Theorem 12.11.19ECB2B21B601

Proof. For $0<\epsilon '<\epsilon $ the difference of the two backgrounds is the positive form $(\epsilon -\epsilon ')h$, so the monotonicity in proposition 12.2.2 shows that $\lambda _\mu (\theta +\epsilon h)$ decreases as $\epsilon \downarrow 0$. Its limit is therefore the infimum over $\epsilon >0$. Infima over $\epsilon >0$ and over Gauduchon metrics commute, and for a fixed metric the quotient is affine in $\epsilon $. Hence

Here the second equality holds because the coefficient of $\epsilon $ is positive, so the inner infimum is the value at $\epsilon =0$. This proves the first identity, and its right side is the definition of $\lambda _\mu (\theta )$, which does not involve $h$. The second identity follows by raising the first to the $n$th power, since $\lambda _\mu (\theta +\epsilon h)^ n=c_{\theta +\epsilon h}(\mu )$ by theorem 12.2.3. $\square$