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Value sharing of an entire function and its derivatives

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Ping LI
Ping LI
Ping LI
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Chung-Chun YANG
Chung-Chun YANG
Chung-Chun YANG
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Abstract

In this paper, when an entire function f and the linear combination of its derivatives L(f) with small functions as its coefficients share one value CM and another value IM is studied. We also resolved the question when an entire function f and its derivative f′f^{\prime} share two values CM jointly. Some of the results remain to be valid if f is meromorphic and satisfying N(r,f)=o(T(r,f)) as r→∞ r\rightarrow\infty and the values a, b are replaced by small functions of f(z) .

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... Then f and L( f ) share 1 IM and 2 CM but f L( f ). Example 1.2 [7]. Let f = 1 2 e z + 1 2 e −z and L( f ) = f (2) + f (1) . ...
... Although one IM shared value and one CM shared value cannot ensure the equality of an entire function with a linear differential polynomial generated by it, Li and Yang [7] exhibited two possibilities in the following theorem. ...
... Theorem E [7]. Let f be a nonconstant entire function and ...
AN ENTIRE FUNCTION SHARING TWO VALUES WITH ITS LINEAR DIFFERENTIAL POLYNOMIAL
Article
  • Oct 2017
We consider the uniqueness of an entire function and a linear differential polynomial generated by it. One of our results improves a result of Li and Yang [‘Value sharing of an entire function and its derivatives’, J. Math. Soc. Japan51 (4) (1999), 781–799].
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... It was shown in [4] that the result in Theorem B is not true if we remove the condition r(a,) > (n + 2)/(n + 3). It is conjectured, [10], that this number can be replaced by 1/2. ...
... In this paper, by further counting the zeros and poles of the auxiliary functions and using some of our earlier results (see [4]), we are able to improve Theorem C by proving the following main result: THEOREM 1.1. Suppose that f is a nonconstant meromorphic function satisfying N(r,f) = S(r, f), and g = L(f) is the linear differential polynomial defined in (1). ...
... [4]). Suppose that f is a nonconstant meromorphic function satisfying N(r,f) = S(r,f), and g = L(f) is the linear differential polynomial defined in(1). ...
Further results on meromorphic functions that share two values with their derivatives
Article
  • Feb 2005
  • J AUST MATH SOC
In this paper we continue our previous studies and derive all possible expressions for a meromorphic function and its differential polunomials when they share two finite distinct values a1, a2, CM (counting multiplicities) in majority.
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... Studying meromorphic functions in view of sharing of a set with its derivatives and nding a possible relationship between them is an interesting topic in the value distribution theory of Nevanlinna. For example, in case of sharing a set of two elements by entire functions f and their derivatives f ′ , in 1999, Li and Yang [17] proved ...
... In view of the above results of Li and Yang [17], and Li [16], it turns out that the following question is inevitable. Question 1.2. ...
On an open problem of value sharing and differential equations
Article
Full-text available
  • Aug 2024
  • P EDINBURGH MATH SOC
Let f be a non-constant meromorphic function. We define its linear differential polynomial Lk[f] L_k[f] by \begin{equation*} L_k[f]=\displaystyle b_{-1}+\sum_{j=0}^{k}b_jf^{(j)}, \text{where}\; b_j (j=0, 1, 2, \ldots, k) \; \text{are constants with}\; b_k\neq 0. \end{equation*} In this paper, we solve an open problem posed by Li [J. Math. Anal. Appl. {310} (2005) 412-423] in connection with the problem of sharing a set by entire functions f and their linear differential polynomials Lk[f] L_k[f] . Further, we study the Fermat-type functional equations of the form fn+gn=1 f^n+g^n=1 to find the meromorphic solutions (f,g) (f, g) which enable us to answer the question of Li completely. This settles the long-standing open problem of Li.
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... If meromorphic functions share a general set, it is not easy to determine these functions. In 1999, Li and Yang [4] deduced that if E f (S) � E f ′ (S) with S contain two distinct constants, then f must have special forms. Fang and Zalcman [5] used the theory of normal family to solve the above problem by proving that there exists a finite set S con- ...
