Periodic behavior of meromorphic functions sharing values with their difference operators in several complex variables

We lay the foundations of Fatou theory in one and several complex variables. We describe the main contributions contained in E. M. Stein’s book Boundary Behavior of Holomorphic Functions, published in 1972 and still a source of inspiration. We also give an account of his contributions to the study of the boundary behavior of harmonic functions. The point of this paper is not simply to exposit well-known ideas. Rather, we completely reorganize the subject in order to bring out the profound contributions of E. M. Stein to the study of the boundary behavior both of holomorphic and harmonic functions in one and several variables. In an appendix, we provide a self-contained proof of a new result which is relevant to the differentiation of integrals, a topic which, as witnessed in Stein’s work, and especially by the aforementioned book, has deep connections with the boundary behavior of harmonic and holomorphic functions.

1. Some problems arising in the study of value distribution of functions of two complex variables. One of the objectives of modern analysis consists in the generalization of methods of the theory of functions of one complex variable in such a way that the procedures in the revised form can be applied in other fields, in particular, in the theory of functions of several complex variables, in the theory of partial differential equations, in differential geometry, etc. In this way one can hope to obtain in time a unified theory of various chapters of analysis. The method of the kernel function is one of the tools of this kind. In particular, this method permits us to develop some chapters of the theory of analytic and meromorphic functions f(z1, * * *, zn) of the class j2(I2n), various chapters in the theory of pseudo-conformal transformations (i.e., of transformations of the domains V2n by n analytic functions of n complex variables) etc. On the other hand, it is of considerable interest to generalize other chapters of the theory of functions of one variable, at first to the case of several complex variables. In particular, the study of value distribution of entire and meromorphic functions represents a topic of great interest. Generalizing the classical results about the zeros of a polynomial, Hadamard and Borel established a connection between the value distribution of a function and its growth. A further step of basic importance has been made by Nevanlinna and Ahlfors, who showed not only that the results of Hadamard and Borel in a sharper form can be obtained by using potential-theoretical and topological methods, but found in this way important new relations, and opened a new field in the modern theory of functions. As one passes from the theory of functions of one variable to the case of two and more variables(2), the question of the value distribution becomes more complex and many directions arise which can be considered as a generalization of the above mentioned chapter of the theory of functions of one variable. However, the formulation of the problems which can be answered, and generalization of methods used in one variable, is not immediate. A function of two complex variables may vanish in a domain on a segment

Previous article Next article A Bound for the Derivative of Positive Real FunctionsF. M. RezaF. M. Rezahttps://doi.org/10.1137/1004005PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] F. M. Reza, A multiplication theorem for positive real functions, Proc. Amer. Math. Soc., 9 (1958), 496–499 MR0096504 0089.28803 CrossrefGoogle Scholar[2A] C. Caratheodory, Theory of functions of a complex variable. Vol. 1, Chelsea Publishing Co., New York, N. Y., 1954xiii+301 MR0060009 0055.30301 Google Scholar[2B] C. Caratheodory, Theory of functions of a complex variable. Vol. 2, Chelsea Publishing Company, New York, 1954, 220– MR0064861 0056.06703 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Applications of the Carathéodory’s Inequality for Driving Point Impedance Functions2 January 2022 | European Journal of Science and Technology Cross Ref Circuit Applications of Schwarz-Pick LemmaIEEE Transactions on Circuits and Systems II: Express Briefs, Vol. 69, No. 1 Cross Ref Some results on the behaviour of transfer functions at the right half planeEvolution Equations & Control Theory, Vol. 11, No. 1 Cross Ref Schwarz lemma for driving point impedance functions and its circuit applications19 February 2019 | International Journal of Circuit Theory and Applications, Vol. 47, No. 6 Cross Ref On boundary analysis for derivative of driving point impedance functions and its circuit applications8 November 2018 | IET Circuits, Devices & Systems, Vol. 13, No. 2 Cross Ref DÜMF Mühendislik Dergisi, Vol. 10, No. 2 Cross Ref Boundary Analysis for the Derivative of Driving Point Impedance FunctionsIEEE Transactions on Circuits and Systems II: Express Briefs, Vol. 65, No. 9 Cross Ref Sharp inequalities for positive functions19 October 2007 | Complex Variables and Elliptic Equations, Vol. 52, No. 2-3 Cross Ref Schwarz's Lemma for n-PortsJournal of the Franklin Institute, Vol. 317, No. 2 Cross Ref Restrictions of the derivatives of positive real functionsJournal of the Franklin Institute, Vol. 312, No. 5 Cross Ref Particular Inequalities Cross Ref Positive real derivatives of driving point functionsJournal of the Franklin Institute, Vol. 287, No. 1 Cross Ref Bounds on the Derivatives of Positive FunctionsJ. L. Goldberg18 July 2006 | SIAM Review, Vol. 8, No. 3AbstractPDF (213 KB)Nonsymmetric Lattice Sections IIEEE Transactions on Circuit Theory, Vol. 12, No. 4 Cross Ref A Note on the Second Derivative of a P. R. FunctionCarl E. Pearson18 July 2006 | SIAM Review, Vol. 7, No. 2PDF (94 KB) Volume 4, Issue 1| 1962SIAM Review History Submitted:07 July 1961Published online:18 July 2006 InformationCopyright © 1962 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1004005Article page range:pp. 40-42ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics

