PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
1992 | 57 | 2 | 165-175
Tytuł artykułu

Uniformly convex functions

Treść / Zawartość
Warianty tytułu
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.
Rocznik
Tom
57
Numer
2
Strony
165-175
Opis fizyczny
Daty
wydano
1992
otrzymano
1991-09-06
Twórcy
autor
  • Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025, U.S.A.
autor
  • 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.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-apmv57z2p165bwm
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.