... Recently years, Nevanlinna characteristic of f(z + c), the value distribution theory for difference analogue, Nevanlinna theory of the difference operator, and the difference analogue of the lemma on the logarithmic derivative had been built, see e.g., [1][2][3][4][6][7][8][9][10][11][12][13][14]. For meromorphic functions f(z), we define its shift by f c (z) � f(z + c) and its difference operators by Theorem 1. ...
Uniqueness Problems about Entire Functions with Their Difference Operator Sharing Sets
Article
Full-text available
  • Nov 2020
In this paper, we study the uniqueness questions of finite order transcendental entire functions and their difference operators sharing a set consisting of two distinct entire functions of finite smaller order. Our results in this paper improve the corresponding results from Liu (2009) and Li (2012). 1. Introduction and Main Results Before proceeding, we spare the reader for a moment and assume some familiarity with the basics of Nevanlinna theory of meromorphic functions in such as the first and second main theorems and the usual notations such as the characteristic function , the proximity function , and the counting function . denotes any quantity satisfying as , except possibly on a set of finite logarithmic measure not necessarily the same at each occurrence, see e.g., [1–3]. Let be a meromorphic functions on . Here, the order is defined byand the exponent of convergence of zeros is defined by For a given , we say that two meromorphic functions and share CM (counting multiplicities) when and have the same a-points. Let be a finite set of some entire functions and an entire function. Then, a set is defined as Assume that is another entire function. We say that and share a set , counting multiplicities (CM), provided that . The uniqueness theory of meromorphic functions sharing sets generalizes that on sharing values and generally is more difficult. If meromorphic functions share a general set, it is not easy to determine these functions. In 1999, Li and Yang [4] deduced that if with contain two distinct constants, then must have special forms. Fang and Zalcman [5] used the theory of normal family to solve the above problem by proving that there exists a finite set containing three distinct elements such that if , then . Recently years, Nevanlinna characteristic of , the value distribution theory for difference analogue, Nevanlinna theory of the difference operator, and the difference analogue of the lemma on the logarithmic derivative had been built, see e.g., [1–4, 6–14]. For meromorphic functions , we define its shift by and its difference operators by By Nevanlinna theory of the difference operator, a natural question to ask whether the derivative can be replaced by the difference operator in the above question ? In 2009, Liu [8] investigated the above question and proved the following result. Theorem 1. Let be a nonzero complex number and be a transcendental entire function with finite order. If and share CM, then for all . In 2012, from Theorem 1, considering the constant in set is replaced by the function, Li [9] proved the following. Theorem 2. If and are two distinct entire functions, then is a nonconstant entire function whose and such that and . If and share CM, then for all . After studying Theorem 2, we propose some questions as follows. Question 1: from Theorem 2, the condition seems more stronger. So, one may ask whether it can be weakened or moved? Question 2: what will happen if the shift be replaced by in Theorem 2? Fortunately, we have recently given a positive answer for Question 1 (see [14]). In this work, we also discuss the above problems and especially for Question 2. Finally, we derive the following results. Theorem 3. Suppose that are two distinct entire functions and is a nonconstant entire function of finite order with such that and . If and share CM, then must take one of the following conclusions:(1), where are two nonzero constants satisfying . Furthermore, .(2). Here, is an entire function and .Using the same method, we improve the above result from the shift to in above theorem and obtain the following result. Theorem 4. Suppose that are two distinct entire functions and is a nonconstant entire function of finite order with such that and . If and share CM, then , where is an entire function such that . 2. Some Lemmas We will introduce some lemmas for the proofs of our theorems in this section. Lemma 1. (see [15]). Let be a meromorphic function of finite order and let be two arbitrary complex numbers such that . Assume that is the order of , then for each , we have Lemma 2 (see [16]). Let be a function transcendental and meromorphic in the plane with order less than 1. Set . Then, there exists an -set such thatuniformly in for . Lemma 3. (see [3]). Suppose that are meromorphic functions and are entire functions satisfying the following conditions:(1).(2).(3)For , , , . Then, . 