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Let A(p) be the class consisting of functions f that are holomorphic in D \ {p}, p ∈ (0, 1) possessing a simple pole at the point z = p with nonzero residue and normalized by the condition f(0) = 0 = f (0)− 1. In this article, we first prove a sufficient condition for univalency for functions in A(p). Thereafter, we consider the class denoted by Σ(p) that consists of functions f ∈ A(p) that are univalent in D. We obtain the exact value for max f∈Σ(p) ∆(r, z/f), where the Dirichlet integral ∆(r, z/f) is given by ∆(r, z/f) = ∫∫ |z|<r | (z/f(z)) ′ | dx dy, (z = x+ iy), 0 < r ≤ 1. We also obtain a sharp estimate for ∆(r, z/f) whenever f belongs to certain subclasses of Σ(p). Furthermore, we obtain sharp estimates of the integral means for the aforementioned classes of functions.

Criteria for univalence, Integral means and Dirichlet integral for Meromorphic functions