On the relationship between the majorization of functions and the majorization of derivatives in certain classes of holomorphic functions. (English) Zbl 0737.30016

Bogowski, F.; Burniak, Cz.

Ann. Univ. Mariae Curie-Skłodowska, Sect. A 43, 1-8 (1989).

Let \(A\) be a given class of functions defined in \(K_ 1=\{z:\;| z|<1\}\). In this paper the following implications are investigated: \[ | f(z)|\leq| F(z)|, \quad z\in K_ 1\Rightarrow| f'(z)|\leq T(r)| F'(z)|, \quad | z|=r\leq r(A), \] where \(f\) is regular in \(K_ 1\) and \(F\in A\). The functions \(T(r)\) and \(r(A)\) are determined for two special classes of functions: \[ \begin{aligned} H &=\{F:\hbox { Re}\{(1-z^ 2)F(z)/z\}>0\hbox { for } z\in K_ 1\},\\ H^* &=\{F:\;\exists\varphi\in S^*,\hbox { Re}\{F'(z)/\varphi(z)\}>0 \hbox { for } z\in K_ 1\}, \end{aligned} \] where \(S^*\) is the well known class of starlike univalent functions. The results are best possible, the extremal functions are given.

Reviewer: J.Stankiewicz (Rzeszów)

MSC:

30C75	Extremal problems for conformal and quasiconformal mappings, other methods
30C45	Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C55	General theory of univalent and multivalent functions of one complex variable