Lipschitz inequality I need to prove that:
$$\left|\frac{d}{dt}\left\langle \varphi(t) , \varphi(t)\right\rangle\right| \le 2L \left\|\varphi(t)\right\|^2$$
and 
$$\left\|x_0\right\|e^{-L(t-t_0)} \leq \left\|\varphi(t)\right\| \le \left\|x_0\right\|e^{L(t-t_0)}.$$
Where $\varphi(t)$ is solution of $x'=f(t,x),$ $x(t_0)=x_0,$ $f:[t_0,\infty)\times D \to \mathbb{R}^n$ is Lipschitz ($|f(t,x)-f(t,y)|\leq L|x-y|$) and $f(t,0)=0$.
For the first inequality i use use Cauchy-Schwarz and get $\left|\frac{d}{dt}\left\langle \varphi(t) , \varphi(t)\right\rangle\right| \le 2L \left\|\varphi(t)\right\|,$ but not the inequality to prove.
For $\left\|\varphi(t)\right\| \le \left\|x_0\right\|e^{L(t-t_0)},$ i used Grönwall's inequality, but $\left\|x_0\right\|e^{-L(t-t_0)} \leq \left\|\varphi(t)\right\|$ i do not know how to do it, and i have not used the hypothesis $f(t,0)=0$.
Could you tell me how can i demonstrate this?.
Thank you
 A: First of all, $\frac{d}{dt} \langle \varphi,\varphi\rangle =2\langle \varphi',\varphi\rangle=2\langle f(t,\varphi),\varphi \rangle $.
Take the equation $\varphi'=f(t,\varphi)$. Then $2\langle \varphi',\varphi \rangle=2\langle f(t,\varphi),\varphi \rangle$. By Cauchy-Schwarz,
$$
|2\langle f(t,\varphi),\varphi \rangle|\leq 2|f(t,\varphi)|\,|\varphi|.
$$
From the Lipschitz condition, (plug in $y=0$), we find $|f(t,\varphi)|\leq L |\varphi|$. This proves the first one in that
$$
\frac{d}{dt}\langle \varphi,\varphi \rangle\leq |\frac{d}{dt}\langle \varphi,\varphi \rangle|\leq 2L\,|\varphi|^2 .
$$
The key is to observe that $\frac{d}{dt} \langle \varphi,\varphi \rangle=\frac{d}{dt}|\varphi|^2=2|\varphi|\frac{d}{dt}|\varphi|.$ But then from above, we have
\begin{align}
&2|\varphi|\frac{d}{dt}|\varphi|\leq 2L\,|\varphi|^2 \\
\implies& \frac{d}{dt}|\varphi|\leq L|\varphi|.
\end{align}
From here, it should be no problem to apply Grönwall's inequality. Note that we also have 
$$
-\frac{d}{dt}\langle \varphi,\varphi \rangle\leq |\frac{d}{dt}\langle \varphi,\varphi \rangle|\leq 2L\,|\varphi|^2,
$$
which leads to
$$
\frac{d}{dt}|\varphi|\geq -L|\varphi|,
$$
hence two applications of Grönwall's.
