Displaying similar documents to “On meromorphic functions defined by a differential system of order 1

Universal sequences for Zalcman’s Lemma and Q m -normality

Shahar Nevo (2005)

Annales Polonici Mathematici

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We prove the existence of sequences ϱ n = 1 , ϱₙ → 0⁺, and z n = 1 , |zₙ| = 1/2, such that for every α ∈ ℝ and for every meromorphic function G(z) on ℂ, there exists a meromorphic function F ( z ) = F G , α ( z ) on ℂ such that ϱ α F ( n z + n ϱ ζ ) converges to G(ζ) uniformly on compact subsets of ℂ in the spherical metric. As a result, we construct a family of functions meromorphic on the unit disk that is Q m -normal for no m ≥ 1 and on which an extension of Zalcman’s Lemma holds.

Meromorphic function sharing a small function with a linear differential polynomial

Indrajit Lahiri, Amit Sarkar (2016)

Mathematica Bohemica

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The problem of uniqueness of an entire or a meromorphic function when it shares a value or a small function with its derivative became popular among the researchers after the work of Rubel and Yang (1977). Several authors extended the problem to higher order derivatives. Since a linear differential polynomial is a natural extension of a derivative, in the paper we study the uniqueness of a meromorphic function that shares one small function CM with a linear differential polynomial, and...

Uniqueness and differential polynomials of meromorphic functions sharing a nonzero polynomial

Pulak Sahoo (2016)

Mathematica Bohemica

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Let k be a nonnegative integer or infinity. For a { } we denote by E k ( a ; f ) the set of all a -points of f where an a -point of multiplicity m is counted m times if m k and k + 1 times if m > k . If E k ( a ; f ) = E k ( a ; g ) then we say that f and g share the value a with weight k . Using this idea of sharing values we study the uniqueness of meromorphic functions whose certain nonlinear differential polynomials share a nonzero polynomial with finite weight. The results of the paper improve and generalize the related results due to...

Pull-back of currents by meromorphic maps

Tuyen Trung Truong (2013)

Bulletin de la Société Mathématique de France

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Let  X and Y be compact Kähler manifolds, and let  f : X Y be a dominant meromorphic map. Based upon a regularization theorem of Dinh and Sibony for DSH currents, we define a pullback operator f for currents of bidegrees ( p , p ) of finite order on  Y (and thus forcurrent, since Y is compact). This operator has good properties as may be expected. Our definition and results are compatible to those of various previous works of Meo, Russakovskii and Shiffman, Alessandrini and Bassanelli, Dinh and Sibony,...

On the meromorphic solutions of a certain type of nonlinear difference-differential equation

Sujoy Majumder, Lata Mahato (2023)

Mathematica Bohemica

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The main objective of this paper is to give the specific forms of the meromorphic solutions of the nonlinear difference-differential equation f n ( z ) + P d ( z , f ) = p 1 ( z ) e α 1 ( z ) + p 2 ( z ) e α 2 ( z ) , where P d ( z , f ) is a difference-differential polynomial in f ( z ) of degree d n - 1 with small functions of f ( z ) as its coefficients, p 1 , p 2 are nonzero rational functions and α 1 , α 2 are non-constant polynomials. More precisely, we find out the conditions for ensuring the existence of meromorphic solutions of the above equation.

Bounds for the derivative of certain meromorphic functions and on meromorphic Bloch-type functions

Bappaditya Bhowmik, Sambhunath Sen (2024)

Czechoslovak Mathematical Journal

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It is known that if f is holomorphic in the open unit disc 𝔻 of the complex plane and if, for some c > 0 , | f ( z ) | 1 / ( 1 - | z | 2 ) c , z 𝔻 , then | f ' ( z ) | 2 ( c + 1 ) / ( 1 - | z | 2 ) c + 1 . We consider a meromorphic analogue of this result. Furthermore, we introduce and study the class of meromorphic Bloch-type functions that possess a nonzero simple pole in 𝔻 . In particular, we obtain bounds for the modulus of the Taylor coefficients of functions in this class.

