a univalent function
/ yoo-NIV-uh-lent /
Imagine pressing a stamp onto paper: a good stamp prints each part of its design onto a distinct spot, never folding the picture over itself. A univalent function is the same idea for a holomorphic map of a region: it is holomorphic AND one-to-one, so two different points of the domain never land on the same image point. The map may stretch, rotate, and bend the region, but it never overlaps it with itself. This is the cleanest, most geometric kind of complex map, and the whole subject of geometric function theory is built around it.
Precisely, a function f holomorphic on a region D is univalent there if f(z_1) = f(z_2) forces z_1 = z_2. The word schlicht is the original German term and means 'simple' or 'plain' in the sense of 'not folded over'. A key fact makes the idea pleasant to work with: for holomorphic functions, being globally one-to-one already implies f'(z) is never zero (a zero of f' would create a local fold, a many-to-one pinch), and a univalent f has a holomorphic inverse on its image. So a univalent map is automatically a conformal map of D onto the region f(D) — angles and small shapes are preserved everywhere.
Be careful to separate two levels. Local univalence near a point z_0 needs only f'(z_0) != 0, by the inverse function theorem; that is easy. Global univalence on all of D is much stronger and genuinely hard to verify — f'(z) != 0 everywhere does NOT guarantee it. The standard example: f(z) = e^z has f'(z) = e^z never zero, yet e^z is not univalent on the whole plane because it is periodic (e^(z + 2 pi i) = e^z). Detecting and bounding global univalence is exactly what the area theorem, the class S, and criteria like Nehari's Schwarzian bound are for.
On the unit disk |z| < 1 the map f(z) = z/(1 - z) is univalent: it is a Mobius transformation, hence one-to-one, and it sends the disk conformally onto the half-plane Re w > -1/2. By contrast f(z) = z^2 is univalent on the upper half of the disk but NOT on the whole disk, since z and -z share an image.
A Mobius map is univalent; z^2 is only locally univalent away from 0 and fails one-to-one globally.
f'(z) != 0 everywhere gives only LOCAL univalence, never global. The periodic e^z is the textbook reminder: a nonvanishing derivative is necessary but far from sufficient for one-to-one.