I'm reading Fangyang Zheng's Complex Differential Geometry. I have a problem with notation while reading the following theorem.
Theorem 7.23 (Yau's Schwarz Lemma). Let $(M, h)$ be a complete Kähler manifold with Ricci curvature $r \geq a$, and $(N, g)$ a Hermitian manifold with holomorphic sectional curvature $H \leq b<0$. Here $a$ and $b$ are constant. If $f: M \rightarrow N$ is a non-constant holomorphic map, then $a<0$, and $\color{red}{f^{*} g \leq \frac{a}{b} h}$.
Denote by $\omega_h$ and $\omega_g$ the Kähler forms of $h$ and $g$, respectively. In the proof, $u$ is a function of $M$ and finally we obtain $$f^*\omega_g\leq u\omega_h\quad\text{and}\quad\sup u\leq\frac ab.$$ Then the proof is over.
This book once defined positive real $(1,1)$-form: A real $(1,1)$-form $\psi$ on $M$ is said to be positive if for any $x\in M$ and any $(1,0)$-type tangent vector $X$ at $x$, $$-\sqrt{-1}\psi(X,\overline X)>0.$$ In this case we will write $\psi>0$. (Similarly for "$\geq$".) So if $\psi=\sqrt{-1}f_{ij}dz^i\wedge d\overline z^j$, then $\psi>0$ (or $\geq0$) iff $(f_{ij})$ is (semi-) positive definite. Thus the notation "$f^*\omega_g\leq u\omega_h$" can be understood explicitly.
However, the book does not provide an explanation for comparing two metrics. My initial understanding was that as two matrices, each pair of corresponding entries satisfies the relationship, i.e., $(f^*g)_{ij}\leq \frac abh_{ij}$. But I cannot obtain this result. The proof suddenly end when we get $\sup u\leq\frac ab$, so I guess $f^{*} g \leq \frac{a}{b} h$ and $f^*\omega_g\leq\frac{a}{b}\omega_h$ is equivalent. Is this correct? Any help would be appreciated!
