Some Lp inequalities for entire functions of exponential type

Let f (z) be an entire function of exponential type tau, D-zeta[f (z)] = tau f (z) + i(1 -zeta)f'(z), and h(f) (theta) = lim sup(r ->infinity) log |f (r e(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) not equal 0 in y = Iz > 0, then sup(-infinity <= x <=infinity) vertical bar D(zeta)f (x)vertical bar <= tau/2 (vertical bar zeta vertical bar e(tau vertical bar y vertical bar) + 1) sup(-infinity <= x <=infinity) vertical bar f (x)vertical bar, for Iz <= 0 and vertical bar zeta vertical bar >= 1. In this paper, we present an extension of this result for entire functions of exponential type belonging to L-p(R), p> 0. It also contains several known results as special cases.