Delta-subharmonic functions of completely regular gamma-growth in a half-plane
The theory of entire functions of completely regular growth (c.r.g.) relative to a function \(\gamma(r)\) close to a power function was developed at the end of 1930s independently by B. Ya. Levin and A. Pfluger [\textit{B. Ya. Levin}, The distribution of the zeros of entire functions (Russian). Moskau: Staatsverlag für technisch-theoretische Literatur (1956; Zbl 0111.07401)] and occupies a significant place in complex analysis. \textit{A. F. Grishin} [Teor. Funkts., Funkts. Anal. Prilozh. 6, 3--29 (1968; Zbl 0215.42701)] transferred the Levin-Pfluger theory onto subharmonic functions in the complex plane. Using the method of Fourier series, \textit{A. A. Kondratyuk} [Math. USSR, Sb. 35, 63--84 (1979; Zbl 0426.30025); Mat. Sb., N. Ser. 113(155), 118--132 (1980; Zbl 0441.30036); Mat. Sb., N. Ser. 120(162), No. 3, 331--343 (1983; Zbl 0516.30021)] generalized the Levin-Pfluger theory of entire functions of c.r.g. functions to meromorphic functions of any \(\gamma\)-type. He made these generalizations in two directions: 1) the growth of functions was measured relative to any function of growth \(\gamma(r)\); 2) the classes of meromorphic functions of c.r.g. in the complex plane were entered and investigated. Counterparts of Kondratyuk's results for \(\delta\)-subharmonic functions were obtained by \textit{Ya. V. Vasil'kiv} in [Ukr. Math. J. 37, 5--10 (1985; Zbl 0594.31006); translation from Ukr. Mat. Zh. 37, No. 1, 8--13 (1985; Zbl 0566.31002)]. In parallel with these investigations, the theory of functions of c.r.g. in the upper complex half-plane \(\mathbb{C}= \{z: \Im z >0\}\) was developed. In the 1960s, \textit{A. F. Grishin} [Teor. Funkts., Funkts. Anal. Prilozh. 7, 59--84 (1968; Zbl 0215.42701); ibid. 8, 126--135 (1969; Zbl 0215.42801)] and \textit{N. V. Govorov} [Sov. Math., Dokl. 6, 697--701 (1965); translation from Dokl. Akad. Nauk SSSR 162, 495--498 (1965; Zbl 0156.29604)] developed independently the Levin-Pfluger theory for the finite-order functions in a half-plane. Whereas Govorov's theory is related to the analytic functions of c.r.g relative to a function \(\gamma(r)=r^\rho\) (\(\rho>0\) is a fixed number), Grishin's theory covers the subharmonic functions of c.r.g in a half-plane relative to a function \(\gamma(r)=r^{\rho(r)} \), where \(\rho\geq 0\), including the classes of subharmonic functions of zero order. The theory of Fourier coefficients of delta-subharmonic functions in a half-plane advanced recently in [\textit{K. G. Malyutin}, Sb. Math. 192, No. 6, 843--861 (2001); translation from Mat. Sb. 192, No. 6, 51--70 (2001; Zbl 0996.31001)], was developed further in the work by \textit{K. G. Malyutin} and \textit{N. Sadik} [Dokl. Math. 64, No. 2, 194--196 (2001); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 380, No. 3, 312--314 (2001; Zbl 1083.30030)], where the concept of a delta-subharmonic function of c.r.g. in a half-plane was introduced. The authors give a short review of principal results of these theories and their interplay. Further, the authors introduce the concept of an indicator of a \(\delta\)-subharmonic function in a half-plane relative to \(\gamma(r)\). They also prove that the indicator belongs to \(L^2[0, \pi]\).
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