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Abstract
In this article, we consider a class denoted by which consists of functions f that are holomorphic in the unit disc
punctured at a point
where f has a simple pole. We prove a sufficient condition for these functions to be univalent in
. By using this condition, we construct the family
of all functions
such that
where
for some
,
. Therefore, functions in the class
are necessarily univalent. We present some basic properties for functions in the class
which include an integral representation formula for such functions and obtain the exact region of variability of the second Taylor coefficient for functions in this class. We also obtain a sharp estimate for the Fekete–Szegö functional defined on the class
along with a subordination result for functions in this family. In addition, we obtain some necessary and sufficient coefficient conditions involving the coefficients
for functions
of the form
to be in the class . We have also obtained sharp bounds for
,
.
AMS Subject Classification:
Acknowledgements
The authors thank Karl-Joachim Wirths for his suggestions and careful reading of the manuscript.
Notes
No potential conflict of interest was reported by the authors.