We have shown algebraic conditions equivalent to saturating the data processing inequality for $1<z\leq\alpha\leq2z$, which generalizes the $\alpha-$ SRD saturation condition from \cite{sandwichConditions}. The techniques in this paper fail for $\alpha<1$ because the H\"{o}lder inequality requires positive powers, so it would be interesting to find a similar result for this case. As mentioned in section \ref{sec: notations}, a quantum channel $\Lambda$ is said to be \emph{sufficient} with respect to $\rho$ and $\sigma$ if there exists a quantum channel $\cR$ such that $(\cR\circ\Lambda)(\rho)=\rho$ and $(\cR\circ\Lambda)(\sigma)=\sigma$. For Umegaki relative entropy, $\alpha$-RRE, and $\alpha$-SRD, saturation of the DPI is equivalent to sufficiency of the quantum channel $\Lambda$. In general, it is not known whether sufficiency of the quantum channel is equivalent to saturation of the $\alpha-z$ RRE DPI. However in \cite{hiaimosonyi}, Hiai and Mosonyi do prove such results for a set of density operators fixed under the quantum channel. (i.e., $\Lambda(\rho)=\rho$ and $\Lambda(\sigma)=\sigma)$. It would be interesting to find a larger class of channels where sufficiency holds. 
\subsection{Acknowledgments}
The author is appreciative to Haonan Zhang for carefully reading this manuscript and asking questions that lead to the modification of Proposition \ref{newPick}. The author is also grateful to Anna Vershynina, who is her advisor, for her comments and suggestions throughout the entirety of this paper. This work is supported by the National Science Foundation (NSF) grant DMS-1812734.
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