Delta-subharmonic functions of completely regular growth in a half-plane.
The theory of entire functions of completely regular growth (c.r.g.) was independently developed by Levin and Pfluger in the late thirties - see thhe classical monograph by [\textit{B. Y. Levin}, Providence, R.I.: American Mathematical Society 1964. VIII, 493 p. (1964; Zbl 0152.06703)]. Regarding further extensions of the theory see, e.g. [ibid, p. 470-491] and \textit{L. I. Ronkin's} paper in [Entire and subharmonic functions, Adv. Sov. Math. 11, 39--74 (1992; Zbl 0777.31004)]. the present authors study \(\delta -\)subharmonic functions in a half-plane of c.r.g. with respect to a growth function \(\gamma (r)\) satisfying the doubling condition as well as some natural smoothness assumptions. a \(\delta -\)subharmonic function \(u\) is said to be of c.r.g. if the limit \[ \lim _{r\rightarrow \infty} \frac{1}{\gamma(r)}\int _{\eta}^{\theta}u(re^{\phi}) \sin \phi d\phi \] exists for all \(\eta ,\theta \in [0,\pi].\) The function has c.r.g if and only if the limits \(\lim _{r\rightarrow \infty} c_k(r,u)/\gamma (r)\) exist for all \(k=1,2,3,\ldots\), where \(c_k4\) are the \(\sin\) -Fourier coefficients of \(u\) over \([0,\pi]\).
scientific article; zbMATH DE number 2074908