- Tytuł:
- Uniformly convex functions II
- Autorzy:
-
Ma, Wancang
Minda, David - Powiązania:
- https://bibliotekanauki.pl/articles/1311788.pdf
- Data publikacji:
- 1993
- Wydawca:
- Polska Akademia Nauk. Instytut Matematyczny PAN
- Tematy:
-
convex functions
coefficient bounds - Opis:
- 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.
- Źródło:
-
Annales Polonici Mathematici; 1993, 58, 3; 275-285
0066-2216 - Pojawia się w:
- Annales Polonici Mathematici
- Dostawca treści:
- Biblioteka Nauki
Artykuł