# Uniformly convex functions

Annales Polonici Mathematici (1992)

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

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topWancang 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

top- [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. Zbl0744.30010
- [G₂] A. W. Goodman, Coefficient problems in geometric function theory, to appear.
- [K] H. Kober, Dictionary of Conformal Representations, Dover, New York 1957. Zbl0049.33508
- [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. Zbl0028.35502
- [Rø] F. Rønning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., to appear. Zbl0805.30012

## Citations in EuDML Documents

top- Frode Rønning, Integral representations of bounded starlike functions
- Wancang Ma, David Minda, Uniformly convex functions II
- D. Vamshee Krishna, B. Venkateswarlu, T. RamReddy, Coefficient inequality for transforms of parabolic starlike and uniformly convex functions
- R. M. El-Ashwah, M. K. Aouf, A. Shamandy, E. E. Ali, Subordination results for some subclasses of analytic functions

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