SOME RESULTS CONCERNING SPIRAL-LIKENESS OF A CLASS OF ANALYTIC FUNCTIONS

O. P. Ahuja
1993 Demonstratio Mathematica  
Silverman [12] observed that for every 7 real, 7 ^ 0,1, there exists an / 6 5 for which g-y(z) 6 S. Some of the above observations about the operators of the form (1.3), have led us to investigate the spiral-likeness and other properties of these operators. Denote by H x (a) the class of A-spiral-like functions of order a in A. An analytic function / in A is in H x (a) if and only if /(0) = 0, /'(0) = 1, and Re(e iX zf'(z)/f((z)) > a cos A for some a, A(0 < a < 1,-t/2 < A < tt/2) and for all z
more » ... A. This class was first studied by Libera [5]. We note that H x (a) C H x ( 0) = H x , the family of A-spiral-like functions. It was shown by Spacek [14] that such functions are univalent in A. Associated with H x (a) is the family C A (a) of analytic functions / for which /(0) = 0, /'(0) = 1 and zf'(z) belongs to H x (a). This family was introduced by Chichra [2]. A function / in C x (a) is called a A-Robertson of order a (see, for example [1]). The class C A (0) was studied by Robertson [9]. It may be noted that a function / in C A (a) is not necessarily univalent in A (see, for example [2], [3], [9]). Also, H°(a) = S*(a) and C°(0) = K.
doi:10.1515/dema-1993-3-416 fatcat:dyqdk2apozhzdheelkcmzrrisi