# 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.

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