Using the technique of differential subordination, in particular, a lemma due to Miller and Mocanu 1, we study a certain differential operator to obtain some sufficient conditions for starlike, convex, strongly starlike and strongly convex functions. In particular, we prove that if f is an element of A(n) (zF'(z)/F(z)(gamma) not equal 0, z is an element of E satisfies gamma (1 + zF ''(z)/F'(z) - zF'(z)/F(z) ) < 2n(1 - alpha)z/(1 - z) (1 + (1 - 2 alpha)z)' 0 <= alpha < 1, z is an element of E, then (zF'(z)/F(z))(gamma) < 1 + (1 - 2 alpha)z/1 - z, z is an element of E, where F(z) = (1 - lambda) f(z) + f'(z), 0 <= lambda <= 1 is univalent and gamma is a non-zero complex number.
展开▼
