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Values shared by an entire function and its derivative
Abstract
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... Rubel and Yang [4] studied the uniqueness problem entire functions that share values with their derivatives. In 1977, they proved if an entire function f shares two finite distinct values CM with f , then f ≡ f . ...
... In 1977, they proved if an entire function f shares two finite distinct values CM with f , then f ≡ f . In 1992, Zheng and Yang [5] further improved the result of Rubel and Yang [4] and they proved if an entire function f shares a, b CM with f (k) (k ≥ 1), then f ≡ f (k) , where a, b ∈ S(f ) are distinct. Again in 1999, Li and Yang [6] improved the result of Zheng and Yang [5] from sharing value b CM to IM and they proved Theorem A [5]. ...
... Rubel and Yang [11] considered the uniqueness of an entire function and its derivative. In 1977, they proved the following result. ...
... Theorem 2C [11]. Let f be a nonconstant entire function and a, b ∈ C such that b = a. ...
... a differential polynomial generated by f of degree ν p = max{ν M j : 1 ≤ j ≤ l} and weight L. Rubel and C. C. Yang [9] first studied the problem of sharing values between entire functions and their derivatives. They proved the following theorem. ...
... Theorem A ( [9]). Let f ∈ E and a ̸ = b ∈ C. If f and f ′ share a and b CM, then f ≡ f ′ . ...
The primary goal of this work is to determine whether the results from [19, 20] still hold true when a differential polynomial is considered in place of a differential monomial. In this perspective, we continue our study to establish the uniqueness theorem for homogeneous differential polynomial of an entire and its higher order derivative sharing two polynomials using normal family theory as well as to obtain normality criteria for a family of analytic functions in a domain concerning homogeneous differential polynomial of a transcendental meromorphic function satisfying certain conditions. Meanwhile, as a result of this investigation, we proved three theorems that provide affirmative responses for the purpose of this study. Several examples are offered to demonstrate that the conditions of the theorem are necessary.
... Rubel and Yang [11] considered the uniqueness of an entire function and its derivative. In 1977, they proved the following result. ...
... Theorem C [11]. Let f (z) be a nonconstant entire function and a, b \in \BbbC such that b \not = a. ...
UDC 517.5 We discuss the problem of uniqueness of a meromorphic function f ( z ) , which shares a 1 ( z ) , a 2 ( z ) , and a 3 ( z ) CM with its shift f ( z + c ) , where a 1 ( z ) , a 2 ( z ) , and a 3 ( z ) are three c -periodic distinct small functions of f ( z ) and c ∈ ℂ ∖ { 0 } . The obtained result improves the recent result of Heittokangas et al. [Complex Var. and Elliptic Equat., 56 , No. 1–4, 81–92 (2011)] by dropping the assumption about the order of f ( z ) . In addition, we introduce a way of characterizing elliptic functions in terms of meromorphic functions sharing values with two of their shifts. Moreover, we show by a number of illustrating examples that our results are, in certain senses, best possible.
... We write f and g share (a, k) to mean that f, g share the value a with weight k. Rubel and Yang Chung-Chun [4] considered the uniqueness of an entire function and its derivative. They proved the following. ...
... Theorem 4 [4]. Let f be a nonconstant entire function. ...
The purpose of this paper is to study the uniqueness problems of certain type of differential-difference polynomials generated by two meromorphic functions. We obtain some results which extend and generalize some recent results due to Husna (J Anal 29(4):1191–1206, 2021).
... We write f and g share (a, k) to mean that f, g share the value a with weight k. Rubel and Yang Chung-Chun [4] considered the uniqueness of an entire function and its derivative. They proved the following. ...
... Theorem 4 [4]. Let f be a nonconstant entire function. ...
In this paper we study the uniqueness of entire functions concerning their difference operator and derivatives. The idea of entire and meromorphic functions relies heavily on this direction. Rubel and Yang considered the uniqueness of entire function and its derivative and proved that if f(z) and share two values a,b counting multilicities then . Later, Li Ping and Yang improved the result given by Rubel and Yang and proved that if f(z) is a non-constant entire function and a,b are two finite distinct complex values and if f(z) and share a counting multiplicities and b ignoring multiplicities then . In recent years, the value distribution of meromorphic functions of finite order with respect to difference analogue has become a subject of interest. By replacing finite distinct complex values by polynomials, we prove the following result: Let be trancendental entire functions of finite order, be integer and and be two polynomials. If and share CM and share IM, then . A non-trivial proof of this result uses Nevanlinna's value distribution theory.
