Sub-Gaussian random variables and Wiman’s inequality for analytic functions - Kuryliak, A.O.; Skaskiv, O.B. (Carpathian Mathematical Publications, 2023)
Let f be an analytic function in {z: |z|<R} of the form f(z)=_n=0^+∞a_n z^n. In the paper, we consider the Wiman-type inequality for random analytic functions of the form f(z,ω)=_n=0^+∞Z_n(ω)a_nz^n, where (Z_n) is a sequence on the Steinhaus probability space of real independent centered sub-Gaussian random variables, i.e. (D>0)(kN)(λR)E(e^λ Z_k)≤ e^D λ^2, and such that (β>0)(n_0N)_n≥ n_0E|Z_n|^-β<+∞. It is proved that for every δ>0 there exists a set E(δ)[0,R) of finite h-logarithmic measure (i.e. olimits_Eh(r)dr<+∞) such that almost surely for all r(r_0(ω),R)E we have \[ M_f(r,ω):={|f(z,ω)||z|=r}≤ h(r)μ_f(r)(^3h(r){h(r)μ_f(r)})^1/4+δ, \] where h(r) is any fixed continuous non-decreasing function on [0;R) such that h(r)≥2 for all r(0,R) and ^R_r_0 h(r) dr =+∞ for some r_0(0,R).