the subordination principle
Subordination is a way of saying that one holomorphic function is 'contained inside' another. Imagine a master function F that paints out some region as its image. A second function f is subordinate to F if everything f does is already done by F but at a slower, reined-in pace — f's values all lie inside F's image, and f starts where F starts. It is a precise sense in which f is a tamed, throttled version of F, and that taming forces f to obey inequalities that F does.
Precisely: let F and f be holomorphic on the unit disk with f(0) = F(0). We say f is subordinate to F, written f is subordinate to F (notation f << F), if there exists a holomorphic 'Schwarz function' w mapping the disk into itself with w(0) = 0 such that f(z) = F(w(z)). The Schwarz function w is the throttle: by the Schwarz lemma it satisfies |w(z)| <= |z|, so the composition f = F composed with w samples F only on the smaller disk |w| <= |z|. The immediate payoffs follow from this and the Schwarz lemma: the image of f is contained in the image of F; the maximum of |f| on the circle |z| = r is at most the maximum of |F| on the same circle (so M_f(r) <= M_F(r)); and the first Taylor coefficients are controlled, for instance |f'(0)| <= |F'(0)|. When F is univalent (one-to-one), subordination is equivalent to the clean geometric statement 'f(0) = F(0) and the image of f lies inside the image of F'.
Subordination is a central tool in geometric function theory — the study of univalent functions, starlike and convex maps, and coefficient problems like the Bieberbach conjecture. It packages 'this function is dominated by that one' into a form you can compose, differentiate, and bound. It is, at heart, the Schwarz lemma wearing work clothes. An honest caveat: subordination requires the SAME starting value f(0) = F(0); without that anchoring condition the Schwarz-function representation can fail, and the inequalities do not follow. And f << F constrains f by F, not the reverse — it is a one-directional domination, not a symmetric relation.
Let F(z) = z and f(z) = z^2 on the disk. Take the Schwarz function w(z) = z^2: then w(0) = 0, |w(z)| = |z|^2 <= |z|, and f(z) = F(w(z)) = w(z) = z^2, so f << F. The conclusions check out: the image of f (the disk) sits inside the image of F (the disk), and M_f(r) = r^2 <= r = M_F(r) for r < 1, exactly as subordination predicts.
f << F via a Schwarz function w with |w(z)| <= |z|; then M_f(r) <= M_F(r).
The equivalence 'f << F iff f(0) = F(0) and image of f is inside image of F' needs F to be univalent. If F is not one-to-one, mere image-containment is weaker than subordination — you genuinely need the Schwarz-function factorization f = F composed with w.