We develop a new technique for studying the boundary limiting behavior of a holomorphic function on a domain $\Omega$ -- both in one and several complex variables. The approach involves two new localized maximal functions. As a result of this methodology, theorems of Calderon type about local boundary behavior on a set of positive measure may be proved in a new and more natural way. We also study the question of nontangential boundedness (on a set of positive measure) versus admissible boundedness. Under suitable hypotheses, these two conditions are shown to be equivalent.

We give a new admissible maximal inequality for analytic functions of several complex variables, and also show that an analytic function which is admissibly bounded at almost all boundary points of a domain D must have an admissible limit a.e.These results are then applied to the boundary behaviour of the Nevanlinna class of D.We shall be concerned with the boundary behaviour of analytic functions in a smooth bounded domain in C. For background and motivation, the reader is referred to Chapter 1 of the book by Stein [3].The aim of this paper is to present two new results, as a corollary of which we shall be able to give very quick and simple proofs of the results in [3, Chapter 2].The author would very much like to express thanks to his adviser Dr. D. A. Edwards for encouragement, to Professor E. M. Stein for pointing out some ambiguity in the proof of Theorem 2 in the original manuscript, and to Mrs. V. Shires for performing the typing.Recall the definition [3, p. 32] of the admissible approach regions &a () at G3Z): &a(0 ={ZED: |(Z -Qv(\ < (1 + a)8e(Z), |Z -||2 < arS(Z)}, (Z) = min(dist of Z from dD, dist of Z from tangent plane at ), Vf = outward drawn normal at .The regions $"() have conical sections in the direction of the "classical tangent" {/uf}, but "tangential parabolic" sections in the 2n -2 complementary real directions (which are n -1 complex directions, of course).Recall also the two types of ball on dD [3, p. 33].The balls Bx are the usual ones of radius p centered at G dD, and the balls B2 are of radius ~p in the classical direction, but of radius ~pl//2 in the complex directions.We consider also the corresponding maximal functions, denoted * and *.Theorem 1.Let u be nonnegative, continuous, andplurisubharmonic in D (we do not require u to be continuous in D), and suppose u has a harmonic majorant, i.e. there is a finite positive measure p on dD so that u(Z) <

The behavior of analytic functions of several complex variables with respect to analytic continuation along certain sets in Cn is studied.Bibliography: 4 titles.

ABSTRACTWe define an analogue of the Baernstein star function for a meromorphic function f in several complex variables. This function is subharmonic on the upper half-plane and encodes some of the main functionals attached to f. We then characterize meromorphic functions admitting a harmonic star function.

Beginning with Valiron [262], the value distribution theory of algebroid functions of one complex variable has been studied extensively. A few contributions are listed under references. The Nevalinna theory of meromorphic functions on a complex manifold M or meromorphic mappings from M into a projective variety has been studied (see Griffith-King [75], Stoll [239], Vitter [265]). Here we will concentrate on the value distribution theory and its application of algebroid functions in several complex variables.

