In this article, we consider a class denoted by A(P) which consists of functions f that are holomorphic in the unit disc ⅅ punctured at a point p ∈ (0, 1) where f has a simple pole. We prove a sufficient condition for these functions to be univalent in ⅅ. By using this condition, we construct the family Up(λ) of all functions f ∈ A(P)such that |(z/f (z))2f’ (z) − 1| < λμ where μ = ((1 − p)/(1 + p))2 for some 0 < λ ≤ 1, z ∈ ⅅ. Therefore, functions in the class Up(λ) are necessarily univalent. We present some basic properties for functions in the class Up(λ) which include an integral representation formula for such functions and obtain the exact region of variability of the second Taylor coefficient for functions in this class. We also obtain a sharp estimate for the Fekete–Szegö functional defined on the class Up(λ) along with a subordination result for functions in this family. In addition, we obtain some necessary and sufficient coefficient conditions involving the coefficients bn for functions f ∈ A(p) of the form (Formula presented.) to be in the class Up(λ). We have also obtained sharp bounds for |bn|, n ≥ 1. .