3. Proof of Theorems Proof of Theorem 1. Due to and share CM, so we setwhere is an entire function. And then it follows from (7) and max that is a polynomial. Using Hadamard Factorization Theorem, we assume that , where is an entire function and is a polynomial which satisfiedSo,Put the forms of and into (7) to yieldTake . We assume that . Then,By Lemma 3, if , , and are not constants, then , a contradiction. So, , , and , (where are three constants). We can get , , and . Hence, is a periodic function. We also know is a polynomial. So, we get (where and are two constants, and ). So, we obtain . Here, is an entire function and . If below, obviously, is a small function of . Rewrite (10) asNote that . Without loss of generality, we set . Suppose that is a zero of , but not a zero of . From (12), we may easily obtain that is a zero of or . We denote by the reduced counting function of those common zeros of and . Similarly, we also denote the reduced counting function of those common zeros of and . Then,which implies that either or . We distinguish the two cases as follows: Case 1: . We may assume that is a common zero of and . It is obvious that is a zero of . If , then a contradiction. Hence, It deduces By Lemma 1, for any , We also get , where is a fixed positive constant. If , using and the above estimates of , It easily gets a contradiction. So, , this means that is a nonzero constant . Then, (16) changes to Also, noting that . Then, by Lemma 2, we get that there exists an -set E, as and such that So, and , and this also means that is a periodic function. If is a nonconstant function, then , a contradiction. Therefore, is a constant. Noting that and is a nonconstant entire function. Then, . Thus, we may set , where are two nonzero constants. Using the assumption of Case 1, one has and as common zeros, which are not zeros of . Suppose that is a common zero of and and not a zero of . Then, is a zero of . Moreover, this implies that . Finally, we deduce , which is the desired result. Case 2: . Suppose which is a common zero of and . Then, it is obvious that is a zero of . If , then a contradiction. Hence, If , then , a contradiction. Thus, . We may set that is a zero of , but not a zero of . It follows from (12) that is a zero of or . We take by the reduced counting function of those common zeros of and . Similarly, we denote by the reduced counting function of those common zeros of and . We obtainIt implies that either or . If , likewise with Case 1, we deduce the desired result. Hence, we set that below. Similarly with Case 2, we also get thatCombining (22) with (24), we deduce thatNote that . Thus, . Again using (24), we have . We rewrite it asThen,By Lemma 3, if , , and are not constants, then ; this is a contradiction. So, , , and (where are three constants). We can get , , and . Hence, is a periodic function. We also know is a polynomial. So, we get (where and are two constants, and ). So, we obtain . Here, is an entire function and . Therefore, the proof of the main Theorem 3 is finished. Proof of Theorem 2. Note that and share CM. So, we also setwhere is an entire function. Furthermore, it deduces from (28) and max that is a polynomial. Using Hadamard Factorization Theorem, we assume that , where is an entire function and is a polynomial satisfyingThen,where are constants. Substituting the forms of and into (28) yieldsSet . Suppose that . Then,By Lemma 3, if , are not constants, then ; a contradiction. So, , , and (where are three constants). We can get , , and . Hence, is a periodic function. We also know is a polynomial. So, we get (where and are two constants, and ). So, we obtain . Here, is an entire function and . If below, obviously, is a small function of . Rewrite (31) asDue to , without loss of generality, we set . Assume that is a zero of , but not a zero of . It deduces from (33) that is a zero of or . We also take the reduced counting function of those common zeros of and . Likewise, we denote by the reduced counting function of those common zeros of and . Then,this implies that either or . We may distinguish the following two cases. Case 1: . We set a common zero of and . Then, it is obvious that is a zero of . If , then a contradiction. Hence, It leads to By Lemma 3, if , , are not constants, then , a contradiction. So, , , (where are three constants). We can get , , . Here, is a periodic function. We also know is a polynomial. So, we get (where and are two constants, and ). Finally, we get , where is an entire function and . Case 2: . Suppose is a common zero of and . Then, it is easy to see that is a zero of . If , then a contradiction. Thus, If , then , a contradiction. Thus, . We assume that is a zero of , but not a zero of . It deduces from (33) that is a zero of or . We take by the reduced counting function of those common zeros of and . Similarly, we denote by the reduced counting function of those common zeros of and . Then,and this implies that either or . If . Similarly, as the same way in Case 1, we get the desired result. So, we assume that as follows. Similarly, as the way in Case 2, we can get thatIt follows from (39) and (41) thatNote that . Thus, . Again by (41), one has . We also rewrite it asBy Lemma 3, if , are not constants, then , a contradiction. So, , (where are three constants). We also get , , . Hence, is a periodic function. We also know is a polynomial. So, we get (where and are two constants, and ). Finally, we get . Here, is an entire function and . Proof of Theorem 4 is completed. Data Availability No data were used to support this study. Conflicts of Interest The authors declare that they have no conflicts of interest. Authors’ Contributions All authors typed, read, and approved the final manuscript. Acknowledgments The work presented in this paper was supported by the Plateau Disciplines in Shanghai, Leading Academic Discipline Project of Shanghai Dianji University (16JCXK02), and Philosophy and Social Sciences Planning Project of the Ministry of Education (Grant no. 18YJC630120).
