Growth of power series with square root gaps

Abstract

For entire functionsf whose power series have Hadamard gaps with ratio ≥1+α>1, Gaier has shown that the condition |f(x)|≤e ^x forx≥0 implies |f(z)|≤C _αe^|z| (*) for allz. Here the result is extended to the case of square root gaps, that is,$f(z) = \sum {b_{pk} z^{pk} }$, with$Pk + 1 - Pk \geqslant \alpha \sqrt {Pk}$, where α>0. Smaller gaps cannot work. In connection with his proof of the general high indices theorem for Borel summability, Gaier had shown that square root gaps imply$b_n = \mathcal{O}\left( {e^{c\sqrt n } /n!} \right)$. Having such an estimate, one can adapt Pitt’s Tauberian method for the restricted Borel high indices theorem to show that, in fact,$\left| {b_n } \right| \leqslant c_\alpha \sqrt n /n!$, which implies (*). The author also states an equivalent distance formula involving monomialsx ^pke^−xinL ^∞ (0, ∞).