General univalence criterion associated with the \(n\)th derivative (Q448662)

Let \(A\) denote the class of functions \(f(z)= z+\sum^\infty_{n= 2} a_nz^n\) that are analytic in \(\mathcal{U}= \{z:|z|< 1\}\) and nonzero in \(\mathcal{U}\setminus\{0\}\). The authors prove a general and interesting criterion for univalence involving the \(n\)th derivative of \(z/f(z)\):NEWLINENEWLINETheorem. Let \(f\in A\), \(f(z)\neq 0\) in \(\mathcal{U}\setminus\{0\}\), and let \(g\in A\) be bounded in \(\mathcal{U}\) satisfying NEWLINE\[NEWLINEm= \inf\left\{\left|\frac{g(z_1)- g(z_2)}{z_1- z_2}\right|: z_1,z_2\in \mathcal{U}\right\}> 0.NEWLINE\]NEWLINE For any \(n\in \{3,4, \dots\}\), if NEWLINE\[NEWLINE\left|\frac{d^n}{dz^n}\left(\frac{z}{f(z)}- \frac{z}{g(z)}\right)\right|\leq K\quad (z\in \mathcal{U}),NEWLINE\]NEWLINE where NEWLINE\[NEWLINEK= \frac{n!}{n-1}\left(\frac{m}{M^2}-\sum\limits_{k=2}^{n-1}\frac{k-1}{k!}|\alpha_k|\right),\quad \alpha_k=\left.\frac{d^k}{dz^k}\left(\frac{z}{g(z)}- \frac{z}{f(z)}\right)\right|_{z=0},\quad M = \sup\{|g(z)|: z\in \mathcal{U}\},NEWLINE\]NEWLINE then \(f\) is univalent in \(\mathcal{U}\).NEWLINENEWLINEThe case \(n=2\) reduces to the univalence criterion due to \textit{D. Yang} and \textit{J. Liu} [Int. J. Math. Math. Sci. 22, No. 3, 605--610 (1999; Zbl 0963.30008)]. Other special cases and examples are given.