Let $$\mathcal {V}_p(\lambda )$$ Vp(λ) be the collection of all functions f defined in the open unit disk $$\mathbb {D}$$ 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))2f′(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∈Vp(λ) has the following Taylor expansion: $$\begin{aligned} f(z)=z+\sum _{n=2}^{\infty }a_n z^n, \quad |z|<p. \end{aligned}$$ f(z)=z+∑n=2∞anzn,|z|<p. In Bhowmik and Parveen (Bull Korean Math Soc 55(3):999–1006, 2018), we conjectured that $$\begin{aligned} |a_n|\le \frac{1-(\lambda p^2)^n}{p^{n-1}(1-\lambda p^2)}\quad \text{ for }\quad n\ge 3, \end{aligned}$$ |an|≤1-(λp2)npn-1(1-λp2)forn≥3, and the above inequality is sharp for the function $$k_p^{\lambda }(z)=-\,p z/(z-p)(1-\lambda p z)$$ kpλ(z)=-pz/(z-p)(1-λpz) . In this article, we first prove the above conjecture for all $$n\ge 3$$ n≥3 where p is lying in some subintervals of (0, 1). We then prove the above conjecture for the subordination class of $$\mathcal {V}_p(\lambda )$$ Vp(λ) for $$p\in (0, 1/3]$$ p∈(0,1/3] . Next, we consider the Laurent expansion of functions $$f\in \mathcal {V}_p(\lambda )$$ f∈Vp(λ) valid in $$|z-p|<1-p$$ |z-p|<1-p and determine the exact region of variability of the residue of f at $$z=p$$ 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 $$\mathcal {V}_p(\lambda )$$ Vp(λ) are also obtained. Next, we prove that $$\mathcal {V}_p(\lambda )$$ Vp(λ) does not contain the class of concave univalent functions for $$\lambda \in (0,1]$$ λ∈(0,1] and vice-versa for $$\lambda \in ((1-p^2)/(1+p^2),1]$$ λ∈((1-p2)/(1+p2),1] . Finally, we show that there are some sets of values of p and $$\lambda $$ λ for which $$\overline{\mathcal {\mathbb {C}}}\setminus {k_p}^{\lambda }(\mathbb {D})$$ C¯\kpλ(D) may or may not be a convex set.