Uniformly convex functions II

Wancang Ma; David Minda

Wancang Ma ; David Minda

Annales Polonici Mathematici, Tome 58 (1993), p. 275-285 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $f^{- 1} (w) = w + d ₂ w ² + d ₃ w ³ + . . .$ . 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.

Publié le : 1993-01-01

Zbl 0792.30008

EUDML-ID : urn:eudml:doc:262304

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     title = {Uniformly convex functions II},
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     volume = {58},
     year = {1993},
     pages = {275-285},
     zbl = {0792.30008},
     language = {en},
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Wancang Ma; David Minda. Uniformly convex functions II. Annales Polonici Mathematici, Tome 58 (1993) pp. 275-285. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv58z3p275bwm/

Bibliographie

[000] [A] L. V. Ahlfors, Complex Analysis, 3rd ed., McGraw-Hill, New York, 1979. | Zbl 0395.30001

[001] [G] A. W. Goodman, On uniformly convex functions, Ann. Polon. Math. 56 (1991), 87-92. | Zbl 0744.30010

[002] [L] A. E. Livingston, The coefficients of multivalent close-to-convex functions, Proc. Amer. Math. Soc. 21 (1969), 545-552. | Zbl 0186.39901

[003] [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. | Zbl 0464.30019

[004] [MM] W. Ma and D. Minda, Uniformly convex functions, Ann. Polon. Math. 57 (1992), 165-175. | Zbl 0760.30004

[005] [P] Ch. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975.

[006] [Rø₁] F. Rønning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc. 118 (1993), 189-196. | Zbl 0805.30012

[007] [Rø₂] F. Rønning, On starlike functions associated with parabolic regions, Ann. Univ. Mariae Curie-Skłodowska Sect. A 45 (1991), 117-122.

[008] [T] S. Y. Trimble, A coefficient inequality for convex univalent functions, Proc. Amer. Math. Soc. 48 (1975), 266-267. | Zbl 0283.30014