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Let $f: I \to I$ be a one-dimensional differentiable function of bounded distortion with distortion constant $M$, where $I$ is a compact interval in $\mathbb{R}$. That is, $I$ can be partitioned such that for each $J$ in the partition there exists an interger $n_J$ such \begin{align} \max_{x, y \in J} \log \frac{Df^k(x)}{Df^k(y)} \leq M, \quad 1 \leq k \leq n_J. \end{align} I've been told that this implies for any subintervals $A, B \subset I$ \begin{align} \frac{|A|}{|B|} \leq M \frac{|f(A)|}{|f(B)|} \end{align} where $|A|$ denotes the size of the interval $A$. The problem is that I don't know how to go about showing this true, does anyone know how to show this?

Let $f: I \to I$ be a one-dimensional differentiable function of bounded distortion with distortion constant $M$, where $I$ is a compact interval in $\mathbb{R}$. That is, $I$ can be partitioned such that for each $J$ in the partition there exists an interger $n_J$ such \begin{align} \max_{x, y \in J} \log \frac{Df^k(x)}{Df^k(y)} \leq M, \quad 1 \leq k \leq n_J. \end{align} I've been told that this implies for any subintervals $A, B \subset I$ \begin{align} \frac{|A|}{|B|} \leq e^M \frac{|f(A)|}{|f(B)|} \end{align} where $|A|$ denotes the size of the interval $A$. The problem is that I don't know how to go about showing this true, does anyone know how to show this?

Let $f: I \to I$ be a one-dimensional function of bounded distortion with distortion constant $M$, where $I$ is a compact interval in $\mathbb{R}$. That is, $I$ can be partitioned such that for each $J$ in the partition there exists an interger $n_J$ such \begin{align} \max_{x, y \in J} \log \frac{Df^k(x)}{Df^k(y)} \leq M, \quad 1 \leq k \leq n_J. \end{align} I've been told that this implies for any subintervals $A, B \subset I$ \begin{align} \frac{|A|}{|B|} \leq M \frac{|f(A)|}{|f(B)|} \end{align} where $|A|$ denotes the size of the interval $A$. The problem is that I don't know how to go about showing this true, does anyone know how to show this?

Let $f: I \to I$ be a one-dimensional differentiable function of bounded distortion with distortion constant $M$, where $I$ is a compact interval in $\mathbb{R}$. That is, $I$ can be partitioned such that for each $J$ in the partition there exists an interger $n_J$ such \begin{align} \max_{x, y \in J} \log \frac{Df^k(x)}{Df^k(y)} \leq M, \quad 1 \leq k \leq n_J. \end{align} I've been told that this implies for any subintervals $A, B \subset I$ \begin{align} \frac{|A|}{|B|} \leq M \frac{|f(A)|}{|f(B)|} \end{align} where $|A|$ denotes the size of the interval $A$. The problem is that I don't know how to go about showing this true, does anyone know how to show this?

Let $f: I \to I$ be a one-dimensional function of bounded distortion with distortion constant $D$, where $I$ is a compact interval in $\mathbb{R}$. That is, $I$ can be partitioned such that for each $J$ in the partition there exists an interger $n_J$ such \begin{align} \max_{x, y \in J} \log \frac{Df^k(x)}{Df^k(y)} \leq D, \quad 1 \leq k \leq n_J. \end{align} I've been told that this implies for any subintervals $A, B \subset I$ \begin{align} \frac{|A|}{|B|} \leq D \frac{|f(A)|}{|f(B)|} \end{align} where $|A|$ denotes the size of the interval $A$. The problem is that I don't know how to go about showing this true, does anyone know how to show this?

Let $f: I \to I$ be a one-dimensional function of bounded distortion with distortion constant $M$, where $I$ is a compact interval in $\mathbb{R}$. That is, $I$ can be partitioned such that for each $J$ in the partition there exists an interger $n_J$ such \begin{align} \max_{x, y \in J} \log \frac{Df^k(x)}{Df^k(y)} \leq M, \quad 1 \leq k \leq n_J. \end{align} I've been told that this implies for any subintervals $A, B \subset I$ \begin{align} \frac{|A|}{|B|} \leq M \frac{|f(A)|}{|f(B)|} \end{align} where $|A|$ denotes the size of the interval $A$. The problem is that I don't know how to go about showing this true, does anyone know how to show this?