  We explore the class of probability distributions on the real line whose
Laplace transform admits a strong upper bound of subgaussian type. Using
Hadamard's factorization theorem, we extend the class $\mathfrak L$ of Newman
and propose new sufficient conditions for this property in terms of location of
zeros of the associated characteristic functions in the complex plane. The
second part of this note deals with Laplace transforms of strictly subgaussian
distributions with periodic components. This subclass contains interesting
examples, for which the central limit theorem with respect to the R\'enyi
entropy divergence of infinite order holds.
