$L^ p$ inequalities for entire functions of exponential type

Abstract:

Let $f$ be an entire function of exponential type $\tau$ belonging to ${L^p}$ on the real line. It has been known since a long time that \[ \int _{ - \infty }^\infty {{{\left | {f’(x)} \right |}^p}dx \leq {\tau ^p}\int _{ - \infty }^\infty {{{\left | {f(x)} \right |}^p}dx\quad {\text {if}}\;p \geq 1.} } \] We prove that the same inequality holds also for $0 < p < 1$. Various other estimates of the same kind have been obtained.