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Extremal problems for a class of functions of positive real part and applications

Part of: Geometric function theory

Published online by Cambridge University Press:  09 April 2009

V. V. Anh  and
P. D. Tuan
V. V. Anh
Affiliation:
Department of Mathematics, Queensland Institute of Technology, P. O. Box 2434 Brisbane, Queensland 4001, Australia
P. D. Tuan
Affiliation:
First Interstate Bank of California, 600 South Spring Street, Los Angeles, California 90014, U.S.A.
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Abstract

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In this paper we determine the lower bound on |z| = r < 1 for the functional Re{αp(z) + β zp′(z)/p(z)}, α ≧0, β ≧ 0, over the class Pk (A, B). By means of this result, sharp bounds for |F(z)|, |F',(z)| in the family and the radius of convexity for are obtained. Furthermore, we establish the radius of starlikness of order β, 0 ≦ β < 1, for the functions F(z) = λf(Z) + (1-λ) zf′ (Z), |z| < 1, where ∞ < λ <1, and .

MSC classification

Secondary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 41 , Issue 2 , October 1986 , pp. 152 - 164
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Al-Amiri, H. S., ‘On the radius of β-convexity of starlike functions of order α’, Proc. Amer. Math. Soc. 39 (1973), 101–109.Google Scholar
[2]Bajpai, P. L. and Singh, P., ‘The radius of starlikeness of certain analytic functions’, Proc. Amer. Math. Soc. 44 (1974), 395–402.CrossRefGoogle Scholar
[3]Bernardi, S. D., ‘The radius of univalence of certain analytic functions’, Proc. Amer. Math. Soc. 24 (1970), 312–318.CrossRefGoogle Scholar
[4]Carathéodory, C., Theory of functions of a complex variable, Vol.2 (Chelsea Publishing Company, New York, 1960).Google Scholar
[5]Duren, P. L., Univalent Functions (Springer Verlag, New York, 1983).Google Scholar
[6]Goel, R. M. and Singh, V., ‘On the radii of univalence of certain analytic functions’, Indian J. Pure Appl. Math. 4 (1973), 402–421.Google Scholar
[7]Janowski, W., ‘Some extremal problems for certain families of analytic functions I’, Ann. Polon. Math. 28 (1973), 298–326.CrossRefGoogle Scholar
[8]Libera, R. J., ‘Some classes of regular univalent functions’, Proc. Amer. Math. Soc. 16 (1965), 755–758.CrossRefGoogle Scholar
[9]Libera, R. J. and Livingston, A. E., ‘On the univalence of some classes of regular functions’, Proc. Amer. Math. Soc. 30 (1971), 327–336.CrossRefGoogle Scholar
[10]Livingston, A. E., ‘On the radius of univalence of certain analytic functions’, Proc. Amer. Math. Soc. 17 (1966), 352–357.CrossRefGoogle Scholar
[11]Padmanabhan, K. S., ‘On the radius of univalence of certain classes of analytic functions’, J. London Math. Soc. (2) 1 (1969), 225–231.CrossRefGoogle Scholar
[12]Singh, V. and Goel, R. M., ‘On radii of convexity and starlikeness of some classes of functions’, J. Math. Soc. Japan 23 (1971), 323–339.CrossRefGoogle Scholar
[13]Zawadzki, R., ‘On some theorems on distortion and rotation in the class of κ-symmetric starlike functions of order α’, Ann. Polon. Math. 24 (1971), 169–185.CrossRefGoogle Scholar
[14]Zmoroviˇ, V. A., ‘On bounds of convexity for starlike functions of order α in the circle |z| < 1 and in the circular region 0 < |z| < 1’, Mat. Sb 68(110) (1965), 518–526, English transl., Amer. Math. Soc. Transl. (2) 80 (1969), 203–213.Google Scholar