A particular family of absolutely monotone functions and relations to branching processes

It is shown that the map z ↦ log(1 − c⁻¹ log(1 − z)) is absolutely monotone on [0, 1) if and only if c ≥ 1. The proof is based on an integral representation for the associated Taylor coefficients and on one of Gautschi’s double inequalities for the quotient of two gamma functions. The result is used to verify that, for every c ≥ 1 and α ∈ (0, 1], the map z ↦ 1 − exp(c − c(1 − c⁻¹ log(1 − z))^α) is absolutely monotone on [0, 1). The proof exploits a continuous-time discrete state space branching process.