Recently, A. W. Goodman introduced the geometrically defined class UCV of uniformly convex functions on the unit disk; he established some theorems and raised a number of interesting open problems for this class. We give a number of new results for this class. Our main theorem is a new characterization for the class UCV which enables us to obtain subordination results for the family. These subordination results immediately yield sharp growth, distortion, rotation and covering theorems plus sharp bounds on the second and third coefficients. We exhibit a function k in UCV which, up to rotation, is the sole extremal function for these problems. However, we show that this function cannot be extremal for the sharp upper bound on the nth coefficient for all n. We establish this by obtaining the correct order of growth for the sharp upper bound on the nth coefficient over the class UCV and then demonstrating that the nth coefficient of k has a smaller order of growth.
@article{bwmeta1.element.bwnjournal-article-apmv57z2p165bwm,
author = {Wancang Ma and David Minda},
title = {Uniformly convex functions},
journal = {Annales Polonici Mathematici},
volume = {57},
year = {1992},
pages = {165-175},
zbl = {0760.30004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv57z2p165bwm}
}
Wancang Ma; David Minda. Uniformly convex functions. Annales Polonici Mathematici, Tome 57 (1992) pp. 165-175. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv57z2p165bwm/
[000] [D] P. Duren, Univalent Functions, Grundlehren Math. Wiss. 259, Springer, New York 1983.
[001] [G] G. M. Goluzin, On the majorization principle in function theory, Dokl. Akad. Nauk SSSR 42 (1935), 647-650 (in Russian).
[002] [G₁] A. W. Goodman, On uniformly convex functions, Ann. Polon. Math. 56 (1991), 87-92. | Zbl 0744.30010
[003] [G₂] A. W. Goodman, Coefficient problems in geometric function theory, to appear.
[004] [K] H. Kober, Dictionary of Conformal Representations, Dover, New York 1957. | Zbl 0049.33508
[005] [P] Ch. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen 1975.
[006] [R] W. Rogosinski, On the coefficients of subordinate functions, Proc. London Math. Soc. 48 (1943), 48-82. | Zbl 0028.35502
[007] [Rø] F. Rønning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., to appear. | Zbl 0805.30012