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Results in Mathematics

, 74:195 | Cite as

On Some Results for a Subclass of Meromorphic Univalent Functions with Nonzero Pole

  • Bappaditya BhowmikEmail author
  • Firdoshi Parveen
Article
  • 75 Downloads

Abstract

Let \(\mathcal {V}_p(\lambda )\) be the collection of all functions f defined in the open unit disk \(\mathbb {D}\), having a simple pole at \(z=p\) where \(0<p<1\) and analytic in \(\mathbb {D}\setminus \{p\}\) with \(f(0)=0=f'(0)-1\) and satisfying the differential inequality \(|(z/f(z))^2 f'(z)-1|< \lambda \) for \(z\in \mathbb {D}\), \(0<\lambda \le 1\). Each \(f\in \mathcal {V}_p(\lambda )\) has the following Taylor expansion:
$$\begin{aligned} f(z)=z+\sum _{n=2}^{\infty }a_n z^n, \quad |z|<p. \end{aligned}$$
In Bhowmik and Parveen (Bull Korean Math Soc 55(3):999–1006, 2018), we conjectured that
$$\begin{aligned} |a_n|\le \frac{1-(\lambda p^2)^n}{p^{n-1}(1-\lambda p^2)}\quad \text{ for }\quad n\ge 3, \end{aligned}$$
and the above inequality is sharp for the function \(k_p^{\lambda }(z)=-\,p z/(z-p)(1-\lambda p z)\). In this article, we first prove the above conjecture for all \(n\ge 3\) where p is lying in some subintervals of (0, 1). We then prove the above conjecture for the subordination class of \(\mathcal {V}_p(\lambda )\) for \(p\in (0, 1/3]\). Next, we consider the Laurent expansion of functions \(f\in \mathcal {V}_p(\lambda )\) valid in \(|z-p|<1-p\) and determine the exact region of variability of the residue of f at \(z=p\) and find the sharp bounds of the modulus of some initial Laurent coefficients for some range of values of p. The growth and distortion results for functions in \(\mathcal {V}_p(\lambda )\) are also obtained. Next, we prove that \(\mathcal {V}_p(\lambda )\) does not contain the class of concave univalent functions for \(\lambda \in (0,1]\) and vice-versa for \(\lambda \in ((1-p^2)/(1+p^2),1]\). Finally, we show that there are some sets of values of p and \(\lambda \) for which \(\overline{\mathcal {\mathbb {C}}}\setminus {k_p}^{\lambda }(\mathbb {D})\) may or may not be a convex set.

Keywords

Meromorphic functions univalent functions growth and distortion theorem laurent coefficients taylor coefficients 

Mathematics Subject Classification

30C45 30C50 30C55 30C80 

Notes

Acknowledgements

The authors thank Karl-Joachim Wirths for his suggestions and careful reading of the manuscript.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology KharagpurKharagpurIndia

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