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On the Taylor Coefficients of a Subclass of Meromorphic Univalent Functions

On the Taylor Coefficients of a Subclass of Meromorphic Univalent Functions Let $${\mathcal {V}}_p(\lambda )$$ V p ( λ ) be the collection of all functions f defined in the unit disc $${{\mathbb {D}}}$$ D having a simple pole at $$z=p$$ z = p where $$0<p<1$$ 0 < p < 1 and analytic in $${{\mathbb {D}}}\setminus \{p\}$$ D \ { p } with $$f(0)=0=f'(0)-1$$ f ( 0 ) = 0 = f ′ ( 0 ) - 1 and satisfying the differential inequality $$|(z/f(z))^2 f'(z)-1|< \lambda $$ | ( z / f ( z ) ) 2 f ′ ( z ) - 1 | < λ for $$z\in {{\mathbb {D}}}$$ z ∈ D , $$0<\lambda \le 1$$ 0 < λ ≤ 1 . Each $$f\in {\mathcal {V}}_p(\lambda )$$ f ∈ V p ( λ ) has the following Taylor expansion: $$\begin{aligned} f(z)=z+\sum _{n=2}^{\infty }a_n(f) z^n, \quad |z|<p. \end{aligned}$$ f ( z ) = z + ∑ n = 2 ∞ a n ( f ) z n , | z | < p . We recently conjectured that $$\begin{aligned} |a_n(f)|\le \frac{1-(\lambda p^2)^n}{p^{n-1}(1-\lambda p^2)}\quad \text{ for }\quad n\ge 3, \end{aligned}$$ | a n ( f ) | ≤ 1 - ( λ p 2 ) n p n - 1 ( 1 - λ p 2 ) for n ≥ 3 , while investigating functions in the class $${\mathcal {V}}_p(\lambda )$$ V p ( λ ) . In the present article, we first obtain a representation formula for functions in this class. Using this, we prove the aforementioned conjecture for $$n=3,4,5$$ n = 3 , 4 , 5 whenever p belongs to certain subintervals of (0, 1). Also we determine non sharp bounds for $$|a_n(f)|,\,n\ge 3$$ | a n ( f ) | , n ≥ 3 and for $$|a_{n+1}(f)-a_n(f)/p|,\,n\ge 2$$ | a n + 1 ( f ) - a n ( f ) / p | , n ≥ 2 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Malaysian Mathematical Sciences Society Springer Journals

On the Taylor Coefficients of a Subclass of Meromorphic Univalent Functions

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References (13)

Publisher
Springer Journals
Copyright
Copyright © 2018 by Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia
Subject
Mathematics; Mathematics, general; Applications of Mathematics
ISSN
0126-6705
eISSN
2180-4206
DOI
10.1007/s40840-018-0685-y
Publisher site
See Article on Publisher Site

Abstract

Let $${\mathcal {V}}_p(\lambda )$$ V p ( λ ) be the collection of all functions f defined in the unit disc $${{\mathbb {D}}}$$ D having a simple pole at $$z=p$$ z = p where $$0<p<1$$ 0 < p < 1 and analytic in $${{\mathbb {D}}}\setminus \{p\}$$ D \ { p } with $$f(0)=0=f'(0)-1$$ f ( 0 ) = 0 = f ′ ( 0 ) - 1 and satisfying the differential inequality $$|(z/f(z))^2 f'(z)-1|< \lambda $$ | ( z / f ( z ) ) 2 f ′ ( z ) - 1 | < λ for $$z\in {{\mathbb {D}}}$$ z ∈ D , $$0<\lambda \le 1$$ 0 < λ ≤ 1 . Each $$f\in {\mathcal {V}}_p(\lambda )$$ f ∈ V p ( λ ) has the following Taylor expansion: $$\begin{aligned} f(z)=z+\sum _{n=2}^{\infty }a_n(f) z^n, \quad |z|<p. \end{aligned}$$ f ( z ) = z + ∑ n = 2 ∞ a n ( f ) z n , | z | < p . We recently conjectured that $$\begin{aligned} |a_n(f)|\le \frac{1-(\lambda p^2)^n}{p^{n-1}(1-\lambda p^2)}\quad \text{ for }\quad n\ge 3, \end{aligned}$$ | a n ( f ) | ≤ 1 - ( λ p 2 ) n p n - 1 ( 1 - λ p 2 ) for n ≥ 3 , while investigating functions in the class $${\mathcal {V}}_p(\lambda )$$ V p ( λ ) . In the present article, we first obtain a representation formula for functions in this class. Using this, we prove the aforementioned conjecture for $$n=3,4,5$$ n = 3 , 4 , 5 whenever p belongs to certain subintervals of (0, 1). Also we determine non sharp bounds for $$|a_n(f)|,\,n\ge 3$$ | a n ( f ) | , n ≥ 3 and for $$|a_{n+1}(f)-a_n(f)/p|,\,n\ge 2$$ | a n + 1 ( f ) - a n ( f ) / p | , n ≥ 2 .

Journal

Bulletin of the Malaysian Mathematical Sciences Society – Springer Journals

Published: Oct 12, 2018

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