$| |t+1|^p - |t|^p - 1 | \leq k ( |t|^{p-1} + |t| )$ for some $k$

Question

Let $1<p<\infty$. Also assume that $t \in [-1,1]$. Prove that there exists a constant $k \in \mathbb{R}$ that depends on $p$, such that: $$| |t+1|^p - |t|^p - 1 | \leq k ( |t|^{p-1} + |t| ).$$ Which would be equivalent to: $$ \sup_{t\in[-1,1]} \left\{ \frac{||t+1|^p - |t|^p -1|}{|t|^{p-1} + |t|} \right\} < \infty.$$ Any hint would be appreciatied.

Answer 1

For $t\ne0$, $$ \frac{||t+1|^p - |t|^p -1|}{|t|^{p-1} + |t|} \leq\frac{||t+1|^p - 1|}{|t|^{p-1} + |t|} + \frac{|t|^p }{|t|^{p-1} + |t|} \leq\frac{||t+1|^p - 1|}{|t|} + \frac{|t|^p }{|t|^{p-1}} =\frac{||t+1|^p - 1|}{|t|} + |t|. $$ On the interval $[-1/2,3/2]$ the function $t\mapsto (t+1)^p$ is differentiable and $|t+1|=t+1$ (the choice is arbitrary, it only matters that it is $>-1$). By the Mean Value Theorem, there exist numbers $c_t$ between $t+1$ and $1$ (so $c_t\in[0,2]$) with $$ (t+1)^p-1=pc_t^{p-1}\,t, $$ and so $$ |\,|t+1|^p-1|=|(t+1)^p-1|=|pc_t^{p-1}t|\leq p2^{p-1}|t|, $$ i.e. $$ \frac{|(t+1)^p-1|}{|t|}\leq p2^{p-1}. $$ For $t<-1/2$, $$ \frac{|\,|t+1|^p-1|}{|t|}\leq\frac{|t+1|^p+1}{1/2}\leq 2(2^p+1). $$ Going back to the first inequality, we get $$ \frac{||t+1|^p - |t|^p -1|}{|t|^{p-1} + |t|} \leq \max\{p2^{p-1},2(2^p+1)\} +1. $$