A New Class of Salagean-Type Multivalent Harmonic Functions Defined by Subordination
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RESEARCH PAPER
A New Class of Salagean-Type Multivalent Harmonic Functions Defined by Subordination Serkan Çakmak1 • Sibel Yalçın1
•
S¸ahsene Altınkaya1
Received: 3 February 2020 / Accepted: 26 September 2020 Ó Shiraz University 2020
Abstract Utilizing the concepts of subordination we have introduced a generalized class Salagean-Type of complex-valued multivalent harmonic functions. We construct some properties of our class. The results obtained here include a number of known and new results as their special cases. Keywords Harmonic functions Multivalent functions Salagean differential operator Subordination Mathematics Subject Classification 30C45 30C80
1 Introduction, Main Notations and Definitions A continuous complex-valued function f ¼ u þ iv defined in a simply connected complex domain D C is said to be harmonic in D if u and v are both real harmonic in D . Consider the functions U and V analytic in D so that u ¼ ReU and v ¼ ImV. Then the harmonic function f can be expressed by f ðzÞ ¼ hðzÞ þ gðzÞ
ðz 2 DÞ;
where h ¼ ðU þ VÞ=2 and g ¼ ðU VÞ=2. In particular, h is called the analytic part and g is called the co-analytic part of f. It is known (see Clunie and SheilSmall 1984) that the function f ¼ h þ g is locally univalent and sense-preserving in the open unit disk U ¼ fz : z 2 C and jzj\1g if and only if jg0 ðzÞj\jh0 ðzÞj ðz 2 DÞ. The coefficient estimations, distortion theorems, integral expressions, Jacobi estimates and growth condition in geometric properties of covering theorem of the co& Sibel Yalc¸ın [email protected] Serkan C¸akmak [email protected] S¸ ahsene Altınkaya [email protected] 1
Department of Mathematics, Faculty of Arts and Sciences, Bursa Uludag University, 16059 Bursa, Turkey
analytic part can be obtained by using the analytic part of harmonic functions (see Ahuja 2014; Ahuja et al. 2019; Aouf and Seoudy 2020; Baksa et al. 2020; Dixit et al. 2011 and the bibliography therein). For a fixed positive integer p 1, let SH(p) denote the class of all multivalent harmonic functions f ¼ h þ g which are sense-preserving in the open unit disk U and are of the form 1 1 X X f ðzÞ ¼ zp þ a n zn þ bn zn jbp j\1 : ð1Þ n¼p
n¼pþ1
Recent interest in the study of multivalent harmonic function prompted the publication of several articles such as Ahuja and Jahangiri (2001, 2002), Jahangiri et al. (2003, 2009), Yas¸ ar and Yalc¸ın (2011, 2013, 2015) and the references cited therein. We say that a function f 2 SH(p) is subordinate to a function F 2 SHðpÞ; and write f ðzÞ FðzÞ; if there exists a complex-valued function w which maps U into oneself with wð0Þ ¼ 0, such that f ðzÞ ¼ FðwðzÞÞ ðz 2 UÞ: Further, for functions f1 ; f2 2 SHðpÞ of the forms: 1 1 X X an z n þ bn zn ð t ¼ 1; 2Þ; ft ðzÞ ¼ zp þ n¼p
n¼pþ1
we define convolution or Hadamard product of f1 and f2 by ðf1 f2 ÞðzÞ ¼ zp þ
1 X n¼pþ1
a1;n a2;n zn þ
1 X
b1;n b2;n zn :
n¼p
123
Iran J Sci Technol Trans Sci
Note that the class SH(p) for p ¼ 1 was defined
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