Univalent functions having univalent derivatives
Let T denote the family of functions \(f(z)=z-\sum^{\infty}_{n=2}a_ nz^ n\), \(a_ n\geq 0\), which are analytic and univalent in the unit disk \(\Delta =\{| z| <1\}\). The author [Proc. Am. Math. Soc. 51, 109-116 (1975; Zbl 0311.30007)] has shown that functions of the above type are in T if and only if \(\sum^{\infty}_{n=2}na_ n\leq 1\), with extreme points being z and \(z-z^ n/n\) \((n=2,3,...)\). In this paper, the author introduces the subfamily \(T_ 1\subset T\) for which f'(z) is univalent in \(\Delta\) whenever \(f\in T_ 1\). For \(T_ 1\), only a sufficient condition is found. Theorem: If \[ f(z)=z- \sum^{\infty}_{n=2}a_ nz^ n\in T,\quad a_ 2>0, \] then \(f\in T_ 1\) if \[ (*)\quad \sum^{\infty}_{3}(n-1)na_ n\leq 2a_ 2. \] The condition (*) is also necessary for the subclass consisting of the cubic polynomials. Theorem: If \[ f(z)=z-a_ 2z^ 2-a_ 3z^ 3\quad (a_ 2>0,\quad a_ 3\geq 0), \] then \(f\in T_ 1\) if and only if \(3a_ 3\leq \min \{1-2a_ 2,a_ 2\}\). Also \(a_ 3\leq 1/9\). Equality holds for \[ f(z)=z-1/3z^ 2-1/9z^ 3. \] However, for the entire class \(T_ 1\), the author shows that * is not the sufficient and necessary condition for functions to belong to \(T_ 1\) and 1/9 is not the upper bound for \(a_ 3\) but rather 1/6 does. The classes \(T_ m\) and \(T_{\infty}\) are also introduced. \(T_ m\) consists of functions in T for which the first m derivatives are univalent, while \(T_{\infty}\) has all derivatives univalent in \(\Delta\). For \(T_ m\) the author obtains a sufficient condition which generalizes (*). Also a sharp bound for \(a_ 2\) for functions in \(T_{\infty}\) is obtained; namely \(a_ 2<\) which surprisingly has no extremal function.
scientific article; zbMATH DE number 3985565
scientific article