Estimating the kth coefficient of (fn(z)) when k is not too large

We derive asymptotic estimates for the coefficient of zk in (fn(z)) when n→∞ and k is of order nδ, where 0<δ<1, and f(z) is a power series satisfying suitable positivity conditions and with f(0)≠0, f′(0)=0. We also show that there is a positive number ε<1 (easily computed from the pattern of non-zero coefficients of f(z)) such that the same coefficient is positive for large n and ε<δ<1, and admits an asymptotic expansion in inverse powers of k. We use the asymptotic estimates to prove that certain finite sums of exponential and trigonometric functions are non-negative, and illustrate the results with examples.