Some inequalities for entire functions of exponential type

Abstract:

If $f(z)$ is an asymmetric entire function of exponential type $\tau$, \[ \left \| f \right \| = \sup \limits _{ - \infty < x < \infty } |f(x)|,\] then according to a well-known result of R. P. Boas, \[ \left \| {f’} \right \| \leq \frac {\tau }{2}\left \| f \right \|\] and \[ |f(x + iy)| \leq \frac {{({e^{\tau |y|}} + 1)}}{2}\left \| f \right \|,\quad - \infty < x < \infty , - \infty < y \leq 0.\] Both of these inequalities are sharp. In this paper we generalize the above two inequalities of Boas by proving a sharp inequality which, besides giving as special cases the above two inequalities of Boas, yields some other results as well.