... R. Nevanlinna launched the value distribution theory in the early nineteenth century with his famous five value and four value theorems, which served as the foundation for the uniqueness theory. In [6], Rubel and Yang first investigated the uniqueness of non-constant entire function f and f ′ sharing two values. This investigation was very important as it first exhibited that in the uniqueness theory, the number of sharing values can be reduced from 5 to 2, for the special class of functions. ...
... They proved the following result. Theorem A [6]. Let f be a nonconstant entire function. ...
In this paper, we investigate the uniqueness of meromorphic functions considering two sharing values concerning differential-difference polynomial and its kth derivative. Accordingly, we have proved some existing results, which improves and generalizes the several earlier results due to A. Banerjee and S. Maity.
... Li and Qiao [16] proved that the five-value theorem remains valid if five values are replaced with five small functions. Rubel and Yang [22] considered the uniqueness problem that for an entire function f , if f shares two finite values CM with f ′ , then f = f ′ . The result was considered into the case of meromorphic functions by Mues and Steinmetz [19] and Gundersen [8]. ...
... al. [7]. As we mentioned above, a large number of research works on uniqueness problem have been studied in complex plane(see e.g., [3,5,7,8,11,16,18,22,25,26]). One may ask whether there exist some corresponding uniqueness results for meromorphic functions sharing values with their shifts or difference operators in the case of higher dimension? ...
The aim of this paper is to deal with the uniqueness problem on meromorphic functions in 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.
... (3) ν(r, F ) = O(log r) if ̺(f ) < ∞. Rubel and Yang (see [9]) considered the uniqueness of an entire function when it shares two values CM with its first derivative. In 1977, the authors proved the following well-known theorem. ...
... Theorem A( [9]). Let a, b ∈ C such that b = a and let f ∈ E (C). ...
... In 1929, Nevanlinna [25] proved the famous five-value theorem that if two non-constant meromorphic functions f and g share five distinct values I M (that is, ignoring multiplicities), then f (z) ≡ g(z). In 1977, Rubel and Yang [29] showed that if a non-constant entire function f and its first derivative f share two distinct values C M (that is, counting multiplicities), then they are identical. Mues and Steinmetz [24] and Gundersen [12] extended this result for meromorphic functions and their derivatives. ...
... By using the difference version of the second main theorem [14,Thm. 2.4], in 2009, Heittokangas-Korhonen-Laine-Rieppo-Zhang [17] considered the shift analogue of the result of Rubel and Yang [29]. They replaced f by the shift f (z + c) and obtained the following result. ...
In this paper, making use of the value distribution theory for meromorphic functions in several complex variables and its difference analogues, we mainly consider the uniqueness problem for meromorphic functions in several complex variables sharing values or small functions with their shifts or difference operators, and weaken the condition of growth of hyper-order of meromorphic functions, which generalize the corresponding results of one complex variable. © 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.
... To study the uniqueness of meromorphic functions with its derivatives is a very important problem in the uniqueness theory. Rubel and Yang [15], Mues and Steinmetz [16], and Gundersen [17] considered about it and obtained the following result. Frank and Weißenborn [18] and Frank and Ohlenroth [19] and Li and Yang [20] considered whether the aforementioned result is valid or not if ′ f is changed to f k ( ) and improved Theorem E as follows. ...
In this article, we study the unicity of meromorphic functions concerning small functions and derivatives-differences. The results obtained in this article extend and improve some results of Chen et al. [Uniqueness problems on difference operators of meromorphic functions] and Chen and Huang [Uniqueness of meromorphic functions concerning their derivatives and shifts with partially shared values].
... Indeed, the number of shared values can be reduced if f (z) and g(z) are related. For example, Rubel and Yang [13] showed that if a non-constant entire function f (z) and its first derivative f (z) share two distinct values CM, then they are identical. Mues and Steinmetz [11] and Gundersen [6] extended this result for meromorphic functions and their derivatives. ...
