Let Vp(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)) 2f′(z) – 1 | < λ for z∈ D, 0 < λ≤ 1. Each f∈ Vp(λ) has the following Taylor expansion: f(z)=z+∑n=2∞anzn,|z|<p.In Bhowmik and Parveen (Bull Korean Math Soc 55(3):999–1006, 2018), we conjectured that |an|≤1-(λp2)npn-1(1-λp2)forn≥3,and the above inequality is sharp for the function kpλ(z)=-pz/(z-p)(1-λpz). 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 Vp(λ) for p∈ (0 , 1 / 3]. Next, we consider the Laurent expansion of functions f∈ Vp(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 Vp(λ) are also obtained. Next, we prove that Vp(λ) does not contain the class of concave univalent functions for λ∈ (0 , 1] and vice-versa for λ∈ ((1 – p2) / (1 + p2) , 1]. Finally, we show that there are some sets of values of p and λ for which C¯kpλ(D) may or may not be a convex set.