$}}}
( \cos 2 \theta_{\odot})_{\mbox{}_{\rm MAX}}$. In the general case this condition might not be satisfied. Let us parametrize the experimental value of $\!\!$, $({\mbox{$\left|  < \! m  \! > \right| \ $}}_{\rm \!\!exp})_{\mbox{}_{\rm MIN}}$, obtained using [*the largest nuclear matrix element*]{}, as follows: $$({\mbox{$\left|  < \! m  \! > \right| \ $}}_{\rm \!\!exp})_{\mbox{}_{\rm MIN}} = 
y \sqrt{{\mbox{$(\Delta  m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}} (\cos 2 \theta_{\odot})_{\mbox{}_{\rm MIN}},~~~~~~~y \geq 1.$$ Using eq. (\[meffpar01\]) we get: $${\mbox{$\left|  < \! m  \! > \right| \ $}}\!\! 
= \zeta y \sqrt{{\mbox{$(\Delta  m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}} (\cos 2 \theta_{\odot})_{\mbox{}_{\rm MIN}}.$$ The requirement that takes values in the region allowed in the case of neutrino mass spectrum with inverted hierarchy, translates into an interval of allowed values of $y$: $$1 \leq y \leq  \frac{\sqrt{{\mbox{$(\Delta  m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}}}{\sqrt{{\mbox{$(\Delta  m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}}}
\frac{1}{(\cos 2 \theta_{\odot})_{\mbox{}_{\rm MIN}}}.$$ The necessary conditions for establishing CP-violation due to the Majorana CP-violating phases, eqs. (\[suffcondinv01a\]) and (\[suffcondinv01\]), lead to the following constraints on the parameters $y$ and $\zeta$: $$\begin{aligned}
\frac{\sqrt{{\mbox{$(\Delta  m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}}}{\sqrt{{\mbox{$(\Delta  m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}}}
\frac{(\cos 2 \theta_{\odot})_{\mbox{}_{\rm MAX}}}
{(\cos 2 \theta_{\odot})_{\mbox{}_{\rm MIN}}} < y <
\frac{1}{ (\cos 2 \theta_{\odot})_{\mbox{}_{\rm MIN}}} \label{suffcondinv02}
\\
