  We provide a mathematical description, based on d-wave Eliashberg theory, of
the strong correlation between the experimentally observed softening of Raman
modes associated with in-plane oxygen motions and the corresponding
superconducting critical temperature $T_c$, as a function of oxygen doping $x$,
in YBa$_2$Cu$_3$O$_x$. The theoretical model provides a direct link between
physical trends of soft optical $A_g$ (in-plane) oxygen modes, the level of
oxygen doping $x$, and the superconducting $T_c$. Different regimes observed in
the trend of $T_c$ vs doping can be related to corresponding regimes of optical
phonon softening in the Raman spectra. These results provide further evidence
related to the physical origin of high-temperature superconductivity in
rare-earth cuprate oxides and to the significant role of electron-phonon
coupling therein.
