Gap theorems for entire functions (Q1330241)

The author proves for an angular domain some results which are similar to those in work of \textit{J. M. Anderson} and \textit{K. G. Binmore} [Glasgow Math. J. 12, 89-97 (1971; Zbl 0237.30006)]. Let \(\varphi\) be a convex function on \([0, + \infty)\) for which \(\lim_{t \to + \infty} t^{-1} \varphi (t) = + \infty\), and \(\overline \varphi\) denote its inverse on \([t_1, + \infty)\). If \(f\) is an entire function, then its \(\varphi\)- order \(\rho\) is \(\lim_{t \to + \infty} \sup t^{- 1} \overline \varphi [\log M (e^t,f)]\) where \(M(r,f) = \max |f(z) |\), \(|z |= r\). The author shows there is an entire function \(f\) defined by a gap power series \(f(z) = \sum^\infty_{n = 0} a_n z^{\lambda_n}\) which is bounded in the angular domain \(B(a) = \{z = re^{i \theta} \mid |\theta |\leq a\) for \(0 \leq a < \pi\}\) and has finite \(\varphi\)-order at most \(\rho\) if and only if \(\lim_{r \to + \infty} \inf r^{-1} k(\varphi (r)) \geq \rho^{-1}\) where \(k(r) = \lambda (r) - {2a \over \pi} \log^+ r\) and \(\lambda (r) = \sum_{\lambda_n \leq r} (2/ \lambda_n)\) for \(r \geq \lambda_1\) and zero for \(r < \lambda_1\). The proofs are different in character from those in the Anderson and Binmore work cited and rely on a careful use of Carlemen's formula.