Abstract
The Al-Refai--Darus criterion guarantees univalence of a normalized function \( f \) when a weighted sum of \( |(z/f)^{(k)}(0)| \), \( 2\le k\le n-1 \), and of \( \sup|(z/f)^{(n)}| \) does not exceed \( 1 \). For \( n\ge3 \) we study the subclass \( \mathcal C_n \) with \( a_2=\dots=a_n=0 \), in which the whole allowance goes to the \( n \) th derivative. The map \( f\mapsto(z/f)^{(n)} \) is a bijection of \( \mathcal C_n \) onto a closed ball of \( H^\infty \), 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 \( z/(1+cz^{n+j}) \); the exact region of \( (a_{n+1},a_{n+2}) \), whose boundary is attained by M\"obius parameters written in closed form; and an exact radius for dilations of products \( f_1f_2/z \), sharp both for Aksentev's inequality and for univalence. The \( n \)-fold symmetric members of \( \mathcal C_n \) correspond to \( g(z^n) \) with \( g \) in the same ball, which gives the exact region of \( (a_{n+1},a_{2n+1}) \). For the larger class defined by \( a_2=0 \) 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.