Let P, d p ( X ) denote the family of functionsf, normalized by f(0) = 0 = f' (0) -1, that are analytic in the open unit disc A = {z : Izi < 11, f(z)#O for z E A\{0) and such that for some ru # 0, E C and g E A with g(z) # 0 in 0 < Izj < 1. The main object of this paper is to study this class and to find conditions on a , and on the function g such that each function f in P,,,:,(X) belongs to a family which is contained in the family of univalent functions in the unit disc A. We also find the exact value of max{Iz/f (z) -1 / } in the class P(X) := P,,o,2(X) for fixed z E A. Further, we also determine condition on X for functions f in P(X) 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 ~y / 2 with y E (O,l]. 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.

Abstract. Let U(λ) denote the family of analytic functions f (z), f (0) = 0 = f (0) − 1, in the unit disk D, which satisfy the condition z/f (z) 2 f (z) − 1 < λ for some 0 < λ ≤ 1. The logarithmic coefficients γ n of f are defined by the formula log(f (z)/z) = 2 ∞ n=1 γ n z n . In a recent paper, the present authors proposed a conjecture that if f ∈ U(λ) for some 0 < λ ≤ 1, then |a n | ≤ n−1 k=0 λ k for n ≥ 2 and provided a new proof for the case n = 2. One of the aims of this article is to present a proof of this conjecture for n = 3, 4 and an elegant proof of the inequality for n = 2, with equality for f (z) = z/[(1 + z)(1 + λz)]. In addition, the authors prove the following sharp inequality for f ∈ U(λ):where Li 2 denotes the dilogarithm function. Furthermore, the authors prove two such new inequalities satisfied by the corresponding logarithmic coefficients of some other subfamilies of S.

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A sharp coefficient estimate is obtained for a class D(a) of functions univalent in the open unit disc. The radius of convexity and an arclength result are also determined for the class.

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