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... In this paper we consider the problem of extending Theorem C to the higher order derivative and to use CMW in place of CM sharing of sets. We use a methodology that is similar to [3] but with necessary modifications. ...
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Full-text available
  • Sep 2021
In the paper we consider the uniqueness problem of an entire function f when it shares a doubleton with its derivative f(k)f(k)f^{(k)}, k≥1k≥1k \ge 1. Our result extends a result of Li and Yang (J Math Soc Japan 51(4):781–799, 1999).
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... Li and Yang [6] gave an example to show that the "CM" in Theorem C cannot be replaced by "IM". However if 0 is a Picard exceptional value of f and f (k) , Zheng and Wang [7] proved the following result. ...
... On the other hand we see that f and f share − 1 6 IM , but f ≡ f . ≡ 1, ∞), a 2 (z)( ≡ 0, ∞) and b(z)( ≡ 0, ∞) be small functions with respect to f and let g(z) = a 1 ...
Uniqueness of meromorphic functions sharing one or two small functions with its derivative
Article
  • Sep 2016
In this paper we study the problem of uniqueness of meromorphic functions f(z) that share two values with g(z)=a1(z)f(z)+a2(z)f(k)(z)g(z)=a_{1}(z)f(z)+a_{2}(z)f^{(k)}(z), where a1(z)(≢1,∞)a_{1}(z) (\not \equiv 1, \infty ) and a2(z)(≢0,∞)a_{2}(z)(\not \equiv 0, \infty ) are small functions with respect to f and obtain some results which improve and generalize the result due to Yao and Li (J Math Anal Appl 322:133–145, 2006). We also solve an open problem as posed in Liu and Qi (Kyungpook Math J 49:235–243, 2009). In this paper we provide some examples to show that the conditions in our results are the best possible.
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... Mues and Steinmetz [9], and Gundersen [4] improved this result and proved the following theorem. An example given in [8] shows that the "CM" in Theorem B cannot be replaced by "IM." However, if 0 is a Picard exceptional value of f and f (k) , Zheng and Wang [12] proved the following theorem. ...
... From (29), (4), (8) and the second fundamental theorem, we can deduce that ...
Meromorphic functions sharing two small functions with its derivative
Article
  • Oct 2006
In this paper, we find all the forms of meromorphic functions f(z) that share the value 0 CM∗, and share b(z)IM∗ with g(z)=a1(z)f(z)+a2(z)f′(z). And a1(z), a2(z) and b(z) (a2(z),b(z)≢0) be small functions with respect to f(z). As an application, we show that some of nonlinear differential equations have no transcendental meromorphic solution.
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... 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]. Let f be a nonconstant entire function and a, b ∈ S(f ) be distinct. ...
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... Li Ping and Yang Chung-Chun [5] improved Theorem 1 and proved. ...
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  • Apr 2023
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).
