MODIFIED ROPER-SUFFRIDGE OPERATOR FOR SOME SUBCLASSES OF STARLIKE MAPPINGS ON REINHARDT DOMAINS

In this note,the author introduces some new subclasses of starlike mappings SΩn,p2,…,pn(β,A,B)={f ∈H(Ω):|itan β +(1-itan β)2/ρ(z)ρ/z(z)Jf-1(z)f(z)-1-AB/1-B2|<B-A/1-B2},on Reinhardt domains Ωn,p2,…,pn={z ∈Cn:|z1|2+∑n j=2 |zj|pj<1},where -1 ≤ A<B<1,q=min{p2,···,pn }≥1,l=max{p2,···,pn }≥2 and β∈(-π/2,π/2).Some diferent conditions for P are established such that these classes are preserved under the following modified Roper-Sufridge operator F(z)=(f(z1)+f′(z1)Pm(z0),(f′(z1))1/mz0)′,where f is a normalized biholomorphic function on the unit disc D,z=(z1,z0) ∈Ωn,p2,…,pn,z0=(z2,···,zn) ∈Cn-1.Another condition for P is also obtained such that the above generalized Roper-Sufridge operator preserves an almost spirallike function of type β and order α.These results generalize the modified Roper-Sufridge extension oper- ator from the unit ball to Reinhardt domains.Notice that when p2=p3=···=pn=2,our results reduce to the recent results of Feng and Yu.