Multivalent and Meromorphic Functions of Bounded Boundary Rotation

Leach, Ronald J.

Canadian journal of mathematics, Tome 27 (1975) no. 1, pp. 186-199

Résumé

The classVk(p). We generalize the class Vk of analytic functions of bounded boundary rotation [8] by allowing critical points in the unit disc U. Definition. Let f(z) = aqzq + . . . (q 1) be analytic in U. Then f(z) belongs to the class Vk(p) if for r sufficiently close to 1, and We note that (1.1) implies that / has precisely p — 1 critical points in U.

Détail

DOI : 10.4153/CJM-1975-024-9

Bibliographie

[1] 1. Brannan, D. A., On functions of bounded boundary rotation. I, Proc. Edinburgh Math. Soc. 16 (1969), 339–347. Google Scholar

[2] 2. Brannan, D. A. and Kirwan, W. E., On some classes of bounded analytic functions, J. London Math. Soc. 2 (1969), 431–443. Google Scholar

[3] 3. Goluzin, G. M., Geometric theory of functions of a complex variable (Amer. Math. Soc, Providence, R.I., 1969). Google Scholar

[4] 4. G∞dman, A. W., On the Schwarz-Christoffel transformation and p-valent functions, Trans. Amer. Math. Soc. 68 (1950), 204–223. Google Scholar

[5] 5. Hayman, W. K., Multivalent functions (Cambridge University Press, Cambridge, 1958). Google Scholar

[6] 6. Hummel, J. A., Extremal properties of weakly starlike p-valent starlike functions, Trans. Amer. Math. Soc. 180 (1968), 544–551. Google Scholar

[7] 7. Leach, R. J., Odd functions of bounded boundary rotation, Can. J. Math. 26 (1974), 551–564. Google Scholar

[8] 8. Lehto, O., On the distortion of conformai mappings with bounded boundary rotation, Ann. Acad. Sci. Fenn. Ser. A, 1, 124 (1952), 14 pp. Google Scholar

[9] 9. Livingston, A. E., p-valent close-to-convex functions, Trans. Amer. Math. Soc. 115 (1965), 161–179. Google Scholar

[10] 10. Livingston, A. E., Meromorphic multivalent close-to-convex functions, Trans. Amer. Math. Soc. 119 (1965), 167–177. Google Scholar

[11] 11. Livingston, A. E., The coefficients of multivalent close-to-convex functions, Proc. Amer. Math. Soc. 21 (1969), 545–552. Google Scholar

[12] 12. N∞nan, J., Asymptotic behavior of functions with bounded boundary rotation, Trans. Amer. Math. Soc. 164 (1972), 397–410. Google Scholar

[13] 13. Pfaltzgralï, J. A. and Pinchuk, B., Constrained extremal problems for classes of meromorphic functions, Bull. Amer. Math. Soc. 75 (1969), 379–383. Google Scholar

[14] 14. Pommerenke, C., On convex and starlike functions, J. London Math. Soc. 37 (1962), 209–224. Google Scholar

[15] 15. Schiffer, M. and Tammi, O., On the fourth coefficient of univalent functions with bounded boundary rotation, Ann. Acad. Sci. Fenn. Ser. A, 1, 396 (1967), 26 pp. Google Scholar

[16] 16. Umezawa, T., Analytic functions convex in one direction, J. Math. Soc. Japan 4 (1952), 194–202. Google Scholar

[17] 17. Umezawa, T., On the theory of univalent functions, Tohoku Math. J. 7 (1955), 212–223. Google Scholar