One obvious way to give a meromorphic function on c is to write down a rational function, the quotient of two polynomials. Although the principle is false in general see, many authors proved normality criterion for families of meromorphic functions corresponding to liouvillepicardtypetheoremsee. Then, there exists a sequence of disks of the form where. On the uniqueness of meromorphic functions sharing three. Tr,f is an increasing convex function of log r for any meromorphic function fz. In the mathematical field of complex analysis, a meromorphic function on an open subset d of the complex plane is a function that is holomorphic on all of d except for a set of isolated points, which are poles of the function. The term comes from the ancient greek meros, meaning part. Value distribution of meromorphic functions and their. The present paper is thus concerned with attempting to improve the conclusions of theorem a to give a lower estimate for the number of zeros, some results in this. Hayman, meromorphic functions, clarendon press, oxford. In 1959, hayman 3 proved the following seminal result, which has come to be known as hayman s alternative. T and hayman tpoints of meromorphic functions for small.
Meromorphic functions paperback january 1, 1964 by w. The major result concerning the order of growth of meromorphic solutions of. On the other hand, if f is a nonconstant rational function such that fz 6 0 for z 2 c. Goldberg, the sum of defects of a meromorphic function and its. On the meromorphic solutions of an equation of hayman. We will confirm the existence of hayman t directions in some angular domains. Corrigendum to filling disks of hayman type of meromorphic functions nanwu 1 andzuxingxuan 2 department of mathematics, school of science, china university of mining and technology, beijing,china beijing key laboratory of information service engineering. On meromorphic functions that share one small function of differential polynomials with their derivatives.
The yearbook of the renowned italian scientific institution, including an historical sketch of its history, the list of all past and present members as well as a wealth of information about its academic and scientific activities. Padic analytic functions describes the definition and properties of padic analytic and meromorphic functions in a complete algebraically closed ultrametric field various properties of padic exponentialpolynomials are examined, such as the hermitelindemann theorem in a padic field, with a new proof. We consider whether a family meromorphic functions f is normal in d, if for every pair of functions f and g in f, f n f and g n g share a nonzero value a. Pdf all admissible meromorphic solutions of haymans. On the counting function for the avalues of a meromorphic. On the product of a meromorphic function and its derivative. We find all nonrational meromorphic solutions of the equation ww. Let fbe a meromorphic function, where here and henceforth meromorphic should be taken to mean meromorphic on the whole complex plane, unless explicitly stated otherwise. Uniqueness of meromorphic functions sharing small functions. We prove that, under the condition, there exists a sequence of filling disks of hayman type, and these filling disks can determine a hayman direction. Oct 27, 2011 in the article, we study the normality of families of meromorphic functions concerning shared values. Finally, the p adic nevanlinna theory is widely explained, with various applications. Apart from special choices of the coefficient functions, the general solution is not meromorphic and contains movable branch points.
The first author is supported in part by the grant nos. In the article titled filling disks of hayman type of meromorphic functions, the following statement was missing from the acknowledgements section. Then the only meromorphic functions fwith in nitely many poles, satisfying flf 6 0, are of the form fh0. Introduction and results for a meromorphicfunction, disks of the form. The derivative of a transcendental meromorphic function all of whose zeros are multiple assumes every nonzero complex value infinitely often. Uniqueness on entire functions and their nth order exact differences. Uniqueness of meromorphic functions of differential polynomials sharing two values im.
The order and type of growth for analytic functions are studied, in the whole field and. Starlikeness properties of a new integral operator for meromorphic functions mohammed, aabed and darus, maslina, journal of applied mathematics, 2011. Let be a meromorphic function in the plane of order 0 meromorphic function in the plane and let arg be a borel direction of order. Hayman, meromorphic functions, clarendon press, oxford, 1964. The term comes from the ancient greek meros, meaning part every meromorphic function on d can be expressed as the ratio between two holomorphic functions with. Research article filling disks of hayman type of meromorphic. Walter kurt hayman frs 6 january 1926 1 january 2020 was a british mathematician known for contributions to complex analysis. A complete bibliography provides the reader with references on the work to date on each problem. On meromorphic functions that share one small function of differential polynomials with their derivatives authors. A function f meromorphic in d is called a normal function 4, in the sense of lehto and virtanen, if there exist a constant mf such that 1. If fz is a nonconstant meromorphic function in izi meromorphic functions. We shall develop in this course nevanlinnas theory of meromorphic functions. This book contains over 500 research problems in complex analysis, known as hayman s list.
