I am trying to generalizate a result of Paul Andi Nagy which says that in a almost Kähler manifold with parallel torsion we have $\langle \rho,\Phi-\Psi\rangle $ is a nonnegative number; in fact $$ 4\langle \rho,\Phi-\Psi\rangle = |\Phi|^2 + |\Psi|^2, $$ where $\rho$ is the Ricci form defined by $\rho=\langle Ric'J\cdot,\cdot\rangle$ ($Ric'$ is the $J$-invariant part of the Ricci tensor) and \begin{eqnarray} \Psi(X,Y)&=& \sum_{i=1}^{2m} \langle (\nabla_{e_i}J)JX,(\nabla_{e_i}J)Y\rangle \\ \Phi(X,Y)&=& \frac{1}{2}\sum_{i=1}^{2m} \langle (\nabla_{JX}J)e_{i},(\nabla_{Y}J)e_{i}\rangle , \end{eqnarray} here, $\nabla$ is the Levi-Civita connection and $\{e_i,1\leq i \leq 2m\}$ is some local orthonormal basis.
I think that for any almost Kähler manifold \begin{eqnarray} |Ric|^2-\frac{1}{2}\langle \rho,\Phi-\Psi\rangle \geq 0 \end{eqnarray} but I don't know how to relate $|Ric|^2-\frac{1}{2}\langle \rho,\Phi-\Psi\rangle $ with the norm of well-known tensors. Thank you for your answers.
I am really sorry for lack of clarity. I would like to show that $|Ric|^2-\frac{1}{2}\langle \rho,\Phi-\Psi\rangle $ is a nonnegative number by finding a formula for $|Ric|^2-\frac{1}{2}\langle \rho,\Phi-\Psi\rangle $ in terms of the norm of well-known tensors (e.g. Weyl Tensor, $Ric$, $Ric'$, $\Psi$, $\Phi$).
