| Abstrakt: |
Let ER be the class of analytic functions f represented by power series of the form f(z) = +∞ ∑ n=0 anz n with the radius of convergence R := R(f) ∈ (0; +∞]. For r ∈ [0, R) we denote the maximum modulus by Mf (r) = max{| f(z)|: |z| = r} and the maximal term of the series by µf (r) = max{|an|r n : n ≥ 0}. We also denote by HR, 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 r0 h(r)d ln r = +∞ for some r0 ∈ (0, R). In particular, the following statements are proved. 1 0 . If h ∈ HR and f ∈ ER, then for any δ > 0 there exist E(δ, f, h) := E ⊂ (0, R), r0 ∈ (0, R) such that ∀ r ∈ (r0, R)\E: Mf (r) ≤ h(r)µf (r) { ln h(r)ln(h(r)µf (r)}1/2+δ and⎰ E h(r)d ln r < +∞ 2 0 . If we additionally assume that the function f ∈ ER is unbounded, then ln Mf (r) ≤ (1 + o(1))ln(h(r)µf (r)) holds as r → R, r ∈/ E. Remark, that assertion 10 at h(r) ≡ const implies the classical Wiman-Valiron theorem for entire functions and at h(r) ≡ 1/(1 − r) theorem about the Kovari-type inequality for analytic functions ¨ in the unit disc. From statement 20 in the case that ln h(r) = o(ln µf (r)), r → R, it follows that ln Mf (r) = (1 + o(1))ln µf (r) holds as r → R, r ∈/ E. [ABSTRACT FROM AUTHOR] |
