Growth of power series with square root gaps

For entire functions $f$ whose power series have Hadamard gaps with ratio $\alpha$ $(>1)$, Gaier has shown that the condition $|f(x)|\le e^x$ for $x\ge 0$ implies $|f(z)|\le C_\alpha e^{|z|}$ $(*)$ for all $z$. Here the result is extended to the case of square root gaps, that is, $f(z)=\sum b_{p_k}z^{p_k}$ with $p_{k+1}-p_k\ge \alpha\sqrt{p_k}$ where $\alpha>0$. 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 $b_n={\cal O}(e^{c\sqrt n}/n!)$. Having such an estimate, one can adaptPitt's Tauberian method for the restricted Borel high indices theorem to show that, in fact,$|b_n|\le c_\alpha\sqrt n/n!$ which implies $(*)$. The author also states an equivalent distance formula involving monomials $x^{p_k}e^{-x}$ in $L^\infty(0,\infty)$.