In this paper, we mainly investigate the uniqueness problem on meromorphic functions in $\mathbb{C}^m$ sharing small functions with their difference operators or shifts, and we obtain some interesting results that act as some extensions of previous results from one complex variable to several complex variables.

In this paper, by using the normality criteria forK-quasimeromorphic mapping of several complex variables, we get a normality criteria for families of holomorphic functions and of meromorphic functions family.

In this paper, we study $q$-difference analogues of several central results in value distribution theory of several complex variables such as $q$-difference versions of the logarithmic derivative lemma, the second main theorem for hyperplanes and hypersurfaces, and a Picard type theorem. Moreover, the Tumura-Clunie theorem concerning partial $q$-difference polynomials is also obtained. Finally, we apply this theory to investigate the growth of meromorphic solutions of linear partial $q$-difference equations.

The aim of this paper is to deal with the uniqueness problem on meromorphic functions in $ \mathbb{C}^{m} $ sharing small functions with their difference polynomial, and the results obtained can be seen as some extensions of previous results from one complex variable to several complex variables.

We use transfinite induction to show that if L is an epimorphism of a vector space V and maps a vector subspace W of V into a proper subspace of itself, then there is a smallest subspace E of V containing W such that L(E) E (or a minimal space of surjectivity or solvability) and we give examples where there are infinitely many distinct minimal spaces of solvability. We produce an example showing that if L 1 and L 2 are two epimorphisms of a vector space V which are endomorphisms of a proper subspace W of V such that LI(W) n L2(W is a proper subspace of W, then there may not exist a smallest subspace E of V containing W such that LI(E) E L2(E). While no nonconstant linear partial differential operator maps the field of meromorphic functions onto itself, we construct a locally convex topological vector space of formal power series containing the meromorphic functions such that every linear partial differential operator with constant coefficients maps this space linearly and continuously onto itself. Furthermore, we show that algebraically there is for every linear partial differential operator P(D) with constant coefficients a smallest extension E of the meromorphic functions in n complex variables, where

Preface Geometric value distribution theory Barsegian, G.: A new program of investigations in analysis: Gamma-lines approaches Sukiasyan, G.: On level sets of quasiconforrnal mappings Classical value distribution theory Alonso, A., Fernandez, A. and Perez, J.: On the unintegrated Nevanlinna fundamental inequality for meromorphic functions of slow growth Barsegian, G. and Yang, C.-C.: On some new concept of exceptional values Ciechanowicz, E. and Marchenko, I.: Maximum modulus points, deviations and spreads of meromorphic functions Craven, T. and Csordas, G.: Composition theorems, multiplier sequences and complex zero decreasing sequences Korhonen, R.: Nevanlinna theory in an annulus Marchenko, I. and Nikolenko, I.: On strong asymptotic tracts of functions holomorphic in a disk Complex differential and functional equations Barsegian, G., Sarkisian, A. and Yang, C.-C.: A new trend in complex differential equations: quasimeromorphic solutions Ha, H.K. and Yang, C.-C.: On the functional equation P(f) = Q(g) He, Y.: Value distribution of the higher order analogues of the first Painleve equation Yang, C.-C. and Li, P.: Some further results on the functional equation P(f) = Q(g) Several variables theory Aihara, Y.: Recent topics in uniqueness problem for meromorphic mappings Berenstein, C. and Li, B.Q.: On interpolation problems in Cn Hu, P.-C. and Yang, C.-C. : Jet bundles and its applications in value distribution of holomorphic mappings Tu, Z.-H.: Normal families of meromorphic mappings of several complex variables into the complex projective space

In this work it is shown the dynamical behavior of driven complex variable oscillators on Lyapunov diagrams, through the computation of the Lyapunov exponents spectrum. Here, we consider the largest and the second largest exponents on Lyapunov diagrams to identify regions of chaotic, quasi-periodic and periodic behaviors. In these diagrams we show the coexistence of limit cycles, chaotic attractors and 2-tori. Periodic and quasi-periodic structures embedded in chaotic regions are also observed on the Lyapunov diagrams.