부산대학교 도서관

Summary: ``Let $f(z)$ be an entire function of exponential type $\tau$, $D_\zeta [f(z)] = \tau f(z) + {\rm i}(1 - \zeta)f' (z)$, and $h_f(\theta) = \lim \sup_{r\to \infty}\log |f(r e^{\rm i\theta} )|/r$. Gardner and Govil [Proc Am Math Soc. 1995;123:2757--2761] proved that if $h_f(\pi/2) = 0$, and $f(z) \neq 0$ in $y =\germ F z > 0$, then $$ \underset {-\infty \leq x\leq \infty}\to \sup |D_\zeta f(x)| \leq \frac{\tau}{2} (|\zeta | {\rm e}^{\tau |y|} + 1) \underset{ -\infty \leq x\leq \infty}\to\sup |f(x)|, $$ for $\germ Fz\leq 0$ and $|\zeta|\geq 1$. In this paper, we present an extension of this result for entire functions of exponential type belonging to $L^p(\Bbb R)$, $p > 0$. It also contains several known results as special cases.''