Wiman’s type inequality for analytic and entire functions and h-measure of an exceptional sets - Skaskiv, O.B.; Kuryliak, A.O. (Carpathian Mathematical Publications, 2020)
Let E_R be the class of analytic functions f represented by power series of the form f(z)=_n=0^+∞a_n z^n with the radius of convergence R:=R(f)(0;+∞]. For r[0, R) we denote the maximum modulus by M_f(r)={|f(z)| |z|=r} and the maximal term of the series by μ_f(r)={|a_n| r^nn≥ 0}. We also denote by H_R, R≤ +∞, the class of continuous positive functions, which increase on [0;R) to +∞, such that h(r)≥2 for all r(0,R) and ^R_r_0 h(r) dr =+∞ for some r_0(0,R). In particular, the following statements are proved. 1^0. If hH_R and fE_R, then for any δ>0 there exist E(δ,f,h):=E(0,R), r_0 (0,R) such that r(r_0,R)EM_f(r)≤ h(r) μ_f(r) {h(r)(h(r)μ_f(r))}^1/2+δ and olimits_E h(r) dr < +∞. 2^0. If we additionally assume that the function fE_R is unbounded, then M_f(r)≤(1+o(1))(h(r)μ_f(r)) holds as r→ R, r otin E. Remark, that assertion 1^0 at h(r)const implies the classical Wiman-Valiron theorem for entire functions and at h(r)1/(1-r) theorem about the Kövari-type inequality for analytic functions in the unit disc. From statement 2^0 in the case that h(r)=o(μ_f(r)), r→ R, it follows that M_f(r)=(1+o(1))μ_f(r) holds as r→ R, r otin E.