A Turan-type inequality for entire functions of exponential type

Abstract

Let $f(z)$ be an entire function of exponential type $\tau$ such that $\|f\| = 1$. Also suppose, in addition, that $f(z)\neq 0$ 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 $y = \mathfrak{I}z \leq 0$ $$\|D_{\zeta}[f]\| \leq \frac{\tau}{2} (|\zeta|+1),$$ 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.