Please excuse my usage of Dirac notations and less than informative question title. I have two normalized vectors $|i\rangle, |j\rangle$ and another one $|\psi\rangle = \frac{1}{\sqrt{2}}(|i\rangle + |j\rangle)$ in a $N > 2$ dimensional inner product space. A vector orthogonal to $|\psi\rangle$ is some other vector (also normalized) $|\phi\rangle$. Among these vectors, I know the following inner products hold:
$$ \begin{align*} \langle i | j \rangle &= 0, \\ \langle \phi | \psi \rangle &= 0, \\ \langle i | \psi \rangle &= \frac{1}{\sqrt{2}}, \\ \langle j | \psi \rangle &= \frac{1}{\sqrt{2}}. \\ \end{align*} $$ Then, what can I say about the inner product of $|i\rangle$ and $|\phi\rangle$? I.e., what would be the expression to find the maximum and minimum value of the following inner products?
$$ \langle i | \phi \rangle, \\ \langle j | \phi \rangle. $$ Thanks!