On normal families and differential polynomials for meromorphic functions

We consider the normality criterion for a families F meromorphic in the unit disc Δ, and show that if there exist functions a(z) holomorphic in Δ, a(z) = 1, for each z ∈ Δ, such that there not only exists a positive number ε0 such that |an(a(z)−1)−1| ε0 for arbitrary sequence of integers an(n ∈ N) and for any z ∈ Δ, but also exists a positive number B >0 such that for every f (z) ∈ F, B|f (z)| |f (z)| whenever f (z)f (z) −a(z)(f (z))2 = 0 in Δ. Then {f (z) f (z) : f (z) ∈ F} is normal in Δ. © 2007 Elsevier Inc. All rights reserved.