[Tex/LaTex] How to center a bullet in itemize using LaTeX

Question

I'm trying to create a list of bullets containing some math in LaTeX, but the default settings of LaTeX put the bullets in the beginning of the first line , due to the use of equation the result is very messy….
Here is a working example:

\documentclass{amsart}
\usepackage{amsmath} 
\begin{document}
\begin{itemize}
\item $g^{(k)}(\psi)=\frac{(n^{2}-1)\delta}{(n-(k-1))((1-\psi)k-(n-1)\delta)}$
\item $f^{(k)}=(g^{(k)}\frac{1}{n})$
\item $\underbar{k}=\frac{n(1+\delta)}{2}$.
\item \begin{equation*}
\underbar{f}=\begin{cases}
n\delta&\mbox{If }n<4\mbox{ or }n=4\&\delta>\frac{1}{8}\\
f^{\underbar{k}}&\mbox{ otherwise}
\end{cases}
\end{equation*}
\item \begin{equation*}
\underbar{g}=\begin{cases}
n\delta&\mbox{If }n<4\mbox{ or }n=4\&\delta>\frac{1}{8}\\
g^{\underbar{k}}&\mbox{ otherwise}
\end{cases}
\end{equation*}
\end{itemize}
\end{document}

I need to move the last two bullets vertically to the middle:

Answer 1

Instead of trying to move the two final bullet points to the right, I'd drop the use of the equation* environments and move the math material to the left, so that all bullet points have the same alignment. (Better still, don't use an itemize environment at all. Look into, say, using an align* environment, with all equations aligned on the respective = symbols.)

I can't help but comment on the use of the \underbar instruction: It typesets its argument in an upright font, and it doesn't adjust the position of the bar if the argument is a letter that has a descender component (such as the letter "g"). Can you maybe find a better notation (or at least a more suitable macro, say, \underline)?

\documentclass{amsart}
\begin{document}
\begin{itemize}
\item $g^{(k)}(\psi)=\dfrac{(n^{2}-1)\delta}{(n-(k-1))((1-\psi)k-(n-1)\delta)}$
\item $f^{(k)}=(g^{(k)}\frac{1}{n})$
\item $\underbar{k}=\dfrac{n(1+\delta)}{2}$
\item $ \underbar{f}=\begin{cases}
n\delta&\text{If $n<4$ or $n=4$ \& $\delta>\frac{1}{8}$}\\
f^{\underbar{k}}&\text{otherwise}
\end{cases}$
\item 
$\underbar{g}=\begin{cases}
n\delta&\text{If $n<4$ or $n=4$ \& $\delta>\frac{1}{8}$}\\
g^{\underbar{k}}&\text{otherwise}
\end{cases}$
\end{itemize}
\end{document}