2018
Том 70
№ 4

# On a complete description of the class of functions without zeros analytic in a disk and having given orders

Chyzhykov. I. E.

Abstract

For arbitrary $0 ≤ σ ≤ ρ ≤ σ + 1$, we describe the class $A_{σ}^{ρ}$ of functions $g(z)$ analytic in the unit disk $D = \{z : ∣z∣ < 1\}$ and such that $g(z) ≠ 0,\; ρ_T[g] = σ$, and $ρ_M[g] = ρ$, where $M(r,g) = \max \{|g(z)|:|z|⩽r\},\quad$ $T(r,u) = \cfrac1{2π} ∫_0^{2π} ln^{+}|g(re^{iφ})|dφ,\quad$ $ρ_M[g] = \lim \sup_{r↑1} \cfrac{lnln^{+}M(r,g)}{−ln(1−r)},$ $\quad ρT[g] = \lim \sup_{r↑1} \cfrac{ln^{+}T(r,g)}{−ln(1−r)}$.

English version (Springer): Ukrainian Mathematical Journal 59 (2007), no. 7, pp 1088-1109.

Citation Example: Chyzhykov. I. E. On a complete description of the class of functions without zeros analytic in a disk and having given orders // Ukr. Mat. Zh. - 2007. - 59, № 7. - pp. 979–995.

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