Starlikeness of Hadamard product of certain analytic functions

An interesting criterion was given by Lashin (A.Y. Lashin, Some convolution properties of analytic functions, Appl. Math. Lett. 18 (2005) 135–138) to be starlike for convolution of analytic functions f, g   such that Re[f′(z)]⩾α,Re[g′(z)]⩾β,α,β<1Re[f′(z)]⩾α,Re[g′(z)]⩾β,α,β<1 in the unit disc U. In this paper we shall improve this criterion.