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

Let V-p(lambda) 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 vertical bar(z/f(z))(2) f'(z) - 1 vertical bar < lambda for z is an element of D, 0 < lambda <= 1. Each f is an element of V-p(lambda) has the following Taylor expansion: f(z) = z + Sigma(infinity)(n=2) a(n)z(n), vertical bar z vertical bar < p. In Bhowmik and Parveen (Bull Korean Math Soc 55(3):999-1006, 2018), we conjectured that vertical bar a(n)vertical bar <= 1 - (lambda p(2))(n)/p(n-1)(1 - lambda p(2)) for n >= 3, and the above inequality is sharp for the function k(p)(lambda) (z) = - pz/(z - p)(1 - lambda pz). 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(lambda) for p is an element of (0, 1/3]. Next, we consider the Laurent expansion of functions f is an element of V-p(lambda) valid in vertical bar z - p vertical bar < 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(lambda) are also obtained. Next, we prove that V-p(lambda) does not contain the class of concave univalent functions for lambda is an element of (0, 1] and vice-versa for lambda is an element of ((1 - p(2))/(1 + p(2)), 1]. Finally, we show that there are some sets of values of p and lambda for which <(C)over bar>\k(p)(lambda)(D) may or may not be a convex set.