On the Taylor coefficients of the composition of two analytic functions

We give an asymptotic formula for the Taylor coefficients f(n) of f(z) = l(h(z)) where l(z) is analytic in the unit disc whose Taylor coefficients l(n) vary 'smoothly' and h(z) is analytic in a larger disc. We show that under mild conditions on h(z), f(n) similar to sigma l([sigma n]) as n --> infinity where sigma = 1/h'(l). Applications to renewal theory are also discussed.