Uniformly convex functions II

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 ${\displaystyle f^{-1}\left(w\right)=w+d₂w²+d₃w³+...}$. The series expansion for ${\displaystyle f^{-1}\left(w\right)}$ converges when ${\displaystyle \left|w\right|<{\varrho }_f}$, where ${\displaystyle 0<{\varrho }_f}$ depends on f. The sharp bounds on ${\displaystyle \left|a_n\right|}$ 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 ${\displaystyle \left|a_n\right|}$ 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 ${\displaystyle \left|a_n\right|}$ for n sufficiently large. We also find the sharp estimate on the functional |μa²₂ - a₃| for -∞ < μ < ∞. We give sharp bounds on ${\displaystyle \left|d_n\right|}$ for n = 2, 3 and 4. For ${\displaystyle n=2,k^{-1}}$ 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.