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

• Published in: Resultate der Mathematik 2019-12, Vol.74 (4), Article 195
• Main Authors: Bhowmik, Bappaditya, Parveen, Firdoshi
• Format: Article
• Language: English
• Subjects: Mathematics; Mathematics and Statistics
• ISSN: 1422-6383; 1420-9012
• DOI: 10.1007/s00025-019-1118-4

Let V p ( λ ) be the collection of all functions f defined in the open unit disk D , having a simple pole at z = p where 0 < p < 1 and analytic in D \ { p } with f ( 0 ) = 0 = f ′ ( 0 ) - 1 and satisfying the differential inequality | ( z / f ( z ) ) 2 f ′ ( z ) - 1 | < λ for z ∈ D , 0 < λ ≤ 1 . Each f ∈ V p ( λ ) has the following Taylor expansion: f ( z ) = z + ∑ n = 2 ∞ a n z n , | z | < p . In Bhowmik and Parveen (Bull Korean Math Soc 55(3):999–1006, 2018 ), we conjectured that | a n | ≤ 1 - ( λ p 2 ) n p n - 1 ( 1 - λ p 2 ) for n ≥ 3 , and the above inequality is sharp for the function k p λ ( z ) = - p z / ( z - p ) ( 1 - λ p z ) . In this article, we first prove the above conjecture for all n ≥ 3 where p is lying in some subintervals of (0, 1). We then prove the above conjecture for the subordination class of V p ( λ ) for p ∈ ( 0 , 1 / 3 ] . Next, we consider the Laurent expansion of functions f ∈ V p ( λ ) 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 V p ( λ ) are also obtained. Next, we prove that V p ( λ ) does not contain the class of concave univalent functions for λ ∈ ( 0 , 1 ] and vice-versa for λ ∈ ( ( 1 - p 2 ) / ( 1 + p 2 ) , 1 ] . Finally, we show that there are some sets of values of p and λ for which C ¯ \ k p λ ( D ) may or may not be a convex set.