Finite logarithmic order meromorphic solutions of linear difference/differential-difference equations

Abdelkader Dahmani, Benharrat Belaidi (2025)

Mathematica Bohemica

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Firstly we study the growth of meromorphic solutions of linear difference equation of the form A k ( z ) f ( z + c k ) + + A 1 ( z ) f ( z + c 1 ) + A 0 ( z ) f ( z ) = F ( z ) , where A k ( z ) , ... , A 0 ( z ) and F ( z ) are meromorphic functions of finite logarithmic order, c i ( i = 1 , ... , k , k ) are distinct nonzero complex constants. Secondly, we deal with the growth of solutions of differential-difference equation of the form i = 0 n j = 0 m A i j ( z ) f ( j ) ( z + c i ) = F ( z ) , where A i j ( z ) ( i = 0 , 1 , ... , n , j = 0 , 1 , ... , m , n , m ) and F ( z ) are meromorphic functions of finite logarithmic order, c i ( i = 0 , ... , n ) are distinct complex constants. We extend some previous results...

Twists and resonance of L -functions, I

Jerzy Kaczorowski, Alberto Perelli (2016)

Journal of the European Mathematical Society

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We obtain the basic analytic properties, i.e. meromorphic continuation, polar structure and bounds for the order of growth, of all the nonlinear twists with exponents 1 / d of the L -functions of any degree d 1 in the extended Selberg class. In particular, this solves the resonance problem in all such cases.

2-Cohomology of semi-simple simply connected group-schemes over curves defined over p -adic fields

Jean-Claude Douai (2013)

Journal de Théorie des Nombres de Bordeaux

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Let X be a proper, smooth, geometrically connected curve over a p -adic field k . Lichtenbaum proved that there exists a perfect duality: Br ( X ) × Pic ( X ) / between the Brauer and the Picard group of X , from which he deduced the existence of an injection of Br ( X ) in P X Br ( k P ) where P X and k P denotes the residual field of the point P . The aim of this paper is to prove that if G = G ˜ is an X e t - scheme of semi-simple simply connected groups (s.s.s.c groups), then we can deduce from Lichtenbaum’s results...

Spaces of geometrically generic configurations

Yoel Feler (2008)

Journal of the European Mathematical Society

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Let X denote either ℂℙ m or m . We study certain analytic properties of the space n ( X , g p ) of ordered geometrically generic n -point configurations in X . This space consists of all q = ( q 1 , , q n ) X n such that no m + 1 of the points q 1 , , q n belong to a hyperplane in X . In particular, we show that for a big enough n any holomorphic map f : n ( ℂℙ m , g p ) n ( ℂℙ m , g p ) commuting with the natural action of the symmetric group 𝐒 ( n ) in n ( ℂℙ m , g p ) is of the form f ( q ) = τ ( q ) q = ( τ ( q ) q 1 , , τ ( q ) q n ) , q n ( ℂℙ m , g p ) , where τ : n ( ℂℙ m , g p ) 𝐏𝐒𝐋 ( m + 1 , ) is an 𝐒 ( n ) -invariant holomorphic map. A similar result holds true for mappings of the configuration...

Further generalized versions of Ilmanen’s lemma on insertion of C 1 , ω or C loc 1 , ω functions

Václav Kryštof (2021)

Commentationes Mathematicae Universitatis Carolinae

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The author proved in 2018 that if G is an open subset of a Hilbert space, f 1 , f 2 : G continuous functions and ω a nontrivial modulus such that f 1 f 2 , f 1 is locally semiconvex with modulus ω and f 2 is locally semiconcave with modulus ω , then there exists f C loc 1 , ω ( G ) such that f 1 f f 2 . This is a generalization of Ilmanen’s lemma (which deals with linear modulus and functions on an open subset of n ). Here we extend the mentioned result from Hilbert spaces to some superreflexive spaces, in particular to L p spaces, p [ 2 , ) . We...

Consecutive square-free values of the type x 2 + y 2 + z 2 + k , x 2 + y 2 + z 2 + k + 1

Ya-Fang Feng (2023)

Czechoslovak Mathematical Journal

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We show that for any given integer k there exist infinitely many consecutive square-free numbers of the type x 2 + y 2 + z 2 + k , x 2 + y 2 + z 2 + k + 1 . We also establish an asymptotic formula for 1 x , y , z H such that x 2 + y 2 + z 2 + k , x 2 + y 2 + z 2 + k + 1 are square-free. The method we used in this paper is due to Tolev.

