  Let $(M^{2},g_{0})$ be a compact manifold with boundary, and let $g$ and
$g_{0}$ be conformally related by $g=e^{2f}g_{0}$. We show that the inequality
$$\nu_{1}(g)\geq\Big(\max_{x\in\partial M}e^{-f(x)}\Big)\nu_{1}(g_{0})$$ stated
in Proposition 2 in [1], is only possible when the equality is achieved. In
order to achieve such equality, it is required that the function $f$ be
constant on $\partial M$, as it is mentioned in Remark 3 also in [1]. Hence,
the scope of this inequality is less broad than the one suggested by the
Proposition.
