Functions of order one, subharmonic in the half-plane, and a Tauberian theorem (Q1177378)

Let \(u\) be a subharmonic function in the upper half-plane. A proximate order \(\rho(r) \to \rho\) is called a formal order of \(u\) if \(u(x)<C| z |^{\rho(| z |)}\) and a semiformal order if in addition every sector of the form \((1-\alpha)R\) \(<| z |<(1+\alpha)R\), \(\delta<\arg z<\pi-\delta\), with some fixed \(\alpha\) and \(\delta\) contains a point \(z\) such that \(u(z)>-A | z |^{\rho(| z |)}\) with some constant \(A>0\). Earlier, in the case when \(\rho>1\), the author gave the conditions on Riesz mass distribution and boundary values of \(u\) which are necessary and sufficient for a given \(\rho(r)\) to be a formal or semiformal order. Now he extends his results to the more difficult case of \(\rho=1\). A Tauberian theorem is proved as a corollary.