Subclasses of typically real functions determined by some modular inequalities

Leopold Koczan, Katarzyna Trąbka-Więcław (2010)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let T be the family of all typically real functions, i.e. functions that are analytic in the unit disk Δ : = { z : | z | < 1 } , normalized by f ( 0 ) = f ' ( 0 ) - 1 = 0 and such that Im z Im f ( z ) 0 for z Δ . Moreover, let us denote: T ( 2 ) : = { f T : f ( z ) = - f ( - z ) for z Δ } and T M , g : = { f T : f M g in Δ } , where M > 1 , g T S and S consists of all analytic functions, normalized and univalent in Δ .We investigate  classes in which the subordination is replaced with the majorization and the function g is typically real but does not necessarily univalent, i.e. classes { f T : f M g in Δ } , where M > 1 , g T , which we denote...

On a sequence formed by iterating a divisor operator

Bellaouar Djamel, Boudaoud Abdelmadjid, Özen Özer (2019)

Czechoslovak Mathematical Journal

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Let be the set of positive integers and let s . We denote by d s the arithmetic function given by d s ( n ) = ( d ( n ) ) s , where d ( n ) is the number of positive divisors of n . Moreover, for every , m we denote by δ s , , m ( n ) the sequence d s ( d s ( ... d s ( d s ( n ) + ) + ... ) + ) m -times = d s ( n ) for m = 1 , d s ( d s ( n ) + ) for m = 2 , d s ( d s ( d s ( n ) + ) + ) for m = 3 , We present classical and nonclassical notes on the sequence ( δ s , , m ( n ) ) m 1 , where , n , s are understood as parameters.

Coleff-Herrera currents, duality, and noetherian operators

Mats Andersson (2011)

Bulletin de la Société Mathématique de France

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Let be a coherent subsheaf of a locally free sheaf 𝒪 ( E 0 ) and suppose that = 𝒪 ( E 0 ) / has pure codimension. Starting with a residue current R obtained from a locally free resolution of we construct a vector-valued Coleff-Herrera current μ with support on the variety associated to such that φ is in if and only if μ φ = 0 . Such a current μ can also be derived algebraically from a fundamental theorem of Roos about the bidualizing functor, and the relation between these two approaches is discussed....

On the birational gonalities of smooth curves

E. Ballico (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let C be a smooth curve of genus g . For each positive integer r the birational r -gonality s r ( C ) of C is the minimal integer t such that there is L Pic t ( C ) with h 0 ( C , L ) = r + 1 . Fix an integer r 3 . In this paper we prove the existence of an integer g r such that for every integer g g r there is a smooth curve C of genus g with s r + 1 ( C ) / ( r + 1 ) > s r ( C ) / r , i.e. in the sequence of all birational gonalities of C at least one of the slope inequalities fails.

Traceability in { K 1 , 4 , K 1 , 4 + e } -free graphs

Wei Zheng, Ligong Wang (2019)

Czechoslovak Mathematical Journal

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A graph G is called { H 1 , H 2 , , H k } -free if G contains no induced subgraph isomorphic to any graph H i , 1 i k . We define σ k = min i = 1 k d ( v i ) : { v 1 , , v k } is an independent set of vertices in G . In this paper, we prove that (1) if G is a connected { K 1 , 4 , K 1 , 4 + e } -free graph of order n and σ 3 ( G ) n - 1 , then G is traceable, (2) if G is a 2-connected { K 1 , 4 , K 1 , 4 + e } -free graph of order n and | N ( x 1 ) N ( x 2 ) | + | N ( y 1 ) N ( y 2 ) | n - 1 for any two distinct pairs of non-adjacent vertices { x 1 , x 2 } , { y 1 , y 2 } of G , then G is traceable, i.e., G has a Hamilton path, where K 1 , 4 + e is a graph obtained by joining a pair of non-adjacent vertices in a K 1 , 4 .

Explicit birational geometry of threefolds of general type, I

Jungkai A. Chen, Meng Chen (2010)

Annales scientifiques de l'École Normale Supérieure

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Let V be a complex nonsingular projective 3-fold of general type. We prove P 12 ( V ) : = dim H 0 ( V , 12 K V ) &gt; 0 and P m 0 ( V ) &gt; 1 for some positive integer m 0 24 . A direct consequence is the birationality of the pluricanonical map ϕ m for all m 126 . Besides, the canonical volume Vol ( V ) has a universal lower bound ν ( 3 ) 1 63 · 126 2 .