This note is devoted to applying the principle of subordination in order to explore the Roper-Suffridge extension operator and the Pfaltzgraff-Suffridge extension operator with special analytic properties.First,we prove that both the Roper-Suffridge extension operator and the Pfaltzgraff-Suffridge extension operator preserve subordination.As applications,we obtain that if β∈[0,1],γ∈[0,1/r]and β+γ≤1,then the Roper-Suffridge extension operator Φ_(β+γ)(f)(z)=(f(z_(1)),(f(z_(1))/z_(1))^(β)(f’(z_(1)))^(γ)w),z∈Ω_(p,r) preserves an almost starlike mapping of complex order λ on Ω_(p,r)={z=(z_(1),w)∈C×X:|z_(1)|^(p)+‖w‖_(X)^(r)<1},where 1≤p≤2,r≥1 and X is a complex Banach space.Second,by applying the principle of subordination,we will prove that the Pfaltzgraff-Suffridge extension operator preserves an almost starlike mapping of complex order λ.Finally,we will obtain the lower bound of distortion theorems associated with the Roper-Suffridge extension operator.This subordination principle seems to be a new idea for dealing with the Loewner chain associated with the Roper-Suffridge extension operator,and enables us to generalize many known results from p=2 to 1≤p≤2.
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