On the complete regularity of growth of the Borel transform of the analytic continuation of the associated function, which has a~finite number of singular points

N. V. Govorov ; N. M. Chernykh

Izvestiya. Mathematics , Tome 13 (1979) no. 2, pp. 253-259

Résumé

The following theorem is proved. Let $A(z)$ be an entire function of exponential type, and let its Borel transform $a(z)$ satisfy the following conditions: 1) $a(z)$ can be analytically continued to a certain Riemann surface $R$ with finite number of branch points, and it has only finitely many singularities $z_k$ on $R$; 2) in any plane with cuts by parallel rays issuing from the $z_k$, a branch of $z_k$ satisfies $$ \varlimsup_{z\to\infty}\frac{\ln|a(z)|}{|z|}\leq0. $$ Then $A(z)$ has completely regular growth. From this theorem it follows, in particular, that if $a(z)$ is an algebraic function or a single-valued function with a finite number of singularities, then $A(z)$ has completely regular growth. Bibliography: 6 titles.