Getting Rid of the Negative Schwarzian Derivative Condition

Abstract

In this paper we will show that the assumption on the negative Schwarzian derivative is redundant in the case of C 3 unimodal maps with a non∞at critical point. The following theorem will be proved: For any C 3 unimodal map of an interval with a non∞at critical point there exists an interval around the critical value such that the flrst entry map to this interval has negative Schwarzian derivative. Another theorem proved in the paper provides useful cross-ratio estimates. Thus, all theorems proved only for unimodal maps with negative Schwarzian derivative can be easily generalized.