need a few inequalities and expressions for $\xi$ and $A$. It follows directly from (\[F-xi\]) and (\[A-f\]) that for all $r>0$ $$\begin{aligned}
\label{cond1}
\xi\,\,&\!\!\!\geq&r\,,\;\;\; 0<\xi'\leq1,\;\;\;  0<\mathrm{e}^{\Phi}\leq1,\;\;\; \xi''= \Big(\!\mathrm{e}^{\Phi}\Big)^{\!\prime}\geq0\,,\\
\label{A'}
A'&=& \frac{2}{r}\,A- 2\,\frac{\xi-3m}{r^2}\,\mathrm{e}^{\Phi}\,,\\
\nonumber
A''&=& \frac{2}{r}\,A'- \frac{2}{r^2}\,A + \frac{2}{r^2}\,\frac{\xi-3m}{r}\,\mathrm{e}^{\Phi}- \frac{2}{r} \left(\frac{\xi-3m}{r}\,\mathrm{e}^{\Phi}\right)'\\
\label{A"}
&=& \frac{A'}{r}- \frac{2}{r} \left(\frac{\xi-3m}{r}\,\mathrm{e}^{\Phi}\right)'.\end{aligned}$$

The two following propositions give us an additional characteristic feature, besides the existence or non-existence of an event horizon and an innermost stable circular orbit, that distinguishes SFNSs from SFBHs.

\[proposition1\] In a SFBH spacetime defined by the quadratures (\[F-xi\])–(\[V\]) and the conditions (\[phi\]) and (\[3m&gt;xi0\]), $A(r)$ is a strictly increasing function outside the event horizon.

In the region where $\xi-3m\,\leq\,0$, the monotonicity (outside the horizon) follows directly from (\[A’\]), so we need to consider only the region where $\xi-3m>0$.

An integration by parts in (\[F-xi\]) yields $$\begin{aligned}
\xi(r)=r+\int_{r}^{\,\infty} \left(1\!-\mathrm{e}^{\Phi}\right)\!dr = r\mathrm{e}^{\Phi}+ \int_{r}^{\,\infty} \Big(\!\mathrm{e}^{\Phi}\Big)^{\!\prime}rdr
\nonumber\\
=r\mathrm{e}^{\Phi}+ \int_{0}^{\,\infty} \Big(\!\mathrm{e}^{\Phi}\Big)^{\!\prime}rdr- \int_{0}^{\,r} \Big(\!\mathrm{e}^{\Phi}\Big)^{\!\prime}rdr
\nonumber\\
=r\mathrm{e}^{\Phi}+ \xi(0)- \int_{0}^{\,r} \Big(\!\mathrm{e}^{\Phi}\Big)^{\!\prime}rdr\,,\end{aligned}$$ $$\begin{aligned}
r\mathrm{e}^{\Phi}=\xi(r)-\xi(0)+\int_{0}^{\,r} \Big(\!\mathrm{e}^{\Phi}\Big)^{\!\prime}rdr\,. \nonumber\end{aligned}$$ Thus, for SFBHs ($\xi(0)<3m$), we have the inequality $$\label{ineq1}
\mathrm{e}^{\Phi}\,>\,\frac{\xi-3m}{r} \quad\text{for all}\;\; r>0\,.$$

Integrating by parts in (\[A-f\]) and then applying the identity $(\xi-3m)'=\mathrm{e}^{\Phi}$, we obtain $$\begin{aligned}
A&=&2r^{2}\!\int_{r}^{\,\infty} \frac{\,\xi-3m}{\,r^4}\,\mathrm{e}^{\Phi}dr
\vphantom{\int\limits_{r}^{\infty}}
\nonumber\\
&=&\frac{2}{3} \frac{\,\xi-3m}{\,r}\mathrm{e}^{\Phi}+ \frac{2r^{2}}{3}\!\int_{r}^{\,\infty}\! \frac{\big[(\xi\!-\!3m) \mathrm{e}^{\Phi}\big]^{\!\prime}} {\,r^3}\,dr \vphantom{\int\limits_{r}^{\infty}}
\nonumber\\
&=&\frac{2}{3} \frac{\,\xi-3m}{r}\,\mathrm{e}^{\Phi}+\, \frac{\mathrm{e}^{2\Phi}}{3}\,+\, \frac{\,r^{2}}{3}\!\int_{r}^{\,\infty}
\Big(\!\mathrm{e}^{2\Phi}\Big)^{\!\prime}\frac{\,dr} {\,r^2}
\vphantom{\int\limits_{r}^{\infty}}
