A normality criterion involving rotations and dilations in the argument

We show that a family F of analytic functions in the unit disk D all of whose zeros have multiplicity at least k and which satisfy a condition of the form f(n)(z)f((k))(xz) not equal 1 for all z is an element of D and f is an element of F (where n >= 3, k >= 1 and 0<|x|<= 1) is normal at the origin. The proof relies on a modification of Nevanlinna theory in combination with the Zalcman-Pang rescaling method. Furthermore we prove the corresponding Picard-type theorem for entire functions and some generalizations.