Dynamics of certain smooth one-dimensional mappings I: The   $C^{1+\alpha}$-Denjoy-Koebe distortion lemma
  We prove a technical lemma, the $C^{1+\alpha }$-Denjoy-Koebe distortion lemma, estimating the distortion of a long composition of a $C^{1+\alpha }$ one-dimensional mapping $f:M\mapsto M$ with finitely many, non-recurrent, power law critical points. The proof of this lemma combines the ideas of the distortion lemmas of Denjoy and Koebe. 
Yunping Jiang