a starlike function
A region is star-shaped with respect to a point if you can stand at that point and see every other point of the region in a straight line that never leaves the region — like standing at the centre of a star and looking out along each spike. A starlike function is a univalent map of the disk whose image is exactly such a star-shaped region (with respect to the origin). It is one of the special, well-behaved sub-families of univalent maps where geometry can be read off an algebraic test.
Precisely, f in the class S is starlike if its image f(disk) is star-shaped with respect to 0: for every w in the image, the whole segment from 0 to w stays in the image. The beautiful part is the analytic characterization: f is starlike if and only if the quantity z f'(z) / f(z) has positive real part for all z in the disk, that is Re( z f'(z) / f(z) ) > 0. Intuitively, as you walk a circle |z| = r outward in argument, the image point f(z) also turns steadily counterclockwise and never backtracks in angle as seen from the origin — that monotone turning of the argument is exactly what 'sees every point along a ray' means.
Starlike functions sit in a clean hierarchy: every convex univalent function is starlike, and every starlike function is close-to-convex, and all three are univalent. They are easier to handle than general class-S maps because the test is a single positivity condition, and for them the sharp coefficient bound is the same |a_n| <= n with Koebe extremal (the Koebe function is itself starlike). A caution: starlikeness is always 'with respect to a chosen point'; the standard normalized statement fixes that point at the origin. A region can be starlike about one point and not about another, so the centre matters.
The Koebe function k(z) = z/(1 - z)^2 is starlike: its image is the plane minus a ray to infinity, and from the origin every other point is visible along a straight segment that avoids the slit. Its test value z k'(z)/k(z) = (1 + z)/(1 - z) does have positive real part on the disk, confirming the criterion.
Koebe is starlike about 0; the criterion Re(z f'/f) > 0 holds throughout the disk.
Starlikeness is relative to a centre; the normalized criterion fixes it at 0. Convex implies starlike but not the reverse — a starlike image can have dents that a straight chord between two of its points would leave.