Displaying similar documents to “A constant in pluripotential theory”

On the mean values of an analytic function.

G. S. Srivastava, Sunita Rani (1992)

Annales Polonici Mathematici

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Let f(z), z = r e i θ , be analytic in the finite disc |z| < R. The growth properties of f(z) are studied using the mean values I δ ( r ) and the iterated mean values N δ , k ( r ) of f(z). A convexity result for the above mean values is obtained and their relative growth is studied using the order and type of f(z).

Approximation of values of hypergeometric functions by restricted rationals

Carsten Elsner, Takao Komatsu, Iekata Shiokawa (2007)

Journal de Théorie des Nombres de Bordeaux

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We compute upper and lower bounds for the approximation of hyperbolic functions at points 1 / s ( s = 1 , 2 , ) by rationals x / y , such that x , y satisfy a quadratic equation. For instance, all positive integers x , y with y 0 ( mod 2 ) solving the Pythagorean equation x 2 + y 2 = z 2 satisfy | y sinh ( 1 / s ) - x | log log y log y . Conversely, for every s = 1 , 2 , there are infinitely many coprime integers x , y , such that | y sinh ( 1 / s ) - x | log log y log y and x 2 + y 2 = z 2 hold simultaneously for some integer z . A generalization to the approximation of h ( e 1 / s ) for rational...

Non-degenerate Hilbert cubes in random sets

Csaba Sándor (2007)

Journal de Théorie des Nombres de Bordeaux

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A slight modification of the proof of Szemerédi’s cube lemma gives that if a set S [ 1 , n ] satisfies | S | n 2 , then S must contain a non-degenerate Hilbert cube of dimension log 2 log 2 n - 3 . In this paper we prove that in a random set S determined by Pr { s S } = 1 2 for 1 s n , the maximal dimension of non-degenerate Hilbert cubes is a.e. nearly log 2 log 2 n + log 2 log 2 log 2 n and determine the threshold function for a non-degenerate k -cube.

Explicit lower bounds for linear forms in two logarithms

Nicolas Gouillon (2006)

Journal de Théorie des Nombres de Bordeaux

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We give an explicit lower bound for linear forms in two logarithms. For this we specialize the so-called Schneider method with multiplicity described in []. We substantially improve the numerical constants involved in existing statements for linear forms in two logarithms, obtained from Baker’s method or Schneider’s method with multiplicity. Our constant is around 5 . 10 4 instead of 10 8 .