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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

1. Bazilevič, , On a case of integrability by quadratures of the equation of Loewner-Kufarev, Mat. Sb. 37 (1955), 471–476 (Russian).Google Scholar
2. Biernacki, M., Sur la représentation conforme de domains linéairement accessible, Prace Mat. Fiz. 44 (1936), 293–314.Google Scholar
3. Hummel, J., Multivalent starlike functions, J. Analyse Math. 18 (1967), 133–160.Google Scholar
4. Kaplan, W., Close-to-convex schlicht functions, Michigan Math. J. 1 (1952), 169–185.Google Scholar
5. Keogh, F. and Miller, S., On the coefficients of Bazilevič functions, Mat. Zametki 11 (1972), 509–516; or Math. Notes 77 (1972), 311–315.Google Scholar
6. Lewandowski, Z., Sur l'identité de certaines classes de fonctions univalentes, I, II, Ann. Univ. Mariae Curie-Sklodowska 12 (1958), 131–146, 14 (1960), 19–46.Google Scholar
7. Livingston, A., p-valent close-to-convex functions, Trans. Amer. Math. Soc. 115 (1965), 161–179.Google Scholar
8. Lyzzaik, A., Multivalent linearly accessible functions and close-to-convex functions, Proc. London Math. Soc. 44 (1982), 178–192.Google Scholar
9. Lyzzaik, A. and Styer, D., The geometry of multivalent close-to-convex functions, Proc. London Math. Soc. 57 (1985), 56–76.Google Scholar
10. Lyzzaik, A. and Styer, D., The uniqueness of decomposition of a class of multivalent functions, to appear.Google Scholar
11. Pommerenke, Chr., Über die subordination analytischerfunktionen, J. Reine Angew. Math. 275 (1965), 159–173.Google Scholar
12. Prokhorov, D., A generalization of a class of close-to-convex functions, Mat. Zametki 11 (1972); or Math. Notes 77 (1972), 311–315.Google Scholar
13. Sheil-Small, T., On Bazilevic functions, Quat. J. Math., Oxford 23 (1972), 135–142.Google Scholar
14. Sheil-Small, T., On linear accessibility and the conformal mapping of convex domains, J. Analyse Math. 25 (1972), 259–276.Google Scholar
15. Styer, D., Close-to-convex multivalent functions with respect to weakly starlike functions. Trans. Amer. Math. Soc. 769 (1972), 105–112.Google Scholar