Jensen's inequality applied in proof of Theorem 6.2 of Entropy and heat kernel bounds on a Ricci flow background

Question

Picture below is from Entropy and heat kernel bounds on a Ricci flow background. I want to get the red line. For me, the Jensen's inequality is $$ \varphi\left(\int_\Omega f d\mu\right) \le \int_\Omega \varphi\circ f d\mu $$ when $\varphi$ is convex. Since $x\rightarrow x\ln x$ is convex, by Jensen's inequality, I have $$ \ln(\frac{1}{|B|_0}\int_B u d g_{_{_0}})\frac{1}{|B|_0}\int_B u d g_{_{_0}} = \ln(\int_B \frac{u}{|B|_0} d g_{_{_0}})\int_B \frac{u}{|B|_0} d g_{_{_0}} \\ \le \int_B \frac{u}{|B|_0}\ln(\frac{u}{|B|_0}) d g_{_{_0}} = \frac{1}{|B|_0}\int_B u(\ln u - \ln|B|_0) dg_{_{_0}} \tag{1} $$ since $\int_B u ~d g_{_{_0}} =1$, it is equal to $$ \ln(\frac{1}{|B|_0}\int_B u d g_{_{_0}})\frac{1}{|B|_0}\int_B u d g_{_{_0}} \le \frac{1}{|B|_0}\int_B u\ln u dg_{_{_0}} -\frac{\ln|B|_0}{|B|_0} \tag{2} $$ Obviously, (2) is not same with the red line of picture below. Whether I make some mistake or there is something imply $\ln |B|_0 \ge 1$ ?