A Turan-type inequality for entire functions of exponential type

Abstract

Let $f(z)$ be an entire function of exponential type $\tau$ such that $\Vert f\Vert =1$. Also suppose, in addition, that $f(z)\ne 0$ for $\mathfrak{I}z>0$ and that $h_f(\frac{\pi }{2})=0$. Then, it was proved by Gardner and Govil [Proc. Amer. Math. Soc., 123(1995), 2757-2761] that for $y=\mathfrak{I}z\le 0$ $$\Vert D_{\zeta }[f]\Vert \le \frac{\tau }{2}(|\zeta |+1),$$ where $D_{\zeta }[f]$ is referred to as polar derivative of entire function $f(z)$ with respect to $\zeta$. In this paper, we prove an inequality in the opposite direction and thereby obtain some known inequalities concerning polynomials and entire functions of exponential type.