An extremal problem and inequalities for entire functions of exponential type

We study two variations of the classical one-delta problem for entire functions of exponential type, known also as the Carathéodory–Fejér–Turán problem. The first variation imposes the additional requirement that the function is radially decreasing while the second one is a generalization which involves derivatives of the entire function. Various interesting inequalities, inspired by results due to Duffin and Schaeffer, Landau, and Hardy and Littlewood, are also established.