On the Radii of univalence of Gel'fond-Leont'ev derivatives

Abstract

Let $0 < R < +\infty,$ let $A(R)$ bethe class of functions $$f(z) = \sum_{k=0}^{\infty}f_kz^k,$$ analytic in $\{ z: |z| < R \}$, and let $$l(z) = \sum_{k=0}^{\infty}l_kz^k,\; l_k > 0$$ 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 $|f_k/f_{k+1} \nearrow R,\; k \rightarrow \infty,\; 0 < R < +\infty$, then the sequence $(\rho_n)$ of radii of univalence of the Gel'fondLeont'ev derivatives satisfies the relation $$D^n_lf(z) = \sum_{k=0}^{\infty}\frac{l_kf_{k+n}}{l_{k+n}}z_k$$ The case where the condition $|f_k/f_{k+1}|\nearrow R,\quad k \rightarrow \infty$, is not satisfied is also considered.