On the Radii of univalence of Gel'fond-Leont'ev derivatives - Sheremeta, M. M. (Ukrains’kyi Matematychnyi Zhurnal, 1995)
Let $0 < R < +∞,$ let $A(R)$ bethe class of functions$$f(z) = \sum_{k=0}^{∞}f_kz^k,$$analytic in $\{ z: |z| < R \}$, and let$$l(z) = \sum_{k=0}^{∞}l_kz^k,\; l_k > 0$$be a formal power series. We prove that if $l^2_k/l_{k+1}l_{k-1}$ is a nonincreasing sequence, $f \in A(R)$, and$|f_k/f_{k+1} earrow R,\; k → ∞,\; 0 < R < +∞$, then the sequence $(ρ_n)$ of radii of univalence of the Gel'fondLeont'ev derivativessatisfies the relation$$D^n_lf(z) = \sum_{k=0}^{∞}\frac{l_kf_{k+n}}{l_{k+n}}z_k$$The case where the condition $|f_k/f_{k+1}| earrow R,\quad k → ∞$, is not satisfied is also considered.