On the Zalcman conjecture for starlike and typically real functions
Let S denote the class of functions \(f(z)=z+a_ 2z^ 2+...\) which are analytic and univalent in the unit disc. Recently, L. Zalcman conjectured that \(| a^ 2_ n-a_{2n-1}| \leq(n-1)^ 2\) for all \(n\geq 2\) and each \(f\in S\) and showed that this inequality implies the Bieberbach conjecture. In this paper we investigate the functional \(\lambda a^ 2_ n-a_{2n-1},\) for certain real \(\lambda\), in the classes of typically real functions and starlike functions. This functional arises quite naturally when considering coefficient problems: If \(f\in S\) then the coefficients of \(f(z^ 2)^{{1\over2}}\) and 1/f(1/z) contain expressions of this form. For the class of typically real functions we give upper and lower bounds for \(\lambda a^ 2_ n-a_{2n-1},\) many of them sharp. The Zalcman conjecture is proved for typically real functions when \(n\geq 3\) and an immediate consequence is the truth of the conjecture for real functions in S when \(n\geq 2\). Our results also include a proof of the conjecture for starlike functions in S. We make use of integral representation formulas for these classes of functions together with an interesting coefficient inequality for functions of positive real part due to \textit{A. E. Livingston} [Proc. Am. Math. Soc. 21, 545-552 (1969; Zbl 0186.399)].
scientific article; zbMATH DE number 3859448
scientific article