An Abelian theorem for a class of subharmonic functions

Abstract:

We show that if the Riesz-mass of a subharmonic function u, of finite order $\lambda$, is distributed along a ray, then regular variation (with exponent $\lambda$) of the mean value of $u(r{e^{i\theta }})$ implies regular variation (with exponent $\lambda$) of each of the ${L_s}( - \pi ,\pi )$ means of $u(r{e^{i\theta }})$. This result extends a known theorem of Edrei and Fuchs, but our method differs from theirs. In particular, for the case of integral orders we obtain the theorem for a much more general distribution of the Riesz-mass. A corollary, which appears to be new, on the deficiency of the value zero of entire functions with positive integral order, follows.