201311

Code: 201311

Modified Roper-Suffridge Operator for Some Subclasses of Starlike Mappings on Reinhardt Domains

Jianfei Wang

 

Abstracts

In this note, the author introduces some new subclasses of starlike mappings

<img height="67" border="0" style="vertical-align:bottom" width="519" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0252960213601101-si1.gif">$\begin{array}{l}S*\Omega n,p_2\mathrm{\dots }{P}_n(\beta ,A,B)\\ =\{f\in H(\Omega ):|i\hspace{0.17em}tan\hspace{0.17em}\beta +(1-i\hspace{0.17em}tan\hspace{0.17em}\beta \frac{2}{\rho (z)}\frac{\partial \rho }{\partial z}(z)J_f^{-1}(z)f(z)-\frac{1-AB}{1-B^2}|<\frac{B-A}{1-B^2}\},\end{array}$

on Reinhardt domains <img height="47" border="0" style="vertical-align:bottom" width="316" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0252960213601101-si2.gif">${\Omega }_{n,p_2\mathrm{\dots }{p}_n}=\{z\in {ℂ}^n:|z_1{|}^2+{\sum _{j=2}^n\left|z_j\right|}^{p_j}<\hspace{0.17em}1\hspace{0.17em}\}$ where –1 ≤ A < B < 1, q = min{p₂,…, p_n} ≥ 1, l = max{p₂,…, p_n} ≥ 2 and β ∈ ( <img height="27" border="0" style="vertical-align:bottom" width="42" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0252960213601101-si3.gif">$-\frac{\pi }{2},\hspace{0.17em}\frac{\pi }{2}$) Some different conditions for P are established such that these classes are preserved under the following modified Roper-Suffridge operator

<img height="43" border="0" style="vertical-align:bottom" width="286" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0252960213601101-si4.gif">$F\left(z\right)={\left(f\left(z_1\right)+f^{\prime }\left(z_1\right)P_m\left(z_0\right),{\left(f^{\prime }\left(z_1\right)\right)}^{\frac{1}{m}}{z}_0\right)}^{\prime },$

where f is a normalized biholomorphic function on the unit disc D, z = (z₁, z₀) ∈ Ω_{n,p2,…, pn}, z₀ = (z₂,…, z_n) ∈ ℂⁿ⁻¹. Another condition for P is also obtained such that the above generalized Roper-Suffridge operator preserves an almost spirallike function of type β and order α. These results generalize the modified Roper-Suffridge extension operator from the unit ball to Reinhardt domains. Notice that when p₂ = p₃ = … = p_n = 2, our results reduce to the recent results of Feng and Yu