A subclass with bi-univalence involving ( $${\mathfrak {p}}$$ p , $${\mathfrak {q}}$$ q )- Lucas polynomials and its c
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ORIGINAL ARTICLE
A subclass with bi-univalence involving (p,q)- Lucas polynomials and its coefficient bounds S. Yalçın1
•
K. Muthunagai2
•
G. Saravanan3,4
Received: 3 September 2019 / Accepted: 24 April 2020 Ó Sociedad Matemática Mexicana 2020
Abstract We have constructed a subclass of analytic bi-univalent functions using (p,q)-Lucas polynomials in this research contribution. Bounds for certain coefficients and Fekete–Szego¨ inequalities have been estimated. Keywords Analytic functions Bi-univalent functions (p; q )-Lucas polynomials Fekete-Szego¨ Inequality
Mathematics Subject Classification 30C45 30C15
1 Introduction Let fðzÞ be a normalized analytic function of the form: fðzÞ ¼ z þ
1 X
a n zn ; z 2 U
ð1:1Þ
n¼2
where U ¼ fz : z 2 C; jzj\1g and let A be the class of all such functions. Let S ¼ ffðzÞ 2 A 3 fðzÞ is univalent in Ug. & K. Muthunagai [email protected] S. Yalc¸ın [email protected] G. Saravanan [email protected] 1
Department of Mathematics, Bursa Uludag University, 16059 Bursa, Turkey
2
School of Advanced Sciences, VIT University, Chennai 600 127, Tamil Nadu, India
3
Research Scholar, School of Advanced Sciences, VIT University, Chennai 600 127, Tamil Nadu, India
4
Present Address: Department of Mathematics, Patrician College of Arts and Science, Adyar, Chennai 600 020, Tamil Nadu, India
123
S. Yalçın et al.
For f1 ðzÞ and f2 ðzÞ 2 A, we say that f1 ðzÞ is subordinate to f2 ðzÞ, if there exists a function namely, Schwarz function w(z) with wð0Þ ¼ 0 and jwðzÞj\1 in U, such that f1 ðzÞ ¼ f2 ðwðzÞÞ, and we write f1 ðzÞ f2 ðzÞ. Obtaining sharp bounds for ja3 ga22 j of any compact family of functions is called the Fekete–Szego¨ problem. In particular, when g ¼ 1, the functional represents Schwarzian derivative and the role of Schwarzian derivative in the theory of Geometric functions is remarkable. Let R denote the class of all bi-univalent functions in U. We say that a function fðzÞ in S belongs to R, if both fðzÞ and its inverse have an analytic continuation to jwj\1. Lewin [9] introduced the class of bi-univalent functions in 1967 and gave an estimate for the second coefficient for functions belonging to this class as pffiffiffi ja2 j\1:51. His result was improved by Brannan and Clunie [2] to ja2 j 2. There is an extensive literature on the estimates of the initial coefficients of bi-univalent functions (see [3, 6, 7, 14, 19, 20]). Special polynomials like Lucas polynomials and their generalizations play a vital role in Numerical Analysis, Number Theory, Combinatorics, etc. These polynomials have attracted the attention of several researchers and research has been productive in this field (see, for example, [1, 4, 5, 10–12, 15–18]). Definition 1.1 (see [1, 8]) Consider two polynomials pðxÞ and qðxÞ with real coefficients. The ðp; qÞ-Lucas polynomial Lp;q;n ðxÞ is given by the recurrence relation:
Lp;q;n ðxÞ ¼ pðxÞLp;q;n1 ðxÞ þ qðxÞLp;q;n2 ðxÞ
ðn 2Þ:
The first few Lucas polynomials are listed below: Lp;q;0 ðxÞ ¼ 2; Lp;q;1 ðxÞ ¼ pðxÞ; Lp;q;2 ðxÞ ¼ p2 ðxÞ þ 2qðxÞ
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