Uniformly convex functions

• Autorzy: Wancang Ma , David Minda
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

convex functions; growth and distorsion theorems; coefficient bounds

Kategorie tematyczne

• 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1992
• Tom: 57
• Numer: 2
• Strony: 165-175

Daty

wydano

1992

otrzymano

1991-09-06

Twórcy

autor

Wancang Ma

• Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, U.S.A.

autor

David Minda

• Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, U.S.A.

Bibliografia

• [D] P. Duren, Univalent Functions, Grundlehren Math. Wiss. 259, Springer, New York 1983.
• [G] G. M. Goluzin, On the majorization principle in function theory, Dokl. Akad. Nauk SSSR 42 (1935), 647-650 (in Russian).
• [G₁] A. W. Goodman, On uniformly convex functions, Ann. Polon. Math. 56 (1991), 87-92.
• [G₂] A. W. Goodman, Coefficient problems in geometric function theory, to appear.
• [K] H. Kober, Dictionary of Conformal Representations, Dover, New York 1957.
• [P] Ch. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen 1975.
• [R] W. Rogosinski, On the coefficients of subordinate functions, Proc. London Math. Soc. 48 (1943), 48-82.
• [Rø] F. Rønning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., to appear.