# Uniformly convex functions II

• Volume: 58, Issue: 3, page 275-285
• ISSN: 0066-2216

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## Abstract

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Recently, A. W. Goodman introduced the class UCV of normalized uniformly convex functions. We present some sharp coefficient bounds for functions f(z) = z + a₂z² + a₃z³ + ... ∈ UCV and their inverses ${f}^{-1}\left(w\right)=w+d₂w²+d₃w³+...$. The series expansion for ${f}^{-1}\left(w\right)$ converges when $|w|<{\varrho }_{f}$, where $0<{\varrho }_{f}$ depends on f. The sharp bounds on $|{a}_{n}|$ and all extremal functions were known for n = 2 and 3; the extremal functions consist of a certain function k ∈ UCV and its rotations. We obtain the sharp bounds on $|{a}_{n}|$ and all extremal functions for n = 4, 5, and 6. The same function k and its rotations remain the only extremals. It is known that k and its rotations cannot provide the sharp bound on $|{a}_{n}|$ for n sufficiently large. We also find the sharp estimate on the functional |μa²₂ - a₃| for -∞ < μ < ∞. We give sharp bounds on $|{d}_{n}|$ for n = 2, 3 and 4. For $n=2,{k}^{-1}$ and its rotations are the only extremals. There are different extremal functions for both n = 3 and n = 4. Finally, we show that k and its rotations provide the sharp upper bound on |f”(z)| over the class UCV.

## How to cite

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Wancang Ma, and David Minda. "Uniformly convex functions II." Annales Polonici Mathematici 58.3 (1993): 275-285. <http://eudml.org/doc/262304>.

@article{WancangMa1993,
abstract = {Recently, A. W. Goodman introduced the class UCV of normalized uniformly convex functions. We present some sharp coefficient bounds for functions f(z) = z + a₂z² + a₃z³ + ... ∈ UCV and their inverses $f^\{-1\}(w) = w + d₂w² + d₃w³ + ...$. The series expansion for $f^\{-1\}(w)$ converges when $|w| < ϱ_f$, where $0 < ϱ_f$ depends on f. The sharp bounds on $|a_n|$ and all extremal functions were known for n = 2 and 3; the extremal functions consist of a certain function k ∈ UCV and its rotations. We obtain the sharp bounds on $|a_n|$ and all extremal functions for n = 4, 5, and 6. The same function k and its rotations remain the only extremals. It is known that k and its rotations cannot provide the sharp bound on $|a_n|$ for n sufficiently large. We also find the sharp estimate on the functional |μa²₂ - a₃| for -∞ < μ < ∞. We give sharp bounds on $|d_n|$ for n = 2, 3 and 4. For $n = 2, k^\{-1\}$ and its rotations are the only extremals. There are different extremal functions for both n = 3 and n = 4. Finally, we show that k and its rotations provide the sharp upper bound on |f”(z)| over the class UCV.},
author = {Wancang Ma, David Minda},
journal = {Annales Polonici Mathematici},
keywords = {convex functions; coefficient bounds},
language = {eng},
number = {3},
pages = {275-285},
title = {Uniformly convex functions II},
url = {http://eudml.org/doc/262304},
volume = {58},
year = {1993},
}

TY - JOUR
AU - Wancang Ma
AU - David Minda
TI - Uniformly convex functions II
JO - Annales Polonici Mathematici
PY - 1993
VL - 58
IS - 3
SP - 275
EP - 285
AB - Recently, A. W. Goodman introduced the class UCV of normalized uniformly convex functions. We present some sharp coefficient bounds for functions f(z) = z + a₂z² + a₃z³ + ... ∈ UCV and their inverses $f^{-1}(w) = w + d₂w² + d₃w³ + ...$. The series expansion for $f^{-1}(w)$ converges when $|w| < ϱ_f$, where $0 < ϱ_f$ depends on f. The sharp bounds on $|a_n|$ and all extremal functions were known for n = 2 and 3; the extremal functions consist of a certain function k ∈ UCV and its rotations. We obtain the sharp bounds on $|a_n|$ and all extremal functions for n = 4, 5, and 6. The same function k and its rotations remain the only extremals. It is known that k and its rotations cannot provide the sharp bound on $|a_n|$ for n sufficiently large. We also find the sharp estimate on the functional |μa²₂ - a₃| for -∞ < μ < ∞. We give sharp bounds on $|d_n|$ for n = 2, 3 and 4. For $n = 2, k^{-1}$ and its rotations are the only extremals. There are different extremal functions for both n = 3 and n = 4. Finally, we show that k and its rotations provide the sharp upper bound on |f”(z)| over the class UCV.
LA - eng
KW - convex functions; coefficient bounds
UR - http://eudml.org/doc/262304
ER -

## References

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1. [A] L. V. Ahlfors, Complex Analysis, 3rd ed., McGraw-Hill, New York, 1979. Zbl0395.30001
2. [G] A. W. Goodman, On uniformly convex functions, Ann. Polon. Math. 56 (1991), 87-92. Zbl0744.30010
3. [L] A. E. Livingston, The coefficients of multivalent close-to-convex functions, Proc. Amer. Math. Soc. 21 (1969), 545-552. Zbl0186.39901
4. [LZ] R. J. Libera and E. J. Złotkiewicz, Early coefficients of the inverse of a regular convex function, Proc. Amer. Math. Soc. 85 (1982), 225-230. Zbl0464.30019
5. [MM] W. Ma and D. Minda, Uniformly convex functions, Ann. Polon. Math. 57 (1992), 165-175. Zbl0760.30004
6. [P] Ch. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975.
7. [Rø₁] F. Rønning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc. 118 (1993), 189-196. Zbl0805.30012
8. [Rø₂] F. Rønning, On starlike functions associated with parabolic regions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 45 (1991), 117-122.
9. [T] S. Y. Trimble, A coefficient inequality for convex univalent functions, Proc. Amer. Math. Soc. 48 (1975), 266-267. Zbl0283.30014

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