We obtain a type of Ayón-Beato-García (ABG) related black hole solutions with five parameters: the mass $m$, the charge $q$, and three dimensionless parameters $\alpha$, $\beta$ and $\gamma$ associated with nonlinear electrodynamics. We find that this type of black hole is regular under the conditions: $\alpha \gamma \ge 6$, $\beta \gamma \ge 8$, and $\gamma >0$. Here we focus on the saturated case: $\alpha =6/\gamma$ and $\beta =8/\gamma$, such that only three parameters $m$, $q$ and $\gamma$ remain, which leads to a new family of ABG black holes. For such a family of black holes, we investigate the influence of the charge $q$ and the parameter $\gamma$ on the horizon radius and the Hawking temperature. In addition, we calculate the quasinormal mode frequencies of massless scalar field perturbations by using the sixth-order WKB approximation method and the unstable circular null geodesic method in the eikonal limit. On the one hand, our results show that the increase of the charge $q$ makes the scalar waves decay faster at first and then slowly except for the case of $\gamma =2$. On the other hand, they show that the increase of the parameter $\gamma$ makes the scalar waves decay at first sharply and then slowly. In particular, $\gamma =1$ can be regarded as the critical value for the transition from an unstable configuration to a stable one. Finally, we compute the shadow radius for the new family of ABG black holes and use the shadow data of the $M87^{*}$ black hole detected by the Event Horizon Telescope to provide an upper limit on the charge $q$ of the new black holes. We find that the increase of the charge $q$ makes the shadow radius decrease monotonically, while the increase of the parameter $\gamma$ makes the shadow radius increase at first rapidly and then almost remain unchanged, especially the parameter $\gamma$ has a significant impact on the shadow radius when it is less than one. Moreover, using the shadow data of the $M87^{*}$ black hole, we find that the upper limit of the charge $q$ increases rapidly at first and then slowly but does not exceed the mass of the $M87^{*}$ black hole at last when the parameter $\gamma$ is increasing and going to infinity, and that the data restrict the frequency range of the fundamental mode with $l=1$ to $1.4\times {10}^{-6}\mathrm{Hz}\sim 1.9\times {10}^{-6}\mathrm{Hz}$.

• Received 28 April 2021
• Accepted 24 May 2021

DOI:https://doi.org/10.1103/PhysRevD.103.124050