An inequality for entire functions of exponential type (Q919555)

Using a result of \textit{L. Hörmander} [Math. Scand. 3, 21-27 (1955; Zbl 0065.303)] the author shows: Let f be entire, of exponential type \(\tau\), such that \(| f(x)| \leq 1\) for real x. If \(| f(0)| =\cos a\), \(0\leq a\leq \pi /2\) and \(f'(0)=0\), then \(| f(x)| \leq \sin [\{(\pi -a)^ 2+\tau^ 2x^ 2\}^{1/2}-\pi /2]\) for \(| x| <\{a(2\pi -a)\}^{1/2}/\tau\). Further \(| f''(0)| \leq (\sin a/a)\tau^ 2\).