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Convex sum of univalent functions

Published online by Cambridge University Press:  09 April 2009

Pran Nath Chichra  and
Ram Singh
Pran Nath Chichra
Affiliation:
Punjabi UniversityPatiala, (India)
Ram Singh
Affiliation:
Punjabi UniversityPatiala, (India)
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Let f(z)=z + … be regular in the unit disc |z| < 1 (hereafter called E). In a recent paper Trimble [7] has proved that if f(z) be convex in E, then F(z) = (1 − λ)z + λf(z) is starlike with respect to the origin in E for (2/3) ≦ λ ≦ 1. The purpose of this note is to show that if certain additional restrictions be imposed on f(z), then F(z) becomes starlike for all λ, 0 ≦ λ ≦ 1. Also we consider some related problems.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 14 , Issue 4 , December 1972 , pp. 503 - 507
Copyright
Copyright © Australian Mathematical Society 1972

References

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