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... Li Ping and Yang Chung-Chun [5] improved Theorem 1 and proved. ...
Единственность целых функциях относительно их разностных операторов и производных
Article
  • Mar 2023
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 f′(z)f'(z) share two values a,b counting multilicities then f(z)≡f′(z)f(z)\equiv f'(z). 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 f(k)(z)f^{(k)}(z) share a counting multiplicities and b ignoring multiplicities then f(z)≡f(k)(z)f(z)\equiv f^{(k)}(z). 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 Δf(z)\Delta f(z) be trancendental entire functions of finite order, k≥0 k \geq 0 be integer and P1P_{1} and P2P_{2} be two polynomials. If Δf(z)\Delta f(z) and f(k)f^{(k)} share P1P_{1} CM and share P2P_{2} IM, then Δf≡f(k)\Delta f \equiv f^{(k)}. A non-trivial proof of this result uses Nevanlinna's value distribution theory.
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... The uniqueness problem of entire functions sharing set with their derivatives, shifts, different types of difference operators has been developed as an interesting direction of research in the realm of value distribution theory. In 1999, Li-Yang [2] made a pioneer work by considering the relation between an entire function and its derivative sharing a set with two elements. Following their footsteps, in 2005, Li [3] investigated the same type of problem for linear differential operator. ...
Entire functions of restricted hyper-order sharing a set of two small functions IM with their linear c-shift operators
Article
Full-text available
  • Aug 2021
Purpose The paper aims to build the relationship between an entire function of restricted hyper-order with its linear c-shift operator. Design/methodology/approach Standard methodology for papers in difference and shift operators and value distribution theory have been used. Findings The relation between an entire function of restricted hyper-order with its linear c-shift operator was found under the periphery of sharing a set of two small functions IM (ignoring multiplicities) when exponent of convergence of zeros is strictly less than its order. This research work is an improvement and extension of two previous papers. Originality/value This is an original research work.
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... Thus, it is important and difficult question to find uniqueness or even a relationship between entire functions and their derivatives sharing a set. Dealing with sharing a set of two elements by entire functions f and their derivatives f 0 , in 1999, Li and Yang [14] explored this situation and proved the following result. ...
Meromorphic functions sharing a set with their differential monomials
Article
  • Mar 2022
Let S be a subset of C\mathbb {C} and for a non-constant meromorphic function f, we define Ef(S):=⋃a∈S{z:f(z)−a=0}E_{f}(S):=\bigcup _{a\in S}\{z: f(z)-a=0\}, where each zero is counted according to its multiplicity. If Ef(S)=Eg(S)E_{f}(S)=E_{g}(S), 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 S={w:awn+bw2m+cwm+1=0}\mathcal {S}=\{w:aw^n+bw^{2m}+cw^m+1=0\} with their differential monomials, where n>2mn>2m and a,b,c∈Ca, b, c\in \mathbb {C}. 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.
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... Using the notion of set sharing instead of value sharing, Li and Yang [10] proved the following theorem. ...
Uniqueness of an entire function sharing two values jointly with its differential polynomials
Preprint
Full-text available
  • Jul 2021
In this paper, we continue to investigate the uniqueness problem when an entire function f and its linear differential polynomial L(f) share two distinct complex values CMW (counting multiplicities in the weak sense) jointly. Also, We investigate the same problem when f and its differential monomial M(f) share two distinct complex values CMW. Our results generalize the recent result of Lahiri (Comput. Methods Funct. Theory, https://doi.org/10.1007/s40315-020-00355-4).
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... Li Ping and Yang Chung-Chun [7] improved Theorem A and proved ...
