list below the properties needed in this paper. For more details, we refer the reader to \[\[BB1\],\[BB2\], and \[Huang\]\].

\[prop2.1\] For any $\mu\in\mathcal{M}$ and $p>1$, define the function $$H_p(z)=pz+(1-p)F_\mu(z),\;\;\;\;\;z\in\mathbb{C}^+,$$ the set $\Omega_p=\{z\in\mathbb{C}^+:\Im H_p(z)>0\}$, and the function $f_\mu:\mathbb{R}\to\mathbb{R}_+\cup\{\infty\}$ as $$\label{fmu}
f_\mu(x)=\int_\mathbb{R}\frac{s^2+1}{(s-x)^2}\;d\rho(s),\;\;\;\;\;x\in\mathbb{R},$$ where $\rho$ is the measure in the Nevanlinna representation $(\ref{NeF})$ of $F_\mu$.

1.  [The function $H_p$ is in $\mathcal{H}$ and the set $\Omega_p$ is a simply connected domain whose boundary is the graph of the continuous function $f_p:\mathbb{R}\to[0,\infty)$, where $$f_p(x)=\inf\left\{y>0:\frac{\Im
    E_\mu(x+iy)}{y}>\frac{-1}{p-1}\right\},\;\;\;\;\;x\in\mathbb{R}.$$]{}

2.  [For $x\in\mathbb{R}$, $f_p(x)=0$ if and only if $f_\mu(x)\leq1/(p-1)$, while $z\in\Omega_p$ if and only if $$\int_\mathbb{R}\frac{s^2+1}{|s-z|^2}\;d\rho(s)<\frac{1}{p-1}.$$ Consequently, the functions $E_\mu$ and $H_p$ have continuous extensions to $\overline{\Omega_p}$ which are Lipschitz continuous with Lipschitz constants $1/(p-1)$ and $2$, respectively. Moreover, Eq. $(\ref{NeF})$ holds for $z\in\overline{\Omega_p}$.]{}

3.  [There exists an analytic function $\omega_p:\mathbb{C}^+\to\mathbb{C}^+$ extending continuously to $\mathbb{C}^+\cup\mathbb{R}$ such that $H_p(\omega_p(z))=z$ holds for $z\in\mathbb{C}^+\cup\mathbb{R}$. Consequently, $\Omega_p=\omega_p(\mathbb{C}^+)$, $\omega_p(H_p(z))=z$ holds for $z\in\overline{\Omega_p}$, and $$\frac{|z_1-z_2|}{2}\leq|\omega_p(z_1)-\omega_p(z_2)|,\;\;\;\;\;z_1,z_2\in\mathbb{C}^+\cup\mathbb{R}.$$]{}

4.  [The function $\omega_p$ is analytic in a neighborhood of $x$ wherever $\omega_p(x)\not\in\mathbb{R}$.]{}

5.  [Let $\mu^{\boxplus p}$ be the unique measure in $\mathcal{M}$ whose reciprocal Cauchy transform satisfies $$\label{p-power}
    F_{\mu^{\boxplus
    p}}(z)=\frac{p\omega_p(z)-z}{p-1},\;\;\;\;\;z\in\mathbb{C}^+.$$ Then there exist some $\alpha,\beta>0$ such that $$\phi_{\mu^{\boxplus
    p}}(z)=p\phi_\mu(z),\;\;\;\;\;z\in\Gamma_{\alpha,\beta}.$$ Moreover, the function $\omega_p$ is the subordination function of $\mu^{\boxplus 