S = \left( {\begin{array}{*{20}{c}}
1&0&0\\
0&0&{ - \sigma }\\
0&{ - \sigma }&0
\end{array}} \right),
\label{Eq:mu-tau}\end{aligned}$$ on the $\left(\nu_e,\nu_\mu,\nu_\tau\right)$ basis, which provides $\mu$-$\tau$ flavored CP symmetry for the flavor neutrinos [@FlavoredSym] subjected to the interchange of $\nu_\mu$ and $\nu_\tau$, it is found that $m_D$ of Eq.(\[Eq:MaximalCPmD\]) satisfies that $S^Tm_D = m^\ast_D$ as expected.

\[sec:4\]Summary and Discussions
================================

We are able to derive the useful and simple relations to induce maximal CP violation, which dictate that $a_+$=imaginary and $a_1$=real and/or $b_+$=imaginary and $b_1$=real for both NH and IH. For NH, either $a_+$=imaginary and $a_-$=real or $b_+$=imaginary and $b_-$=real also arises. These relations are limited to hold in specific textures where at least one of $a_{1,+,-}$ and $b_{1,+,-}$ vanishes. If the atmospheric neutrino mixing is also maximal, we have obtained ${a_3} = -\sigma{a_2}^\ast$, ${b_3} = -\sigma{b_2}^\ast$, $a_1$=real and $b_1$=real applicable to more general textures. These relations turn out to be equivalent to the familiar relations of $M_{e\tau}=-\sigma M^\ast_{e\mu}$,$M_{\tau\tau}=M^\ast_{\mu\mu}$, $M_{ee}={\rm real}$ and $M_{\mu\tau}={\rm real}$ for flavor neutrinos.

Our findings about various relations among the Dirac masses giving the maximal CP violation for flavor neutrinos become useful when neutrino physics is affected by phases of the Dirac masses. The immediate such an example is 