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The Multivalent Class of Geometrically Close-to-Convex Functions

Published online by Cambridge University Press:  20 November 2018

Abdallah Lyzzaik
Abdallah Lyzzaik*
Affiliation:
King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia
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The class of univalent close-to-convex functions, K, was introduced by Kaplan [4] and first studied by him. The first important extension to the class of multivalent close-to-convex functions, K(p) where p is a positive integer, was considered by Livingston [7]. Somewhat later, Styer [15] introduced the more general class, Kw(p), of weakly close-to-convex functions by simply taking the closure of Livingston's class K(p) in the topology of locally uniform convergence in B = {z: |z| ≤ 1}.

In 1936 Biernacki [2] introduced his class of linearly accessible functions. A function f is linearly accessible if f is univalent in B, f(0) = 0, and C – f(B) where C is the complex plane, is a union of closed (Euclidean) rays with disjoint interiors. In an interesting result, Lewandowski [6] showed that the classes of univalent close-to-convex functions and linearly accessible functions are equal.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 2 , 01 April 1987 , pp. 297 - 308
Copyright
Copyright © Canadian Mathematical Society 1987

References

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