DEMONSTRATIO MATHEMATICA Vol. XXXVI No 1 2003 D. Blezu, N. N.Pascu, R. N. Pascu INTEGRAL OPERATOR WHICH PRESERVES THE UNIVALENCE IN THE UPPER HALF-PLANE Abstract. In this paper, using the Pfaltzgraff integral operator and the so called "parametric circles method" introduced by N.N.Pascu (1999), we can obtain an univalence criterion for the analytic function defined in the upper half-plane and also for comparison two univalence criteria obtained by a simple composition of functions. Introduction We will denote by D the upper half-plane and by S(D) the class of analytic univalent functions in D which are not necessarily hydrodinamic normalized. S(U) is the class of analytic, and univalent functions in U /(0) = 0, The function ip : U —> D, tp(u) = ij^ maps the unit disk U in D. For 0 < r < 1, the image of the disk Ur = {z € C, \z\ = r } under is the disk DT = {z G C : \z — zr \ < Rr} where 1. Preliminary results It is known that an important problem is the preservation of the uni- valence of function through integral operators. Therefore we mention the classical Pfaltzgraff integral operator which will be used in this paper. T H E O R E M A [PF]. If f e S(U) then for a e C , |A| < \ the function defined f ' ( 0 ) = 1. _ . 1 + r 2 1991 Mathematics Subject Classification: 30C45. Presented at the International Conference on Complex Analysis and Related Topics and the IX-th Romanian-Finnish Seminar August 27-31, 2001, Bra^ov, Romania Unauthenticated Download Date | 2/26/20 9:09 AM