Asymptotic modulus of continuity of subharmonic functions of finite order (Q1821232)

Let u(z) be a subharmonic function in the complex plane for which \(u(z)\leq const(1+| z|^{\rho}),\) \(z\in {\mathbb{C}}\). Necessary and sufficient conditions on the Riesz measure \(\mu_ n\) are found such that \((u(z+hz)-u(z))/| z|^{\rho}\to 0\), \(z\to \infty\) uniformly for \(h\in {\mathbb{C}}\) with \(z+hz\not\in C^ 0\) where \(C^ 0\subset {\mathbb{C}}\) is the set of zero linear density. This result improves earlier theorems by B. Ya. Levin and A. F. Grishin.