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On convex univalent functions. II. (English) Zbl 0542.30015

[For part I see the author in ibid. 33, 221-225 (1976; Zbl 0371.30011).] Let B denote the family of analytic and univalent functions of the form \[ f(z)=z+z^ L/L!+\sum^{\infty}_{n=L+1}A_ nz^ n,\quad L\geq 1,z\in D;\quad | z|<1. \] A function \(f\in B\) is said to be convex univalent in D if f(D) is a convex set. Let S be a subclass of B. The author studies the class of functions \(f\in S\) for which \(| 1+z(f^{L+1}(z)/f^ L(z))-m|<M,\quad | z|<1,\) where L is some integer \(\geq 1\) and \[ (m,M)\in \{(m,M):1>m>1/2,\quad m\geq M>1- m\}\cup \{(m,M):m\geq 1,\quad m\geq M>m-1\}. \] This subclass of S is denoted by K(m,M,L). The author finds: The integral representation for the functions \(f(z)\in K(m,M,L);\) the estimations of the functionals \(| A_ n|\), \(| f^ L(z)|\), \(| f(z)-z|, | f^{L+1}(z)/f^ L(z)|\) defined on K(m,M,L); and the radius of convexity of \(f\in K(m,M,L)\).
Reviewer: L.Mikołajczyk

MSC:

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

Citations:

Zbl 0371.30011
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