Normality through sharing of pairs of functions with derivatives

Kuldeep Singh Charak, Manish Kumar, Anil Singh

Let F ⊂M(D) and let a, b and c be three distinct complex numbers. If, there exist a holomorphic function h on D and a positive constant ρ such that for each f ∈ F , f and f ′ partially share three pairs of functions (a, h), (b, cf ) and (c, df ) on D, where cf and df are some values in some punctured disk D∗ ρ(0), then F is normal in D. This is an improvement of Schwick’s result [Arch. Math. (Basel), 59 (1992), 50-54]. We also obtain several normality criteria which significantly improve the existing results and examples are given to establish the sharpness of results