New criteria and distortion theorems for univalent functions

Let denote the family of functions f, normalized by f(0)=0=f′−1, that are analytic in the open unit disc and such that for some with . The main object of this paper is to study this class and to find conditions on α, β and on the function g such that each function f in belongs to a family which is contained in the family of univalent functions in the unit disc δ. We also find the exact value of in the class for fixed z∊δ. Further, we also determine condition on λ for functions f in P(λ) to be in the class of strongly starlike functions, or in the class of functions whose derivative lies in a sector of angle less than or equal to πγ/2 with γ∊(0,1]. Finally, we also obtain a sufficient condition for an analytic function f to satisfy the analytic univalence criteria of Noshiro-Warschawski. Several examples are stated in support of the sharpness of our criteria.