Topics covered include meromorphic, entire, subharmonic, harmonic, univalent and multivalent functions. Thereafter, the value distribution of derivatives of transcendental functions continued to be studied. Nov 06, 2014 in so doing we answer a question of hayman by showing that all such solutions have finite order. Then fis a rational function if and only if it has at worse a pole at in nity. It follows easily from the ahlforsshimizu characteristic function that if f is a julia exceptional function, then tr, f. Please see wikipedias template documentation for further citation fields that may be required. Concerning the uniqueness and sharing values of meromorphic functions, many results about. On the zeros of the differential polynomial z mdpi. May 12, 2009 this paper is devoted to investigate the singular directions of mero morphic functions in some angular domains. Hayman t directions of meromorphic functions in some. It is shown that if f is a transcendental meromorphic function of nite order and if c is a polynomial which does not vanish identically, then f0f c has in nitely many zeros. Introduction let d denote the unit disk in the complex plane c. Meromorphic continuations of local zeta functions and their applications to oscillating integrals okada, toshihisa and takeuchi, kiyoshi, tohoku mathematical journal, 20.
We have defined the nevalinna characteristic function of a meromorphic function fz. For example, we show that if g is a real transcendental meromorphic function, c. The function tr,f is called the characteristic function of fz. Hayman, meromorphic functions, clarendon press, oxford, 19. Kotus, lu yinian skip to main content accessibility help we use cookies to distinguish you from other users and to provide you with a better experience on our websites. Value distribution of the kth derivatives of meromorphic. In place of 2 the author introduces two characteristics, t. Intuitively, a meromorphic function is a ratio of two wellbehaved holomorphic functions. Hayman, the strength of cartans version of nevanlinna theory. Let be a meromorphic function in the plane and let be a borel direction of order.
On meromorphic functions with growth conditions springerlink. The zeros of exponent convergence of meromorphic function fis said by. Filling disks of hayman type of meromorphic functions core. If for each pair of functions f and g inf, ff and gg share a nonzero value a in d, then f is normal in d. Corrigendum to filling disks of hayman type of meromorphic functions nanwu 1 andzuxingxuan 2 department of mathematics, school of science, china university of mining and technology, beijing,china beijing key laboratory of information service engineering, department ofgeneral education,beijing unionuniversity. The structure of the proof of haymans conjecture in 2, 3, 20 is as follows. If a nonconstant meromorphic function f and its derivative. As you have access to this article, a pdf of this content is available in through the.
May 15, 2003 research has concentrated on the value distribution of meromorphic solutions and their rates of growth. Meromorphic functions sharing a small function wang, songmin and gao, zongsheng, abstract and applied analysis, 2007 common hypercyclic vectors for certain families of differential operators tsirivas, n. On the derivative of meromorphic functions with multiple zeros. We prove the second fundamental theorem for the ahlforssh.
In so doing, we answer a question of hayman by showing that all such solutions have finite order. This theory has proved a tool of unparallelled precision for the study of the roots of equations fz a, f1z b, etc. Let be a meromorphic function in the plane of order. We obtain the existence of the filling disks with respect to hayman directions. All admissible meromorphic solutions of haymans equation. Multiplicities in haymans alternative journal of the. Filling disks of hayman type of meromorphic functions. Pdf research questions on meromorphic functions and complex. Some normality criteria of functions related a hayman.
