Assume that $h:\mathbb{R} \to \mathbb{R}$ is a differentiable function for which there is a number $\lambda \in \mathbb{R}_+^n$ so that: $$\lvert ((dh)(x)(t)\rvert \ge \lambda \lvert t \rvert, \forall t,x\in \mathbb{R} .$$ Prove that:
1.h satisfies the relation $$\lvert (h°h°...°h)(x)-(h°h°...°h)(y)\rvert \ge \lambda^n \lvert x-y \rvert, \forall x,y\in \mathbb{R}, n \in \mathbb{N}^*$$ where h is composite for n times.
2.h is a diffeomorphism.
I think that in the first inequality that "|t|" can be excluded. Therefore, we will have that the absolute value of the diff. is >= than a non-zero value, so it is not zero. This is a first step in showing the diffeomorphism. I thought about using the IFT / LIT.