### Uniformly convex functions II

Wancang Ma, David Minda (1993)

Annales Polonici Mathematici

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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 ${f}^{-1}\left(w\right)=w+d\u2082w\xb2+d\u2083w\xb3+...$. The series expansion for ${f}^{-1}\left(w\right)$ converges when $\left|w\right|<{\varrho}_{f}$, 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...