New Results on Certain Subclasses of Univalent Functions Defined by the nth Derivative of z/f(z)

For n≥3, we study the subclass Cn of univalent functions f with a2=…=an=0 in which the whole allowance of the Al-Refai–Darus criterion is assigned to the nth derivative of z/f. The map f↦(z/f)(n) is a bijection of Cn onto a closed ball of H∞, which turns extremal problems into problems for bounded analytic functions. We prove sharp growth, distortion, covering-radius and coefficient bounds with the (n+j)-fold symmetric extremal functions z/(1+czn+j); the exact region of (an+1,an+2), whose boundary is attained by Möbius parameters written in closed form; an exact radius for dilations of products f1f2/z, sharp both for Aksentev’s inequality and for univalence. The vanishing of a2,…,an does not by itself imply n-fold symmetry; the n-fold symmetric members of Cn correspond exactly to g(zn) with g in the same ball, which provides the exact region of (an+1,a2n+1). For the larger class defined by a2=0 and the criterion, we obtain sharp bounds as well. The comparison with n-fold symmetric univalent functions records a gap; it is not a sharpening of known results.