Theorem c, a celebrated result of gu 6, is related to theorem b via blochs principle. A note on haymans conjecture international journal of. Value distribution and uniqueness results of zeroorder. In 1959 hayman proved an inequality from which it follows that if f is transcendental and meromorphic in the plane then either f takes every finite complex value infinitely often or each derivative f k, k. On the characteristic of functions meromorphic in the unit disk and of their integrals. Suppose that fz is a transcendental integral function in the finite plane, then there exists a direction h.
This gives the maximum principle and applications to the behavior of the spectral radius. Hayman t directions of meromorphic functions in some angular. Two examples show that the conditions in our results are best possible in a sense. If fand gare meromorphic functions satisfying ffz,gz,z. Pdf we study normality for families of meromorphic functions which is related to an extended version of a hayman s conjecture on value distribution.
Haymans alternative and normal families of nonvanishing. Jul 01, 2012 in this article, we consider the singular points of meromorphic functions in the unit disk. Hayman, meromorphic function, clarendon press, oxford. Uniqueness, meromorphic function, differential polynomial. Research open access some normality criteria of functions. Such a function will still be wellbehaved, except possibly at the points where the denominator of the fraction is zero. Pdf all admissible meromorphic solutions of haymans equation. Let f be a transcendental meromorphic function on the complex plane c. Picardhayman behavior of derivatives of meromorphic functions. Every meromorphic function of positive and nite order has a sequence of lling disks of hayman type, which can also determine a hayman direction of order. Padic meromorphic functions are studied, not only on the whole field but also in an open disk and on the complemental of an open disk, using motzkin meromorphic products. An english translation of this book is to be published soon by the ams. Langley abstract this is an appendix to the english translation of the book by a. Heuristially a meromorphic solution is admissible if it is more complicated than the coefficients that appear in the equation.
Article views269 pdf downloads38 cited by0 download pdf. Lang, introduction to complex hyperbolic spaces, springerverlag, new york, 1987. It is natural to wonder to what extent these give all meromorphic functions on c. Research problems in function theory fiftieth anniversary. Research article normal family of meromorphic functions. Remarkable is that the functions f and log f are subharmonic when f is a banachspacevalued analytic function.
Corrigendum to filling disks of hayman type of meromorphic. Hayman directions of meromorphic functions springerlink. The purpose of the present paper is to show that a thorough search will yield a list of all meromorphic solutions of a multiparameter ordinary differential equation introduced by hayman. For the standard notation and terminology of nevanlinna theory, see 10,19. Value distribution of meromorphic functions and their derivatives. Corrigendum corrigendum to filling disks of hayman type of. Working with tropical meromorphic functions of one variable tsai, yenlung, taiwanese journal of mathematics, 2012. Meromorphic function, normal family, normal function, uniformly normal family.
Hayman author see all formats and editions hide other formats and editions. This follows from an argument which can be found in titchmarsh 6, p. Every meromorphic function of positive and finite order has a sequence of filling disks of hayman type, which can also determine a hayman direction of order. We investigate the extent to which these values may be ramified, and we establish. This paper is devoted to investigate the singular directions of mero morphic functions in some angular domains. Ostrovskii, distribution of values of meromorphic functions, moscow, nauka, 1970. Then, a half line is a order borel direction if and only if it satisfies. On the counting function for the avalues of a meromorphic function by joseph miles1 introduction. Nov 21, 2014 in this paper, we will find all admissible meromorphic solutions of the differential equation.
In this paper, we will give suitable conditions on differential polynomials q f such that they take every finite nonzero value infinitely often, where f is a meromorphic function in complex plane. The characteristic function tr,f is continuous in r for every meromorphic function fz. The impetus for the twovolume set subharmonic functions arosefromplansofwalters. Let f be a transcendental meromorphic function in the complex plane c.
Sorry, we are unable to provide the full text but you may find it at the following locations. We say two meromorphic functions fz and gz share the finite complex value. Value distribution of padic meromorphic functions boussaf, kamal and ojeda, jacqueline, bulletin of the. In this paper, we shall prove the existence of the singular directions related to hayman s problems 1.
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