The asymptotic behaviour of certain integral functions

Abstract:

Let$f(z)$ be an integral function satisfying \[ {\int _{}^\infty \{\log m(r,f) - \cos \pi \rho \log M(r,f)\} ^ + }\frac {{dr}}{{{r^{\rho + 1}}}} < \infty \] and \[ 0 < \lim \limits _{\overline {r \to \infty } } \frac {{\log M(r,f)}}{{{r^\rho }}} < \infty \] for some $\rho : 0 < \rho < 1$. It is shown that such functions have regular asymptotic behaviour outside a set of circles with centres ${\zeta _i}$ and radii ${t_i}$ for which \[ \sum \limits _{i = 1}^\infty {\frac {{{t_i}}}{{\left | {{\zeta _i}} \right |}}} < \infty \].