I was trying to solve a problem, and I was stuck in one part of my solution. Firs I should mention that I should have find a orthogonal projective operator $\mathrm{P} \in \mathbb{C}^{2 \times 2}$ that maximizes this expression, $\langle\Psi|\mathrm{P}| \Psi\rangle-\langle\Phi|\mathrm{P}| \Phi\rangle$, where $|\Psi\rangle,|\Phi\rangle \in \mathbb{C}^{2}$. So I find the relative expression and it is:
$f(x,y)=ax+by+b^*\frac{1}{y}-b^*\frac{x^2}{y}+c$
which constants $a,b,c,$ and variable $y$ could be complex number and x is a real number. I thaught I can use $\frac{\partial f}{\partial x}=0 and \frac{\partial f}{\partial y}=0$ to find the maximum of the f, but I think it is wrong. How can I find the maximum value of the f?