norm estimates of the pre–schwarzian derivatives for two certain subclasses of starlike functions

Let α ∈ [π/2, π) and γ1, γ2 ∈ (0, 1]. for a normalized analytic functions f in the open unit disc ∆ we consider m(α) :=( f ∈ a : 1 + α − π 2 sin α < re ( zf0 (z) f(z) ) < 1 + α 2 sin α , z ∈ ∆ ), and s ∗ t (γ1, γ2) := ) f ∈ a : − πγ1 2 < arg) z f 0 (z) f(z)  < πγ2 2 , z ∈ ∆) . in the present paper, we establish a sharp norm estimate of the pre– schwarzian derivative for functions f belonging two these subclasses of analytic and normalized functions