Brück’s conjecture asserts that if a non-constant entire function f(z) with hyper-order ρ2(f)∉N∪{∞} shares one finite value a CM (counting multiplicities) with its derivative, then f′-a=c(f-a), for some non-zero constant c. This conjecture has been affirmed for entire functions with finite order and hyper-order less than one. In this paper, we show that Brück’s conjecture is true for entire functions that satisfy second order differential equations with meromorphic coefficients of finite order.
... In the same way, we can define N L (r, 1 We denote by Q = max{Γ M j − d(M j ) : 1 ≤ j ≤ t} = max{n 1 j + 2n 2 j + ... + kn k j : 1 ≤ j ≤ t}. Thus, L. A. Rubel and C. C. Yang [18], showed that a derivative is worth two values. Since then the study of uniqueness of meromorphic functions sharing values with derivatives and recently with shifts, difference operator became a subject of much interest ( [15], [21], [22]). ...
In this paper, we investigate the uniqueness of meromorphic function with its shift and differential polynomial. The results in this paper generalize and improve the results due to C. Meng and G. Liu[14]. 2010 Mathematics subject classification: primary 30D35.
... Since then, research on uniqueness of meromorphic functions has been widely considered. Motivated by Hayman's remarkable alternative concerning the value distribution of f and f , and Nevanlinna's five-value theorem, Rubel and Yang [18] proved the following theorem. ...
In this paper, we give a complete characterization for meromorphic functions that share three distinct values CM, with their difference operator or shift f(z+c). This provides a difference analogue of the corresponding results of Rubel-Yang, Mues-Steinmetz, and Gundersen. In particular, we prove that if an entire function f and its difference derivative share three distinct values CM, then . And our results show that the conjecture posed by Chen and Yi in 2013 holds for entire functions, and does not hold for meromorphic functions. Compared with many previous papers, our method circumvents the obstacle of the difference logarithmic derivative lemma for meromorphic functions of infinite order, since this method does not depend on the growth of the functions, but requires the knowledge of linear algebra and combinatorics.
... We In 1977 L. A. Rubel and C. C. Yang [10] first considered the problem of value sharing by an entire function with its derivative. Inspired by their work a lot of researchers devoted themselves to explore such problems and extensions to different directions. ...
In connection to Br?ck conjecture we prove a uniqueness theorem for entire functions concerning homogeneous differential polynomials.
... Rubel and Yang [9] were the first to study entire functions that share values with their derivatives. In 1977, they proved if f A EðCÞ shares two finite distinct values CM with f 0 , then f 1 f 0 . ...
... Rubel and Yang [20] considered the uniqueness of a nonconstant entire function when it shares two values with its first derivative. Mues, Steinmetz [17] and Gundersen [12] improved the result to the case of meromorphic functions and obtained the following result. ...
The uniqueness problems of the j-th derivative of a meromorphic function f(z) and the k-th derivative of its shift f(z + c) are investigated in this paper, where j, k are integers with 0 ⩽ j < k. We show that when f (j) (z) and f (k) (z + c) share one IM value and two partially shared values CM, the uniqueness result remains valid under some additional hypotheses. With one CM value and two partially shared values CM, a uniqueness theorem about the j-th derivative of f(z) and the k-th derivative of its shift f(z + c) is also proved.
... The uniqueness theory of entire and meromorphic functions in the purview of sharing values (using value distribution theory of Nevanlinna [29]) has grown up to an extensive subfield of the value distribution theory. Interested readers are referred to the articles [9][10][11]13,21,23,28,30,31] and references therein. After the development of the difference analogue lemma of logarithmic derivatives, by Halburd and Korhonen [14] in 2006, and Chiang and Feng [6], in 2008, independently, the research findings dealing with the sharing value problems between the shifts f (z + c) or with the difference operators c f of meromorphic functions f , gets a new dimension in the literature of meromorphic functions. ...
