ANNALES POLONICI MATHEMATICI 86.1 (2005) Univalence, strong starlikeness and integral transforms by M. Obradovi´ c(Belgrad), S. Ponnusamy (Chennai), and P. Vasundhra (Chennai) Abstract. Let A represent the class of all normalized analytic functions f in the unit disc ∆. In the present work, we first obtain a necessary condition for convex functions in ∆. Conditions are established for a certain combination of functions to be starlike or convex in ∆. Also, using the Hadamard product as a tool, we obtain sufficient conditions for functions to be in the class of functions whose real part is positive. Moreover, we derive conditions on f and µ so that the non-linear integral transform z 0 (ζ/f (ζ )) μ dζ is univalent in ∆. Finally, we give sufficient conditions for functions to be strongly starlike of order α. 1. Introduction. Let H denote the class of all functions analytic in the unit disc ∆ = {z : |z | < 1}, and A the class of all normalized functions f (f (0) = f ′ (0) − 1 = 0) in H. Let S denote the univalent subclass of A, and S ∗ denote the subclass of f ∈S for which f (∆) is starlike with respect to the origin. Recall the prominent subclasses studied in the theory of univalent functions (see [7]), for 0 ≤ β< 1: P (β )=  f ∈A : Re  f (z ) z  > β, z ∈ ∆  , R(β )= {f ∈A : zf ′ ∈P (β )}, S ∗ (β )=  f ∈A : Re  zf ′ (z ) f (z )  > β, z ∈ ∆  , S ∗ β =  f ∈A :     arg  zf ′ (z ) f (z )     < βπ 2 ,z ∈ ∆  , K(β )= {f ∈A : zf ′ ∈S ∗ (β )}. It is well known that K≡K(0)  S ∗ (1/2). Functions in S ∗ β are called strongly starlike of orde r β , while those in S ∗ (β ) are starlike of order β . For β< 0, S ∗ (β )  S , while for 0 <β< 1, S ∗ (β )  S ∗  S , and functions 2000 Mathematics Subject Classification : 30C45, 30C55. Key words and phrases : univalent, starlike, strongly starlike and convex functions. The authors thank the referee for his/her valuable comments. [1]