Kuttner’s problem and a Pólya type criterion for characteristic functions

Abstract:

Let $\varphi : [0,\infty ) \to \mathbb {R}$ be a continuous function with $\varphi (0) = 1$ and $\lim _{t \to \infty } \varphi (t)$ $= 0$. If $t^{-1} (\sqrt {t} \varphi ''(\sqrt {t}) - \varphi ’(\sqrt {t}))$ is convex, then $\psi (t) = \varphi (|t|)$, $t \in \mathbb {R}$, is the characteristic function of an absolutely continuous probability distribution. The criterion complements Pólya’s theorem and applies to characteristic functions with various types of behavior at the origin. In particular, it provides upper bounds on Kuttner’s function $k(\lambda )$, $\lambda \in (0,2)$, which gives the minimal value of $\kappa$ such that $(1-|t|^\lambda )_+^\kappa$ is a characteristic function. Specifically, $k(5/3) \leq 3$. Furthermore, improved lower bounds on Kuttner’s function are obtained from an inequality due to Boas and Kac.