a close-to-convex function

Convex and starlike images are very tidy, but many perfectly nice univalent images are neither — they may have indentations yet still never fold over themselves. Is there a roomier geometric class that still guarantees one-to-one-ness through a simple test? Yes: the close-to-convex functions. Loosely, the image may dent inward, but it is never allowed to make a hairpin turn that doubles its boundary back on itself; the boundary's complement stays accessible by non-crossing rays.

Precisely, f in the class S is close-to-convex if there is some convex univalent function g such that Re( f'(z) / g'(z) ) > 0 for all z in the disk. The geometric meaning, due to Kaplan, is that the boundary of f(disk) never turns back through an angle of pi or more as you traverse it — the tangent direction never reverses by a half-turn. Equivalently the complement of the image is a union of non-crossing rays. This is weaker than starlike (take g(z) = z to recover a sufficient condition close to starlikeness) but, crucially, close-to-convexity still implies univalence: it is a usable sufficient condition for a holomorphic map to be one-to-one.

Close-to-convex functions complete the standard hierarchy convex implies starlike implies close-to-convex implies univalent, widening the net of maps you can certify as univalent by an easy real-part test rather than by checking injectivity directly. The Bieberbach bound |a_n| <= n holds for them too, with Koebe extremal. An honest caveat: the implication is one-way — close-to-convexity is sufficient for univalence but not necessary, so a univalent map can fail the close-to-convex test. It is a convenient sufficient criterion, not a characterization of all univalent functions.