Equilibrium states for potentials with $\sup \varphi -\inf \varphi <\text{htop}(f)$

In the context of smooth interval maps, we study an inducing scheme approach to prove existence and uniqueness of equilibrium states for potentials $\varphi$ with he bounded range' condition $\sup \phi - \inf \phi < \htop$, first used by Hofbauer and Keller. We compare our results to Hofbauer and Keller's use of Perron-Frobenius operators. We demonstrate that this bounded range’ condition on the potential is important even if the potential is H"older continuous. We also prove analyticity of the pressure in this context.