The field of c-periodic meromorphic functions in is defined by and the c-shift linear difference polynomial of a meromorphic function f is defined by
where . It is easy to see that if , then , where is a higher difference operator of f. Let
In this paper, we study the value sharing problem between a meromorphic functions f and their linear difference polynomials and prove a result generalizing several existing results. In addition, we find the class completely which gives the positive answers to a conjecture and an open problem in this direction.
... and f (6) = α 6 + 15cα 5 + 65c 2 α 4 + 90c 3 α 3 + 31c 4 α 2 + c 5 α ( f − a) ...
In the paper we prove a uniqueness theorem that extends a result of Wang and Laine (Comput Method Funct Theory (CMFT) 8(2):327–338, https://doi.org/10.1007/BF03321691, 2008) and gives a general form of its kind.
... Further reduction in the number of shared values can only lead to uniqueness if something more is known about the functions. Rubel and Yang [14] proved that if a non-constant entire function f and its derivative f share two distinct finite values, taking multiplicities into account, then f ≡ f . Mues and Steinmetz [12] and Gundersen [6] extended this result for meromorphic functions. ...
... Rubel and Yang are the first to study the uniqueness problem when entire functions sharing values with one of their derivatives in [6]. Then Li and Yang [4] considered the case that they share values with two of their derivatives. ...
Suppose that F ≡ 0 is an entire function with hyper-order σ 2 ( F ) < 1 and m , n ( n > m ≥ 1 ) are two integers. If F , Δ η m F and Δ η m F share two values IM, then either F ≡ Δ η m F or Δ η m F ≡ Δ η m F . Moreover, if n = km for some integer k ≥ 2, then Δ η m F ≡ Δ η m F .
... Let f and g share (0,1), (∞, 0), (1, ∞). If Rubel and Yang [18] in 1977 initiated the study of entire functions sharing values with their derivatives instead of studying the problem of sharing value of two meromorphic functions f and g. ...
In this paper, we study the value distribution of finite order meromorphic, entire functions and their difference operators sharing CM and IM. Our results in this paper improve and generalizes the corresponding results from Dong-Mei Wei and Zhi-Gang Huang.
... There are many papers about meromorphic functions sharing some values with their derivatives, see e.g., [3,5,12,18,22,26]. For example, Brück [3] raised the following conjecture. ...
Suppose that f(z) is a meromorphic function with hyper order $\sigma_{2}(f)
... Rubel and Yang [17] first investigated the uniqueness of an entire function concerning its derivative, and proved the following result. ...
In this paper, we study the unicity of entire functions concerning their shifts and derivatives and prove: Let f be a non-constant entire function of hyper-order less than 1, let c be a non-zero finite value, and let a, b be two distinct finite values. If f′(z) and f(z+c)f(z+c) share a, b IM, then f′(z)≡f(z+c). This improves some results due to Qi and Yang (Comput Methods Funct Theory 20:159–178, 2020).
... Rubel and Yang [5] proved that if a non-constant entire function f and its derivative f 0 share two distinct finite complex numbers CM, then f ≡ f 0 . What will be the relation between f and f 0 , if an entire function f and its derivative f 0 share one finite complex number CM? Br€ uck [6] made a conjecture that if f is a non-constant entire function satisfying σ 2 ðf Þ < ∞, where σ 2 ðf Þ is not a positive integer and if f and f 0 share one finite complex number a CM, then f 0 − a ¼ cðf − aÞ for some finite complex number c ≠ 0. Br€ uck [6] himself proved the conjecture for a ¼ 0. Br€ uck also proved that the conjecture is true for a ≠ 0 provided that f satisfies the additional assumption N ðr; 1 f 0 Þ ¼ Sðr; f Þ and in this case the order restriction on f can be omitted. ...
Purpose
The purpose of this current paper is to deal with the study of non-constant entire solutions of some non-linear complex differential equations in connection to Brück conjecture, by using the theory of complex differential equation. The results generalize the results due to Pramanik et al.
Design/methodology/approach
39B32, 30D35.
