Uniformly convex functions

Wancang Ma; David Minda

Annales Polonici Mathematici (1992)

  • Volume: 57, Issue: 2, page 165-175
  • ISSN: 0066-2216

Abstract

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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.

How to cite

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Wancang Ma, and David Minda. "Uniformly convex functions." Annales Polonici Mathematici 57.2 (1992): 165-175. <http://eudml.org/doc/262507>.

@article{WancangMa1992,
abstract = {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.},
author = {Wancang Ma, David Minda},
journal = {Annales Polonici Mathematici},
keywords = {convex functions; growth and distorsion theorems; coefficient bounds},
language = {eng},
number = {2},
pages = {165-175},
title = {Uniformly convex functions},
url = {http://eudml.org/doc/262507},
volume = {57},
year = {1992},
}

TY - JOUR
AU - Wancang Ma
AU - David Minda
TI - Uniformly convex functions
JO - Annales Polonici Mathematici
PY - 1992
VL - 57
IS - 2
SP - 165
EP - 175
AB - 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.
LA - eng
KW - convex functions; growth and distorsion theorems; coefficient bounds
UR - http://eudml.org/doc/262507
ER -

References

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  1. [D] P. Duren, Univalent Functions, Grundlehren Math. Wiss. 259, Springer, New York 1983. 
  2. [G] G. M. Goluzin, On the majorization principle in function theory, Dokl. Akad. Nauk SSSR 42 (1935), 647-650 (in Russian). 
  3. [G₁] A. W. Goodman, On uniformly convex functions, Ann. Polon. Math. 56 (1991), 87-92. Zbl0744.30010
  4. [G₂] A. W. Goodman, Coefficient problems in geometric function theory, to appear. 
  5. [K] H. Kober, Dictionary of Conformal Representations, Dover, New York 1957. Zbl0049.33508
  6. [P] Ch. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen 1975. 
  7. [R] W. Rogosinski, On the coefficients of subordinate functions, Proc. London Math. Soc. 48 (1943), 48-82. Zbl0028.35502
  8. [Rø] F. Rønning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., to appear. Zbl0805.30012

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