that if $n$ having all its zeros in $|z| \leq k, k \geq 1$, then for $|\alpha| 1+k+k^n$,$$\begin{aligned}\max _{|z|=1}\left|D_\alpha p(z)\right| & n\left(\frac{|\alpha|-k}{1+k^n}\right) \max _{|z|=1}|p(z)| \\& +n\left(\frac{|\alpha|-\left(1+k+k^n\right)}{1+k^n}\right) \min _{|z|=k}|p(z)| .\end{aligned}$$In this paper, we obtain refinement above inequality. Received: March 16, 2023Accepted: July 4, 2023