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Uniformly convex functions

100%

Ma W., Minda D.

Annales Polonici Mathematici

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1992

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tom 57

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nr 2

165-175

EN

Recently, A. W. Goodman introduced the geometrically defined class UCV of uniformly convex functions on the unit disk; he established some theorems and raised a number of interesting open problems for this class. We give a number of new results for this class. Our main theorem is a new characterization for the class UCV which enables us to obtain subordination results for the family. These subordination results immediately yield sharp growth, distortion, rotation and covering theorems plus sharp bounds on the second and third coefficients. We exhibit a function k in UCV which, up to rotation, is the sole extremal function for these problems. However, we show that this function cannot be extremal for the sharp upper bound on the nth coefficient for all n. We establish this by obtaining the correct order of growth for the sharp upper bound on the nth coefficient over the class UCV and then demonstrating that the nth coefficient of k has a smaller order of growth.

2

Hyperbolically convex functions

100%

Ma W., Minda D.

Annales Polonici Mathematici

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1994-1995

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tom 60

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nr 1

81-100

EN

We investigate univalent holomorphic functions f defined on the unit disk 𝔻 such that f(𝔻) is a hyperbolically convex subset of 𝔻; there are a number of analogies with the classical theory of (euclidean) convex univalent functions. A subregion Ω of 𝔻 is called hyperbolically convex (relative to hyperbolic geometry on 𝔻) if for all points a,b in Ω the arc of the hyperbolic geodesic in 𝔻 connecting a and b (the arc of the circle joining a and b which is orthogonal to the unit circle) lies in Ω. We give several analytic characterizations of hyperbolically convex functions. These analytic results lead to a number of sharp consequences, including covering, growth and distortion theorems and the precise upper bound on |f''(0)| for normalized (f(0) = 0 and f'(0) > 0) hyperbolically convex functions. In addition, we find the radius of hyperbolic convexity for normalized univalent functions mapping 𝔻 into itself. Finally, we suggest an alternate definition of "hyperbolic linear invariance" for locally univalent functions f: 𝔻 → 𝔻 that parallels earlier definitions of euclidean and spherical linear invariance.

3

Uniformly convex functions II

100%

Ma W., Minda D.

Annales Polonici Mathematici

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1993

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tom 58

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nr 3

275-285

EN

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.

Ograniczanie wyników

3 Annales Polonici Mathematici

3 Ma W.

3 Minda D.

1 1994

1 1993

1 1992

Uniformly convex functions

Hyperbolically convex functions

Uniformly convex functions II