Some properties of starlike functions with respect to symmetric-conjugate points

Hassoon Al-Amiri, Dan Coman and Petru T. Mocanu

International Journal of Mathematics and Mathematical Sciences, 1995, vol. 18, 1-6

Abstract:

Let A be tile class of all analytic functions in the unit disk U such that f ( 0 ) = f ′ ( 0 ) − 1 = 0 . A function f ∈ A is called starlike with respect to 2 n symmetric-conjugate points if Re z f ′ ( z ) / f n ( z ) > 0 for z ∈ U , where f n ( z ) = 1 2 n ∑ k = 0 n − 1 [ ω − k f ( ω k z ) + ω k f ( ω k z ˜ ) ¯ ] , ω = exp ( 2 π i / n ] . This class is denoted by S n * , and was studied in [1]. A sufficient condition for starlikeness with respect to symmetric-conjugate points is obtained. In addition, images of some subclasses of S n * under the integral operator I : A → A , I ( f ) = F where F ( z ) = c + 1 ( g ( z ) ) c ∫ 0 z f ( t ) ( g ( t ) ) c − 1 g ′ ( t ) d t , c > 0 and g ∈ A is given are determined.

Date: 1995
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:524353

DOI: 10.1155/S0161171295000597

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