On the normality criteria of Montel and Bergweiler-Langley

A well-known result of Montel states that for a family F of meromorphic functions in a domain D⊂C, if there exist three distinct points a1, a2, a3 in Cˆ and positive integers ℓ1, ℓ2, ℓ3 such that 1ℓ1+1ℓ2+1ℓ3<1 and all zeros of f−ai have multiplicity at least ℓi for all f∈F and i∈{1,2,3}, then F is normal in D. Inspired by this classical result, during the past 100 years, a large number of normality criteria have been established for the case where meromorphic functions (or differential polynomials generated by the members of the family) meet some distinct points with sufficiently large multiplicities. This means that these criteria strictly apply only to the case in which derivatives of functions (differential polynomials, respectively) vanish on respective zero sets. In this paper, we generalize some normality criteria of Montel, Grahl-Nevo, Gu, and Bergweiler-Langley to the case where derivatives are bounded from above on zero sets.