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

### Volume 71 / 2023

Bulletin Polish Acad. Sci. Math. 71 (2023), 135-145
MSC: Primary 26A18; Secondary 39B22, 26A48.
DOI: 10.4064/ba230411-18-6
Published online: 13 July 2023

#### Abstract

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.