Subbundle slope bound from ambient Chern curvature ($\mu (F)\le C\mathrm{area}(X)/(2\pi )$)

Fix a Hermitian metric on a holomorphic vector bundle $E$ over a compact Riemann surface and a positive area form $\omega$. Write $i{F}_E=K\omega$ for its Chern curvature. If every eigenvalue of the Hermitian endomorphism $K$ is at most $C$, the curvature formula for a holomorphic subbundle gives $\mathrm{deg}F\le \mathrm{rank}(F)C{\int }_X\omega /(2\pi )$. The second-fundamental-form term is nonpositive. Compactness supplies a finite $C$ independent of the subbundle. If $F_E=0$, every subbundle has nonpositive degree.