Extensions of sufficient conditions for starlikeness and convexity of order

For analytic functions f ( z ) in the open unit disk U , there are many sufficient conditions for f ( z ) to be starlike and convex in U . In view of the paper by P.T. Mocanu [P.T. Mocanu, Some starlikeness conditions for analytic functions, Rev. Roumaine Math. Pures Appl. 33 (1988) 117–124], some extensions of sufficient conditions for f ( z ) to be starlike of order α and to be convex of order α are discussed.