a convex univalent function
A region is convex if, whenever you pick two points inside it, the straight segment joining them stays entirely inside — no dents, no notches, no missing bites. A disk, a half-plane, and the inside of an ellipse are convex; a crescent or a star is not. A convex univalent function is a one-to-one holomorphic map of the disk whose image is a convex region. These are the most rigid and best-behaved of the geometric sub-classes.
Precisely, f in the class S is convex if f(disk) is a convex set. The analytic test is again clean: f is convex if and only if Re( 1 + z f''(z) / f'(z) ) > 0 for all z in the disk. The quantity 1 + z f''/f' measures how the tangent direction along an image circle turns; demanding its real part stays positive says the boundary curve always turns the same way (never reverses its bending), which is exactly convexity. There is a lovely link to starlikeness, the Alexander theorem: f is convex if and only if z f'(z) is starlike, so the two tests are derivatives of one another.
Convex maps sit at the top of the hierarchy: convex implies starlike implies close-to-convex implies univalent. Because they are so constrained, their sharp coefficient bound is much smaller than the general one — for a convex f, |a_n| <= 1 for all n (extremal: the half-plane map z/(1 - z) = z + z^2 + z^3 + ... , whose coefficients are all 1), far below the |a_n| <= n of the full class S. A caution: 'convex' here is convexity of the IMAGE region, a geometric property of the map; it is unrelated to a function being convex in the real-calculus sense of an upward-curving graph.
The map f(z) = z/(1 - z) sends the unit disk onto the half-plane Re w > -1/2, which is convex. Its series z + z^2 + z^3 + ... has every a_n = 1, the extremal value for convex functions. Note its derivative-relative z f'(z) = z/(1 - z)^2 is exactly the Koebe function, which is starlike — illustrating Alexander's theorem.
The half-plane map is convex with |a_n| = 1; its z f'(z) is the starlike Koebe function (Alexander).
'Convex' refers to the image region being convex, not to a convex graph in real calculus. Convex is strictly stronger than starlike: |a_n| <= 1 for convex versus |a_n| <= n in general.