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Complete monotonicity of solutions of the Abel equation F(ex)=F(x)+1

Hilberdink Titus

Bulletin of the Polish Academy of Sciences. Mathematics

2023

Vol. 71, no 2

135--146

We investigate the functions F : R → R which are C ∞ solutions of the Abel functional equation F(ex) = F(x) + 1. In particular, we determine the asymptotic behaviour of the derivatives and show that no solution can have F′completely monotonic on any interval (α, ∞). We discuss what could be considered the best behaved solution of this equation.