Displaying similar documents to “A free boundary value problem in potential theory”

Plurisubharmonic functions with logarithmic singularities

E. Bedford, B. A. Taylor (1988)

Annales de l'institut Fourier

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To a plurisubharmonic function u on C n with logarithmic growth at infinity, we may associate the Robin function ρ u ( z ) = lim sup λ u ( λ z ) - log ( λ z ) defined on P n - 1 , the hyperplane at infinity. We study the classes L + , and (respectively) L p of plurisubharmonic functions which have the form u = log ( 1 + | z | ) + O ( 1 ) and (respectively) for which the function ρ u is not identically - . We obtain an integral formula which connects the Monge-Ampère measure on the space C n with the Robin function on P n - 1 . As an application we obtain a criterion...

Note on special arithmetic and geometric means

Horst Alzer (1994)

Commentationes Mathematicae Universitatis Carolinae

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We prove: If A ( n ) and G ( n ) denote the arithmetic and geometric means of the first n positive integers, then the sequence n n A ( n ) / G ( n ) - ( n - 1 ) A ( n - 1 ) / G ( n - 1 ) ( n 2 ) is strictly increasing and converges to e / 2 , as n tends to .

Linear forms in the logarithms of three positive rational numbers

Curtis D. Bennett, Josef Blass, A. M. W. Glass, David B. Meronk, Ray P. Steiner (1997)

Journal de théorie des nombres de Bordeaux

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In this paper we prove a lower bound for the linear dependence of three positive rational numbers under certain weak linear independence conditions on the coefficients of the linear forms. Let Λ = b 2 log α 2 - b 1 log α 1 - b 3 log α 3 0 with b 1 , b 2 , b 3 positive integers and α 1 , α 2 , α 3 positive multiplicatively independent rational numbers greater than 1 . Let α j 1 = α j 1 / α j 2 with α j 1 , α j 2 coprime positive integers ( j = 1 , 2 , 3 ) . Let α j max { α j 1 , e } and assume that gcd ( b 1 , b 2 , b 3 ) = 1 . Let b ' = b 2 log α 1 + b 1 log α 2 b 2 log α 3 + b 3 log α 2 and assume that B max { 10 , log b ' } . We prove that either { b 1 , b 2 , b 3 } is c 4 , B -linearly dependent over (with respect to a 1 , a 2 , a 3 )...

On the lower order ( R ) of an entire Dirichlet series

P. K. Jain, D. R. Jain (1974)

Annales de l'institut Fourier

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The estimations of lower order ( R ) λ in terms of the sequences { a n } and { λ n } for an entire Dirichlet series f ( s ) = n = 1 a n e s λ n , have been obtained, namely : λ = max { λ n p } lim inf p λ n p log λ n p - 1 log | a n p | - 1 = max { λ n p } lim inf p ( λ n p - λ n p - 1 ) log λ n p - 1 log | a n p - 1 | a n p | . One of these estimations improves considerably the estimations earlier obtained by Rahman (Quart. J. Math. Oxford, (2), 7, 96-99 (1956)) and Juneja and Singh (Math. Ann., 184(1969), 25-29 ).

Regularity properties of solutions of elliptic equations in R 2 in limit cases

Angela Alberico, Vincenzo Ferone (1995)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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In this paper the Dirichlet problem for a linear elliptic equation in an open, bounded subset of R 2 is studied. Regularity properties of the solutions are proved, when the data are L 1 -functions or Radon measures. In particular sharp assumptions which guarantee the continuity of solutions are given.

Distribution of nodes on algebraic curves in N

Thomas Bloom, Norman Levenberg (2003)

Annales de l’institut Fourier

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Given an irreducible algebraic curves A in N , let m d be the dimension of the complex vector space of all holomorphic polynomials of degree at most d restricted to A . Let K be a nonpolar compact subset of A , and for each d = 1 , 2 , . . . , choose m d points { A d j } j = 1 , . . . , m d in K . Finally, let Λ d be the d -th Lebesgue constant of the array { A d j } ; i.e., Λ d is the operator norm of the Lagrange interpolation operator L d acting on C ( K ) , where L d ( f ) is the Lagrange interpolating polynomial for f of degree d at the points { A d j } j = 1 , . . . , m d . Using techniques...