Another improvement of Montel's criterion

Abstract

Let $\cal F$ be a family of meromorphic functions defined in a domain D $subset$ C, let ψ₁, ψ₂ and ψ₃ be three meromorphic functions such that ψ_i(z) $\not\equiv$ ψ_j(z) (i ≠ j) in D, one of which may be ∞ identically, and let l₁, l₂ and l₃ be positive integers or ∞ with 1/l₁ + 1/l₂ + 1/l₃ < 1. Suppose that, for each f $in$ $\cal F$ and z $in$ D, (1) all zeros of f – ψ_i have multiplicity at least l_i for i = 1,2,3; (2) f(z₀) ≠ ψ_i(z₀) if there exist i, j $in$ {1,2,3} (i ≠ j) and z₀ $in$ D such that ψ_i(z₀) = ψ_j(z₀). Then $\cal F$ is normal in D. This improves and generalizes Montel's criterion.