bounded and $z\in\mathbb{C}^+$) is finite, in which case $\mu(\{\alpha\})=1/F_\mu'(\alpha)$.

Given any measure $\mu\in\mathcal{M}$, the function $E_\mu(z)=z-F_\mu(z)$ is called the energy function associated with $\mu$ and belongs to the following set $$\mathcal{E}=\left\{E|E:\mathbb{C}^+\to\mathbb{C}^-\cup\mathbb{R}\;\;\mathrm{is}\;\;\mathrm{analytic}\;\;\mathrm{and}
\;\;\lim_{y\to\infty}\frac{E(iy)}{iy}=0\right\}.$$ Conversely, any function $E$ in $\mathcal{E}$ is the energy function of some $\mu\in\mathcal{M}$ whose Nevanlinna representation is given by $$\label{NeE}
E(z)=\Re E(i)+\int_\mathbb{R}\frac{1+sz}{z-s}\;d\rho(s),$$ where $\rho$ is some finite positive Borel measure on $\mathbb{R}$. Observe that we have the inclusion $\mathcal{G}\subset\mathcal{E}$. Indeed, for any measure $\mu\in\mathcal{M}$ it was proved in \[\[Maa\]\] that $\mu$ has mean zero and finite variance $\sigma^2$, i.e., $$\int_\mathbb{R}s\;d\mu(s)=0\;\;\;\;\;\mathrm{and}\;\;\;\;\;\int_\mathbb{R}s^2\;d\mu(s)=\sigma^2$$ if and only if there exists some unique $\nu\in\mathcal{M}$ such that $$\label{EMaa} E_\mu=\sigma^2G_\nu.$$ If $\sigma^2=1$, let $\Phi(\mu)$ be the unique measure satisfying $E_\mu=G_{\Phi(\mu)}$. The Eq. (\[EMaa\]) particularly shows that $\mu^{\uplus1/\sigma^2}$ has mean zero and unit variance, i.e., $E_\mu=\sigma^2G_{\Phi(\mu^{\uplus1/\sigma^2})}$.

Next, consider the set $$\mathcal{H}=\left\{H|H:\mathbb{C}^+\to\mathbb{C}\;\;\mathrm{is}\;\;\mathrm{analytic},\;\Im H(z)\leq\Im
z, \;\;z\in\mathbb{C}^+,\;\;\mathrm{and}
\;\;\lim_{y\to\infty}\frac{H(iy)}{iy}=1\right\},$$ which plays an important role in the investigation of the free convolution powers of measures in $\mathcal{M}$. Indeed, for any $H\in\mathcal{H}$ the function $2z-H(z)\in\mathcal{F}$ is the reciprocal Cauchy transform of some measure in $\mathcal{M}$. More importantly, the right inverses of the functions in $\mathcal{H}$ can be used to construct the $p$-th $\boxplus$-convolution power $\mu^{\boxplus p}$, $p\geq1$, of any measure $\mu\in\mathcal{M}$. We 