Findings
In the current paper, we mainly study the Brück conjecture and the various works that confirm this conjecture. In our study we find that the conjecture can be generalized for differential monomials under some additional conditions and it generalizes some works related to the conjecture. Also we can take the complex number a in the conjecture to be a small function. More precisely, we obtain a result which can be restate in the following way: Let f be a non-constant entire function such that σ 2 ( f ) < ∞ , σ 2 ( f ) is not a positive integer and δ ( 0 , f ) > 0 . Let M [ f ] be a differential monomial of f of degree γ M and α ( z ) , β ( z ) ∈ S ( f ) be such that max { σ ( α ) , σ ( β ) } < σ ( f ) . If M [ f ] + β and f γ M − α share the value 0 CM, then M [ f ] + β f γ M − α = c , where c ≠ 0 is a constant.
Originality/value
This is an original work of the authors.
... To reduce the number of shared values, Rubel and Yang appear to be the first to consider the unity of the entire function sharing two values with its first derivative in [21]. They proved that, for a nonconstant entire function f , if f and f share values a, b CM, then f ≡ f . ...
This paper is to consider the unity results on entire functions sharing two values with their difference operators and to prove some results related to 4 CM theorem. The main result reads as follows: Let f(z) f ( z ) be a nonconstant entire function of finite order, and let a 1 , a 2 be two distinct finite complex constants. If f(z) f ( z ) and Δ η n f ( z ) share a 1 and a 2 “CM”, then f ( z ) ≡ Δ η n f ( z ) , and hence f(z) f ( z ) and Δ η n f ( z ) share a 1 and a 2 CM.
... Rubel and Yang [18] in 1977 initiated the study of entire functions sharing values with their derivatives instead of studying the problem of sharing value of two meromorphic functions f and g. ...
Let f and g be two nonconstant meromorphic functions. Shared value problems related to f and g are investigated in this paper. We give sufficient conditions in terms of weighted value sharing which imply that f is a linear transformation or inversion transformation of g. We also investigate the uniqueness problem of meromorphic functions with their difference operators and derivatives sharing some values.
... An active subject in the uniqueness theory is the investigation on the uniqueness of the meromorphic function sharing values with its derivatives, which was initiated by Rubel et al [8]. We first recall the following result by Jank et al [9]. ...
This paper delves into the exploration of value sharing in connection with a transcendental entire function and its differential-difference polynomials. Motivated by some recent results of Dong and Liu (Math Slovaca 67(3):691–700, 2017), we present here some results that can be viewed as the differential-difference analogue of the Brück Conjecture.
This paper addresses the uniqueness problem concerning the j-th derivative of a meromorphic function f(z) and the k-th derivative of its shift, f(z+c), where j,k are integers with In this regard, our work surpasses the achievements of [2], as we have improved upon the existing results and provided a more refined understanding of this specific aspect. We give some illustrative examples to enhance the realism of the obtained outcomes. Denote by E(a,f) the set of all zeros of where each zero with multiplicity m is counted m times. In the paper proved, in particular, the following statement:\\ Let f(z) be a non-constant meromorphic function of finite order, c be a non-zero finite complex number and j,k be integers such that If and have the same points for a finite value and satisfy conditions then (Theorem 6).
In this paper, we study a problem to determine the forms of a non-constant entire function f when it shares a set S of 4 distinct elements with its k-th derivatives. This work is motivated by the work of Chang et al. (Arch Math 89: 561–569, 2007) which pertains to a set of three elements. We have comprehensively extended the same result, as this type of extension has not been addressed earlier.
Suppose that f(z) is a nonconstant meromorphic function of hyper order strictly less than 1, c is a nonzero finite constant, a, b are distinct finite constants, and , are all integers. In this paper, we firstly prove one uniqueness theorem when f(z) and share CM and share b IM with additional conditions and give some further discussions. We also prove another uniqueness theorem when and f(z+c) share CM and share a, b IM. Our main theorems generalize and improve many recent known results.
The objective of this study is to determine whether the existence of shared sets S containing both meromorphic (entire) functions and their higher derivatives, as well as powers of meromorphic functions and their differential polynomials, could result in uniqueness. The main focus is on determining the precise solutions to various differential equations. This problem is being studied in a broader context, specifically in set sharing.
