Complete monotonicity of solutions of the Abel equation $F(e^x)=F(x)+1$

We investigate the functions $F:\mathbb R\to \mathbb R$ which are $C^\infty $ solutions of the Abel functional equation $F(e^x)= F(x)+1$. In particular, we determine the asymptotic behaviour of the derivatives and show that no solution can have $F’$ completely monotonic on any interval $(\alpha ,\infty )$. We discuss what could be considered the best behaved solution of this equation.