I'm currently reading some papers regarding curvature for markov chains in disrecte spaces, which is not my main field in mathematics and got a question. In the paper of Frank Bauer, Paul Horn, Yong Lin, Gabor Lippner, Dan Mangoubi and Shing-Tung Yau with the name Li-Yau inequality on graphs (https://arxiv.org/abs/1306.2561) it is written on page 6 just before Definition 3.9 that "this motivates the following key modification of the CD-inequality".
But I don't see why. How does this equality (3.8) motivates the extra term $- \Gamma \left(f, \frac{\Gamma(f)}{f}\right)$ in the definition of the exponential curvature condition? Especially regarding the fact that in the equality there are square roots.
I think one can show that the equality proves $\frac{\Gamma(f)}{f} = 4\Gamma(\sqrt(f))$ but this seems not helping either.
Thanks for your help!