Fourier series and δ-subharmonic functions of finite γ-type in a half-plane

Let gamma (r) be a growth function and let upsilon (z) be a proper delta -subharmonic function in the sense of Grishin in a complex half-plane, that is upsilon = upsilon (1) - upsilon (2), where upsilon (1) and upsilon (2) are proper subharmonic functions (lim sup(z -->t) v(i)(z) less than or equal to 0, for each real t, i = 1, 2), let lambda = lambda (+) - lambda (-) be the full measure corresponding to upsilon and let T(r, upsilon) be its Nevanlinna characteristic. The class J delta(gamma) of functions of finite gamma -type is defined as follows: upsilon is an element of J delta(gamma) if T(r,upsilon) < A gamma (Br)/r for some positive constants A and B. The Fourier coefficients of upsilon are defined in the standard way: C-k (r, upsilon) = 2/pi integral (pi)(0) upsilon (re(i theta)) sin k theta d theta, r > 0, k is an element of N. The central result of the paper is the equivalence of the following properties: (1) upsilon is an element of J delta(gamma); (2) N(r) less than or equal to A(1)gamma (B(1)r)/r, where N(r) = N(r,lambda (+)) or N(r) = N(r,lambda (-)) and /c(k)(r,upsilon)/ less than or equal to A(2)gamma (B(2)r). It is proved in addition that J delta(gamma) = JS (gamma) - JS(gamma), where JS (gamma) is the class of proper subharmonic functions of finite gamma -type.