Normal families and value distribution in connection with composite functions

This paper concerns composite functions of the form f(g(z)) where f is a transcendental entire function of order less than 1/2, (that is, an analytic function which is not a polynomial and which grows at a certain rate), and g(z) is a nonconstant polynomial. We prove a value distribution result for the derivatives of a function of this form. This result has several interesting corollaries. Furthermore, we use it to prove a new criterion for normal families.