Complete monotonicity of solutions of the Abel equation
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.
