Entire functions of exponential type not vanishing in the half-plane $\mathrm{\Im }z>k$, where $k>0$

Let $P(z)$ be a polynomial of degree $n$ having no zeros in $|z|<k$, $k\le 1$, and let $Q(z):=z^n\overline{P(1/\stackrel{―}{z})}$. It was shown by Govil that if $\underset{|z|=1}{max}|P^{\prime }(z)|$ and $\underset{|z|=1}{max}|Q^{\prime }(z)|$ are attained at the same point of the unit circle $|z|=1$, then $$\underset{|z|=1}{max}|P^{\prime }(z)|\le \frac{n}{1+k^n}\underset{|z|=1}{max}|P(z)|.$$The main result of the present article is a generalization of Govil's polynomial inequality to a class of entire functions of exponential type.