Extensions of sufficient conditions for starlikeness and convexity of order αα

For analytic functions f(z)f(z) in the open unit disk UU, there are many sufficient conditions for f(z)f(z) to be starlike and convex in UU. 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)f(z) to be starlike of order αα and to be convex of order αα are discussed.