The $p-$parabolicity under a decay assumption on the Ricci curvature

We prove that, given $\alpha >0$, if $M$ is a complete Riemannian manifold which Ricci curvature satisfies.$$\underset{x}{\mathrm{Ric}}(v)\ge \alpha {\mathrm{sech}}^2(r(x)))$$ or $$\underset{x}{\mathrm{Ric}}(v)\ge -\frac{h_{\alpha }(r(x))}{r(x{)}^2},$$ where $$h_{\alpha }(r)=\frac{\alpha (\alpha +1)r(x{)}^{\alpha }}{r(x{)}^{\alpha }-1},$$ for all $x\in M\mathrm{\backslash }{B}_R(o)$ and for all $v\in T_{x}M,$ $\parallel v\parallel =1,$ where \ $o$ is a fixed point of $M$, $r(x)=d(o,x)$, $d$ the Riemannian distance in $M$ and $B_R(o)$ the geodesic ball of $M$ centered at $o$ with radius $R>0$, then $M$ is $p-$parabolic for any $p>1$, if satisfies the first inequality, and $M$ is $p-$parabolic, for any $p\ge (\alpha +1)(n-1)+1$, if satisfies the second inequality.