On the univalence radii of successive Gelfond-Leont'ev-Salagean and Gelfond-Leont'ev-Ruscheweyh derivatives

Анотація

For an analytic in the disk $\left\{z:\left|z\right|<1\right\}$ function $f\left(z\right)=z+\underset{i=1}{\overset{\infty }{\sum f_k{z}^k}}$ and formal power series $l\left(z\right)=z+\sum _{k=1}^{\infty }{l}_k{z}^k$ with $l_k>0$ the operator
$D_{l,\left[S\right]}^{n}f\left(z\right)=z+\sum _{k=2}^{\infty }{\left(\frac{l_1{l}_{k-1}}{l_k}\right)}^k{f}_k{z}^k$
is called the Gelfond-Leont'ev-Salagean derivative and the operator
$D_{l,\left[R\right]}^{n}f\left(z\right)=z+\sum _{k=2}^{\infty }{\left(\frac{l_{k-1}{l}_n}{l_{n+k-1}}\right)}^k{f}_k{z}^k$
is called the Gelfond-Leont'ev-Ruscheweyh derivative. By $\rho \left[f\right]$ we denote the radius of the univalence of the function $f$. It is proved, for example, that for each $n\ge 1$
$\frac{\sqrt{2}-1}{\sqrt{2}}\left|\frac{f_1}{f2}\right|{\left(\frac{l_2}{l_1^2}\right)}^n\le \rho \left[D_{l,\left[S\right]}^{n}f\right]\le 2\left|\frac{f_1}{f_2}\right|{\left(\frac{l_2}{l_1^2}\right)}^n$
and
$\frac{\sqrt{2}-1}{\sqrt{2}}\left|\frac{f_1}{f2}\right|\frac{l_{n+1}}{l_1{l}_n}\le \rho \left[D_{l,\left[R\right]}^{n}f\right]\le 2\left|\frac{f_1}{f_2}\right|\frac{l_{n+1}}{l_1{l}_n}.$