\left|\int_{\partial B_r'}huv\,dS'\right|{\leqslant}2{\varepsilon}B(r)+O(1)r^{N-1}H(r)+O(1)r^{N-1}D(r)\quad\text{as }r\to0^+.$$ From Corollary \[c:limpos\], and , choosing $\e=\frac14$, we deduce that, for some constants $C_1,C_2>0$ independent of $r$, $D(r)+C_2H(r)\geq0$ and $$D'(r){\geqslant}\frac12
r^{1-N}B(r)-\frac{C_1}2r^{-1}(D(r)+C_2H(r))\quad\text{for all $r$
  sufficiently small}.$$ The proof is thereby complete.

\[l:limitN\] Let $\mathcal N:(0,r_0)\to{{\mathbb R}}$ be defined in . Then $$\label{eq:23}
  \mathcal N(r)=O(1)\quad\text{as }r\to0^+.$$ Furthermore the limit $$\gamma:=\lim_{r\to 0^+}\mathcal N(r)$$ exists, is finite and $$\gamma{\geqslant}0.$$

Let us consider the set $$\Sigma=\{r\in(0,r_0):D'(r)H(r){\leqslant}H'(r) D(r)\}$$ (which is well-defined up to a zero measure set).

If there exists $r\in(0,r_0]$ such that $|(0,r)\cap\Sigma|_1=0$ (where $|\cdot|_1$ stands for the Lebesgue measure in ${{\mathbb R}}$) we have that $\mathcal N'{\geqslant}0$ a.e. in $(0,r)$ and hence $\mathcal N$ is non-decreasing in $(0,r)$ and admits a limit as $r\to0^+$ which is necessarily finite and non-negative due to .

Let us now assume that, for all $r\in(0,r_0]$, $|(0,r)\cap\Sigma|_1>0$. In view of Lemma \[l:stima\_B\] and we have that, a.e. in $(0,r_0)\cap \Sigma$, $$\begin{aligned}
\label{eq:22}
B(r)&\leq
2 r^{N-1}\frac{H'(r)D(r)}{H(r)}+C_1 r^{N-2}  (D(r)+C_2 H(r))\\
\notag&=4 r^{N-2}\frac{D^2(r)}{H(r)}+C_1  r^{N-2}  (D(r)+C_2 H(r)).\end{aligned}$$ Schwarz inequality implies that the function $\nu_1$ appearing in Lemma \[mono\] is non-negative, hence , Lemma \[l:STIMA\_DI\_nu\_2\], and imply that $$\mathcal N'(r){\geqslant}O(1)\left(
1+\mathcal N(r)+\sqrt{4 \mathcal N^2(r)+C_1(\mathcal
  N(r)+C_2)}\right)$$ as $r\to0^+$, $r\in \Sigma$. Hence there exist $\tilde C,\tilde r>0$ such that $$\mathcal N'(r){\geqslant}-\tilde C\left(
1+\mathcal 