We investigate the uniqueness problems of meromorphic functions and their difference operators by using a new method. It is proved that if a non-constant meromorphic function f shares a non-zero constant and ∞ counting multiplicities with its difference operators Δcf(z) and , then . In particular, we give a difference analogue of a result of Jank–Mues–Volkmann. Our method has two distinct features: (i) It converts the relations between functions into the corresponding vectors. This makes it possible to deal with the uniqueness problem by linear algebra and combinatorics. (ii) It circumvents the obstacle of the difference logarithmic derivative lemma for meromorphic functions of infinite order, since this method does not depend on the growth of the functions. Furthermore, the idea in this paper can also be applied to the case for several variables.
This paper brings out some improvements as well as generalization results of a paper of Qi and Yang [17] [Comput. Methods Funct. Theory 20, 159–178 (2020)], which deals with the uniqueness results of f′(z) and f(z+c). To be more realistic about the obtained results, we exhibit some examples.
In the paper, we use the idea of normal family to investigate the uniqueness problem of entire functions that share two values with their kth derivatives and obtain a result which improves as well as generalizes the recent result due to Li and Yi (Arch Math (Basel) 87(1):52–59, 2006). Also our result solves an unsolved problem of Zhang and Yang (Comput Math Appl 60:2153–2160, 2010).
Let S be a subset of and for a non-constant meromorphic function f, we define , where each zero is counted according to its multiplicity. If , then we say that f and g share the set S CM. The main aim of this paper is to discuss the uniqueness of meromorphic functions sharing the set with their differential monomials, where and . Our key findings in this paper is the precise form of the solutions of certain differential equations obtained in the main result. A number of examples have been exhibited to validate certain claims of the main results. As a consequence, we prove a corollary of the main results which improved the corresponding results of Fang and Zalcman [J. Math. Anal. Appl. 280: 273–283, 2003], and Chang et al. [Arch. Math. 89: 561–569, 2007] in some sense.
This research investigates the uniqueness of meromorphic function f(z) and its order difference operator with two shared values counting multiplicities. It generalizes the results of Yeyang Jiang and Zongxuan Chen [5] and for the case , it admits a very special generalization. Moreover, we give an example to illustrate the validity of our main result.
In this paper, we study the uniqueness of entire function and its differential-difference operators. We prove the following result: let f be a transcendental entire function of finite order, let η be a non-zero complex number, n≥1,k≥0 two integers and let a and b be two distinct finite complex numbers. If f and (Δηnf)(k) share a CM and share b IM, then f≡(Δηnf)(k).
In this paper, we focus on a conjecture concerning uniqueness problem of meromorphic functions sharing three distinct polynomials with their difference operators, which is mentioned in Chen and Yi (Result Math v. 63, pp. 557–565, 2013), and prove that it is true for meromorphic functions of finite order. Also, a result of Zhang and Liao, obtained for entire functions (Sci China Math v. 57, pp. 2143–2152, 2014), we generalize to the case of meromorphic functions.
In this paper, we study the unicity of entire functions and their derivatives and obtain the following result: let f be a non-constant entire function, let a 1 , a 2 , b 1 , and b 2 be four small functions of f such that a 1 ≢ b 1 , a 2 ≢ b 2 , and none of them is identically equal to ∞ . If f and f ( k ) share ( a 1 , a 2 ) CM and share ( b 1 , b 2 ) IM, then ( a 2 − b 2 ) f − ( a 1 − b 1 ) f ( k ) ≡ a 2 b 1 − a 1 b 2 . This extends the result due to Li and Yang [Value sharing of an entire function and its derivatives, J. Math. Soc. Japan. 51 (1999), no. 7, 781–799].
In this article, we dealt with the solutions of the difference analogue of Fermat-type equation of the form and proved a result which generalizes a result of Han and L\"u [{J. Contm. Math. Anal. 2019}] and Ma. et. al. [{J. Func. Spaces, Volume 2020, Article ID 3205357}]. Furthermore, we have been able to explore the class of the function satisfying the Fermat-type difference equation. A considerable number of examples have been exhibited throughout the paper pertinent with the different issues. In the last section, we have characterized solutions of the Fermat-type difference equation .
