On certain subclasses of univalent functions of complex order associated with poisson distribution series
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ORIGINAL ARTICLE
On certain subclasses of univalent functions of complex order associated with poisson distribution series Aslıhan Altunhan1 • Sevtap Su¨mer Eker2 Received: 12 December 2019 / Accepted: 12 June 2020 Ó Sociedad Matemática Mexicana 2020
Abstract In the present paper, making a connection between some subclasses of univalent functions of complex order and Poisson distribution series, we give some conditions for Poisson distribution series belonging to these subclasses. Keywords Univalent functions Complex order Poisson Distributions Coefficient bounds and coefficient estimates
Mathematics Subject Classification 30C45 30C50 30C55
1 Introduction Let U ¼ fz : z 2 C and jzj\1g. Let A stand for the standard class of analytic functions f : U ! C normalized by f ðzÞ ¼ z þ
1 X
aj z j :
ð1Þ
j¼2
Denote by S the class of functions in A which are univalent. A function f 2 A is said to be starlike of complex order c ðc 2 C ¼ C n f0g) if and only if f ðzÞ z 6¼ 0; z 2 U, and & Sevtap Su¨mer Eker [email protected] 1
Institute of Natural and Applied Science, Dicle University, Diyarbakir, Turkey
2
Department of Mathematics, Faculty of Science, Dicle University, Diyarbakir, Turkey
123
A. Altunhan, S. S. Eker
1 zf 0 ðzÞ 1 [ 0; Re 1 þ c f ðzÞ
ðz 2 UÞ:
ð2Þ
We denote by SðcÞ the class of all such functions. The class SðcÞ introduced by Nasr and Aouf [1] (See also [13]). For a function f 2 A, we say that it is convex function of order c ðc 2 C ), that is f 2 CðcÞ, if and only if f 0 ðzÞ 6¼ 0 in U and 1 zf 00 ðzÞ Re 1 þ [ 0; ðz 2 UÞ: ð3Þ c f 0 ðzÞ The class CðcÞ was introduced by Wiatrowski [2] (See also [14–16]). We note that f 2 CðcÞ () zf 0 2 SðcÞ. For a function f 2 A, we say that it is close-to-convex function of order c ðc 2 C ), that is f 2 RðcÞ, if and only if 1 ð4Þ Re 1 þ ðf 0 ðzÞ 1Þ [ 0; ðz 2 UÞ: c The class RðcÞ was introduced by Halim [8] and Owa [9]. If the special values of c are taken in the inequality (2), various well-known subclasses of univalent functions are obtained. For example, for c ¼ 1, c ¼ 1 a; ð0 a\1Þ, c ¼ cos heih ðh is real and jhj\ p2Þ and c ¼ ð1 ih ðh is real and jhj\ p2Þ, we obtain the class of starlike functions, S , aÞcoshe class of starlike functions of order a, S ðaÞ, the class of spirallike functions S h and the class of spirallike functions of order a, S ha , respectively. Also, for the special values of c in the inequality (3), we have other well-known subclasses of univalent functions. For example, for c ¼ 1, c ¼ 1 a ð0 a\1Þ and c ¼ cos heih ðh is real and jhj\ p2Þ , we obtain the class of convex functions, C, the class of convex functions of order a, CðaÞ and the class of h-Robertson functions S h , respectively. Let us define by T the subclass of functions f in A of the form f ðzÞ ¼ z
1 X
aj z j ;
ðaj 0Þ:
ð5Þ
j¼2
Altıntas¸ et al. [5] defined following subclasses of AðnÞ which consist of the funcP j tions of the form f ðzÞ ¼ z 1 j¼nþ1 aj z : Definition 1 [5] Let S n ðc; k; bÞ denote the subclass of AðnÞ co
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