An inequality for functions of exponential type not vanishing in a half-plane

Abstract

Let f ( z ) f(z) be an entire function of order 1, type τ \tau having no zero in Im z > 0 \operatorname {Im} \;z > 0 . If h f ( − π / 2 ) = τ , h f ( π / 2 ) ⩽ 0 {h_f}( - \pi /2) = \tau , {h_f}(\pi /2) \leqslant 0 then it is known that sup − ∞ > x > ∞ | f ′ ( x ) | ⩾ ( τ / 2 ) sup − ∞ > x > ∞ | f ( x ) | {\sup _{ - \infty > x > \infty }}|f’(x)| \geqslant (\tau /2){\sup _{ - \infty > x > \infty }}|f(x)| . In this paper we consider the case when f ( z ) f(z) has no zero in Im z > k , k ⩽ 0 \operatorname {Im} \;z > k, k \leqslant 0 and obtain a sharp result.