In the paper we consider the uniqueness problem of an entire function f when it shares a doubleton with its derivative f(k), k≥1. Our result extends a result of Li and Yang (J Math Soc Japan 51(4):781–799, 1999).
Let f(z) be a non-constant meromorphic (entire) function of hyper-order strictly less than 1, n ≥ 3 (n ≥ 2) be an integer. It is shown that if fn(z) and fn(z + c) share a(≠ 0) ∈ ℂ and ∞ CM, then f(z) = t1f(z + c) or f(z) = t2f(z + 2c), where t1 and t2 satisfy , (i = 1, 2). Some examples are provided to show the sharpness of our results. In addition, we mainly obtain two uniqueness results of f(z) with their n-th order differences Δnf. For example, let f(z) be transcendental entire, and let a(≠ 0) ∈ ℂ. We show that, if f(z) and Δnf(≢ 0) share 0 CM and a IM, then f(z) = Δnf. And this research extends earlier results by Chen et al.
{Das Hauptergebnis der Arbeit ist folgender Satz: Im Einheitskreis sei eine unendliche Menge von Systemen von je p regulären Funktionen ohne Nullstellen gegeben, für die identisch erfüllt ist. Man kann eine Teilmenge auswählen, die eine der beiden folgenden Eigenschaften besitzt:par I. Die Nummern 1,dots,p zerfallen in zwei Sorten derart, daß (1) der Quotient zweier beliebiger Funktionen erster Art konvergiert, (2) der Quotient einer beliebigen Funktion zweiter Art durch eine Funktion erster Art gegen Null konvergiert, (3) als Folge von (2) der Quotient der Summe aller Funktionen erster Art durch eine beliebige derselben gegen Null konvergiert. Es gibt stets mindestens zwei Nummern erster Art. Die Nummern zweiter Art können fehlen.par II. Es gibt zwei Gruppen von Nummern, deren jede mindestens zwei Nummern umfaß t, dazu evtl. einige Nummern, die keiner der beiden Gruppen angehören. In jeder Gruppe kann man wieder zwei Nummernsorten unterscheiden. Für diese gelten wieder die Aussagen (1), (2), (3) wie unter I, nur daß (3) keine Folge von (2) mehr ist.par Besonders eingebend werden die Fälle und erörtert. Unter den unmittelbaren Anwendungen sei hervorgehoben das Studium von Familien von Funktionenpaaren, derart, daß alle Funktionen frei von Nullstellen sind, und daß die Summe eines jeden Paares den Wert Eins ausläß t; oder derart, daß die Summe gegen Null konvergiert. In diesem letzteren Falle z. B. kann man eine Teilfolge auswählen, in der entweder die beiden Funktionen f und g des Paares einzeln gegen Null streben, oder gegen konvergiert.par Weiter ergeben sich gewisse Verallgemeinerungen der Sätze von it Schottky und it Landau.par Es folgen Anwendungen auf den it Pólya-Nevanlinnaschen Unitätssatz der in der Umgebung von z=infty meromorphen Funktionen, wobei namentlich die Fälle von 2, 3, 4 Stellensorten diskutiert werden. Die Beweise stützen sich auf einen hier erstmalig bewiesenen Satz von it A. Bloch. Verf. gibt ihm die folgende präzise Fassung: In der z-Ebene seien n beliebige Stellen gegeben; h sei eine gegebene positive Zahl. Die Stellen z, für die gilt, gehören dem Innern von höchstens n Kreisen an, deren Radiensumme 2eh nicht übertrifft. Dieser Satz wird verschiedentlich verallgemeinert und auch auf einige funktionentheoretische Fragen angewendet. Hervorgehoben sei z. B. die folgende Verschärfung eines Satzes von it S. Mandelbrojt: Es sei f(z) in regulär und daselbst ; gegeben seien zwei beliebige positive Zahlen und gamma. Dann gehört dazu eine von f unabhängige positive Zahl A so, daß fraclog vert f(x)vertlog vert f(y)vert} leqq A für jedes Wertepaar aus gilt, ausgenommen die x-Stellen aus gewissen Kreisen der Radiensumme gamma.par Besprechung: P. Montel; Bulletin sc. Math. 53 (1929), 44-48.