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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

[1]Bernardi, S. D., ‘Convex and starlike univalent functions’, Trans. Amer. Math. Soc. 135 (1969), 429–446.CrossRefGoogle Scholar
[2]Libera, R. J., ‘Some classes of regular univalent functions’, Proc. Amer. Math. Soc. 16 (1965), 755–758.CrossRefGoogle Scholar
[3]Noshiro, K., ‘On the univalency of certain analytic functions’, J. Fac. Sci. Hokkaido. Imp. Univ. 2 (1934), 89–101.Google Scholar
[4]Singh, Ram, ‘On a class of starlike functions’, Composition Mathematica 19 (1967), 78–82.Google Scholar
[5]Sakaguchi, K., ‘On a certain univalent mapping’, J. Math. Soc. Japan. 11 (1959), 72–86.CrossRefGoogle Scholar
[6]Strohhacker, E., ‘Beitrage zur Theorie derschlichten Funktionen’, Math. Z. 37 (1933), 356–380.CrossRefGoogle Scholar
[7]Trimble, S. Y., ‘The convex sum of convex functions’, Math. Z. 109 (1969), 112–114.CrossRefGoogle Scholar