The context is that of the mean curvature flow, more precisely, concerning Type I singularities and the rescaling procedure. The text I am following is by Mantegazza: "Lecture notes on Mean Curvature Flow". We rescale the moving hypersurfaces $\varphi$ around $\hat{p}=\lim_{t\rightarrow T}\varphi (p,t)$ in this way \begin{equation*} \widetilde{\varphi}(q,s)=\frac{\varphi(q,t(s))-\hat{p}}{\sqrt{2(T-t(s))}}, \hspace{1cm} s=s(t)=-\frac{1}{2}\log(T-t). \end{equation*} Then, we rescale also the monotonicity formula in order to get information on these hypersurfaces. In the following $\tilde{\mu}_s=\frac{\mu_t}{[2(T-t)]^{n/2}}$ will be the canonical measure associated to the rescaled hypersurface $\tilde{\varphi}_s$ which satisfies \begin{equation}\frac{d}{ds}\tilde{\mu}_s=(n-\widetilde{H}^2)\tilde{\mu}_s.\end{equation} Rescaled Monotonicity Formula We have \begin{equation} \frac{d}{ds}\int_Me^{-\frac{|y|^2}{2}}d\widetilde{\mu}_s=-\int_Me^{-\frac{|y|^2}{2}}\left|\widetilde{H}+\langle y|\widetilde{\nu}\rangle\right|^2d\widetilde{\mu}_s\leq 0 \end{equation} which integrated becomes \begin{equation} \int_Me^{-\frac{|y|^2}{2}}d\widetilde{\mu}_{s_1}-\int_Me^{-\frac{|y|^2}{2}}d\widetilde{\mu}_{s_2}=\int_{s_1}^{s_2}\int_Me^{-\frac{|y|^2}{2}}\left|\widetilde{H}+\langle y|\widetilde{\nu}\rangle\right|^2d\widetilde{\mu}_s ds. \end{equation} One of the consequences of the rescaled monotonicity formula is the following lemma, credited to Stone (it can be found in "A density function and the structure of singularities of the mean curvature flow", or Lemma 3.2.7 of Mantegazza).
Lemma. There is a uniform constant $C=C(n,\text{Area}(\varphi_0),T)$ such that, for any $p\in M$ and for all $s\in\bigg[-\frac{1}{2}\log T,+\infty\bigg)$, \begin{equation*} \int_M e^{-|y|}d\widetilde{\mu}_s\leq C. \end{equation*}
In the proof these passages appears \begin{align*} \frac{d}{ds}\int_Me^{-|y|}d\widetilde{\mu}_s&=\int_M\bigg\lbrace n-\widetilde{H}^2-\frac{1}{|y|}\langle y|\widetilde{H}\widetilde{\nu}+y\rangle\bigg\rbrace e^{-|y|}d\widetilde{\mu}_s \\ &\leq \int_M\lbrace n-\widetilde{H}^2-|y|+|\widetilde{H}|\rbrace e^{-|y|}d\widetilde{\mu}_s \\ &<\int_M\lbrace n+1-|y|\rbrace e^{-|y|}d\widetilde{\mu}_s \\ &\leq (n+1)\bigg\lbrace \int_{\widetilde{\varphi}_s(M)\cap B_{n+1}(0)}e^{-|y|}d\widetilde{\mathcal{H}}^n-\int_{\widetilde{\varphi}_s(M)\setminus B_{2n+2}(0)}e^{-|y|}d\widetilde{\mathcal{H}}^n\bigg\rbrace. \end{align*} The first and second steps are clear ($\frac{d}{ds}\tilde{\mu}_s=(n-\widetilde{H}^2)\tilde{\mu}_s$ is applied). But how do I obtain the last two inequalities?