PDM KG-Coulomb particles in cosmic string rainbow gravity spacetime and a uniform magnetic field

Klein-Gordon (KG) particles in cosmic string rainbow gravity spacetime and a uniform magnetic field are studied in the context of the so called, effectively speaking, position-dependent mass (PDM) settings. We show that the corresponding KG-equation collapses into a two-dimensional radial Schrödinger-Coulomb like model. The exact textbook solution of which is used to find the energies and wave functions of KG-Coulomb particles (both constant mass and PDM ones). In so doing, we consider, with $y=E/{E}_{P}$, four pairs of rainbow functions: (a) ${g}_{{}_{0}}\left(y\right)=1$, ${g}_{{}_{1}}\left(y\right)=\sqrt{1-ϵ{y}^{2}}$, (b) ${g}_{{}_{0}}\left(y\right)=1$, ${g}_{{}_{1}}\left(y\right)=\sqrt{1-ϵy}$, (c) ${g}_{{}_{0}}\left(y\right)={g}_{{}_{1}}\left(y\right)={\left(1-ϵy\right)}^{-1}$, and (d) ${g}_{{}_{0}}\left(y\right)=\left({e}^{ϵy}-1\right)/ϵy$, ${g}_{{}_{1}}\left(y\right)=1$. Interestingly, we observe that the first pair in (a) introduces the Planck energy ${E}_{p}$ as a maximum possible KG-particle/antiparticle energy value.