An inequality for functions of exponential type not vanishing in a half-plane
Copyright policy )An inequality for functions of exponential type not vanishing in a half-plane
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.
Special classes of entire functions of one complex variable and growth estimates, Inequalities in the complex plane
Special classes of entire functions of one complex variable and growth estimates, Inequalities in the complex plane
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