Let $L_1, L_2$ be two line bundles on a smooth projective variety $X$. Consider that there exists a non-zero homomorphism $f:L_1 \to L_2$. Over curve this gives us $\text{deg}(L_1) \leq \text{deg}(L_2)$ (Please correct me if this is wrong).
If we now consider $X$ to be a higher dimensional variety (say a surface), $H$ be an ample divisor on $X$ and a non-zero morphism $f:E_1 \to E_2$ between two rank $r$ bundles. Then do we have the analogous result that $\text{slope}_H(E_1) \leq \text{slope}_H(E_1)$?
If this is not true in general, then what can be said about their slope realtion?
Thanks in Advance