14.14.15 The remaining eigenvalue configuration.57C7A82680F3

Suppose now that $\lambda _ n<-\delta \lambda $, keeping the same constants. Since $f_ n\geq \mathcal F/n$ and $|\chi +W|\leq C\sqrt K$,

In (14.14.11), discard $\mathcal C$ and the term with $\psi ''$. Since $\psi $ is now fixed, $|\psi '\sum _ k f_ kv_{k\bar k}|\leq C\lambda \mathcal F$. Using $g_{1\bar1,k}=a_ k+O(\lambda )$ and Young's inequality, we bound the last error term by $\frac12\sum _ k f_ k|a_ k|^2/\lambda ^2+C\mathcal F$. This leaves

By (14.14.4) and the scalar inequality $\tfrac 32|a+b|^2\leq 2|a|^2+6|b|^2$,

Since $\lambda \geq K$, combining the last two inequalities and dividing by $\lambda \mathcal F>0$ gives $\lambda /K\leq C$. This finishes both cases. Since $H$ attains its maximum at the chosen point, the bound for $\lambda _1/K$ there gives a bound at all points, and $|\lambda _ i|\leq (n-1)\lambda _1$ then bounds every entry of the complex Hessian, as asserted.