THE GENERALIZED ROPER-SUFFRIDGE EXTENSION OPERATOR ON REINHARDT DOMAIN Dp

We define the generalized Roper-Suffridge extension operator Φn,β₂,γ₂,...,βn,γn(f) on Reinhardt domain Dp as ${\phi _n},{\beta _2},{\gamma _2}, \cdots ,{\beta _n},{\gamma _n}\left( f \right)\left( z \right) = \left( {f\left( {{z_1}} \right),{{\left( {\frac{{f\left( {{z_1}} \right)}}{{{z_1}}} \right)}^{{\beta _2}}}{{\left( {f'\left( {{z_1}} \right)} \right)}^{{\gamma _2}}}{z_2}, \ldots ,{{\left( {\frac{{f\left( {{z_1}} \right)}}{{{z_1}}} \right)}^{{\beta _n}}}{{\left( {f'\left( {{z_1}} \right)} \right)}^{{\gamma _n}}}{z_n}} \right)$; for z = (z₁, z₂, · · · ,zn) ∊ Dp, where ${D_p} = \left\{ {\left( {{z_1},{z_2}, \ldots ,{z_n}} \right) \in{C^n}:\sum\limits_{j = 1}^n {{{\left| {{z_j}} \right|}^{{p_j}}} < 1} \right\}$ , p = (p₂, p₂, · · · , pn), pj > 0, 0 ≤ γj ≤ 1 - βj, 0 ≤ βj ≤ 1, j = 1,2, · · · , n, and we choose the branch of the power functions such that ${\left( {\frac{{f\left( {{z_1}} \right)}}{{{z_1}}} \right)^{{\beta _j}}}{|_{{z_1} = 0}} = 1$ and ${\left( {f'\left( {{z_1}} \right)} \right)^{{\gamma _j}}}{|_{{z_1} = 0}} = 1$, j = 2, · · · , n. In the present paper, we show that the operator Φn,β₂,γ₂,...,βn,γn(f) preserves almost spirallike mapping of type β and order α and spirallike mapping of type β and order α on Dp for some suitable constants βj, γj, pj. The results improve the corresponding results of earlier authors.