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    Wyświetlanie 1-5 z 5
    Tytuł:
    Hyperbolically convex functions II
    Autorzy:
    Ma, William
    Minda, David
    Powiązania:
    https://bibliotekanauki.pl/articles/1294054.pdf
    Data publikacji:
    1999
    Wydawca:
    Polska Akademia Nauk. Instytut Matematyczny PAN
    Tematy:
    hyperbolic convexity
    two-variable characterization
    Schwarzian derivative
    distortion theorem
    Opis:
    Unlike those for euclidean convex functions, the known characterizations for hyperbolically convex functions usually contain terms that are not holomorphic. This makes hyperbolically convex functions much harder to investigate. We give a geometric proof of a two-variable characterization obtained by Mejia and Pommerenke. This characterization involves a function of two variables which is holomorphic in one of the two variables. Various applications of the two-variable characterization result in a number of analogies with the classical theory of euclidean convex functions. In particular, we obtain a uniform upper bound on the Schwarzian derivative. We also obtain the sharp lower bound on |f'(z)| for all z in the unit disk, and the sharp upper bound on |f'(z)| when |z| ≤ √2 - 1.
    Źródło:
    Annales Polonici Mathematici; 1999, 71, 3; 273-285
    0066-2216
    Pojawia się w:
    Annales Polonici Mathematici
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    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
    Tytuł:
    Uniformly convex functions
    Autorzy:
    Ma, Wancang
    Minda, David
    Powiązania:
    https://bibliotekanauki.pl/articles/1312032.pdf
    Data publikacji:
    1992
    Wydawca:
    Polska Akademia Nauk. Instytut Matematyczny PAN
    Tematy:
    convex functions
    growth and distorsion theorems
    coefficient bounds
    Opis:
    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.
    Źródło:
    Annales Polonici Mathematici; 1992, 57, 2; 165-175
    0066-2216
    Pojawia się w:
    Annales Polonici Mathematici
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
    Tytuł:
    Hyperbolically convex functions
    Autorzy:
    Ma, Wancang
    Minda, David
    Powiązania:
    https://bibliotekanauki.pl/articles/1311659.pdf
    Data publikacji:
    1994
    Wydawca:
    Polska Akademia Nauk. Instytut Matematyczny PAN
    Tematy:
    hyperbolic convexity
    distortion theorem
    growth thoerem
    linear invariance
    Opis:
    We investigate univalent holomorphic functions f defined on the unit disk such that f() is a hyperbolically convex subset of ; there are a number of analogies with the classical theory of (euclidean) convex univalent functions. A subregion Ω of is called hyperbolically convex (relative to hyperbolic geometry on ) if for all points a,b in Ω the arc of the hyperbolic geodesic in 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 into itself. Finally, we suggest an alternate definition of "hyperbolic linear invariance" for locally univalent functions f: → that parallels earlier definitions of euclidean and spherical linear invariance.
    Źródło:
    Annales Polonici Mathematici; 1994-1995, 60, 1; 81-100
    0066-2216
    Pojawia się w:
    Annales Polonici Mathematici
    Dostawca treści:
    Biblioteka Nauki
    Artykuł
      Wyświetlanie 1-5 z 5

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