Unicity on entire function concerning its differential polynomials in Several complex variables
Preprint
  • Mar 2021
In this paper, we study the uniqueness of the differential polynomials of entire functions. We prove the following result: Let f(z) be a nonconstant entire function on Cn\mathbb{C}^{n} and g(z)=b−1+∑i=0nbif(ki)(z)g(z)=b_{-1}+\sum_{i=0}^{n}b_{i}f^{(k_{i})}(z), where b−1b_{-1} and bi(i=0…,n)b_{i} (i=0\ldots,n) are small meromorphic functions of f, ki≥0(i=0…,n)k_{i}\geq0 (i=0\ldots,n) are integers. Let a1(z)≢∞,a2(z)≢∞a_{1}(z)\not\equiv\infty, a_{2}(z)\not\equiv\infty be two distinct small meromorphic functions of f(z). If f(z) and g(z) share a1(z)a_{1}(z) CM, and a2(z)a_{2}(z) IM. Then either f(z)≡g(z)f(z)\equiv g(z) or a1=2a2a_{1}=2a_{2}, f(z)≡a2(e2p(z)−2ep(z)+2),f(z)\equiv a_{2}(e^{2p(z)}-2e^{p(z)}+2), and g(z)≡a2ep(z),g(z)\equiv a_{2}e^{p(z)}, where p(z) is a nonzero entire function satisfying a2(z)p′(z)=1a_{2}(z)p'(z)=1.
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... Li Ping and Yang Chung-Chun [9] improved Theorem A and proved ...
Uniqueness of Entire Functions With Respect To Their Shifts Concerning Derivatives
Preprint
  • Sep 2020
In this paper, we study the uniqueness of entire function that sharing small functions with their shifts concerning its k−thk-th derivatives. We prove that: Let f(z) be a transcendental entire function of finite order, let c be a nonzero finite value, k be a positive integer, and let a(z)≢∞,b(z)≢∞a(z)\not\equiv\infty, b(z)\not\equiv\infty be two distinct small functions of f(z+c) and f(k)(z)f^{(k)}(z). If f(k)(z)f^{(k)}(z) and f(z+c) share a(z) CM, and share b(z) IM, then f(k)(z)≡f(z+c)f^{(k)}(z)\equiv f(z+c). The result improves some conclusions due to Qi and Yang \cite {qy}.
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... Li Ping and Yang Chung-Chun [9] improved Theorem B and proved ...
Unicity on entire function concerning its differential-difference operators
Preprint
  • Sep 2020
In this paper, we study the uniqueness of the differential-difference polynomials of entire functions on Cn\mathbb{C}^{n}. We prove the following result: Let f(z) be a transcendental entire function on Cn\mathbb{C}^{n} of hyper-order less than 1 and g(z)=b−1+∑i=0nbif(ki)(z+ηi)g(z)=b_{-1}+\sum_{i=0}^{n}b_{i}f^{(k_{i})}(z+\eta_{i}), where b−1b_{-1} and bi(i=0…,n)b_{i} (i=0\ldots,n) are small meromorphic functions of f on Cn\mathbb{C}^{n}, ki≥0(i=0…,n)k_{i}\geq0 (i=0\ldots,n) are integers, and ηi(i=0…,n)\eta_{i} (i=0\ldots,n) are finite values. Let a1(z)≢∞,a2(z)≢∞a_{1}(z)\not\equiv\infty, a_{2}(z)\not\equiv\infty be two distinct small meromorphic functions of f(z) on Cn\mathbb{C}^{n}. If f(z) and g(z) share a1(z)a_{1}(z) CM, and a2(z)a_{2}(z) IM. Then either f(z)≡g(z)f(z)\equiv g(z) or a1=2a2=2a_{1}=2a_{2}=2, f(z)≡e2p−2ep+2,f(z)\equiv e^{2p}-2e^{p}+2, and g(z)≡ep,g(z)\equiv e^{p}, where p(z) is a non-constant entire function on Cn\mathbb{C}^{n}. Especially, in the case of g(z)=(Δηnf(z))kg(z)=(\Delta_{\eta}^{n}f(z))^{k}, we obtain f(z)≡(Δηnf(z))kf(z)\equiv (\Delta_{\eta}^{n}f(z))^{k}.
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... where c is a non-zero complex number. Li and Yang [21] also deduced that if E f (S) = E f ′ (S), where S = {a, b} with a + b = 0, then either f (z) = Ae z or f (z) = Ae z + a + b, where A is a non-zero complex number. Later, Fang and Zalcman [10], using the theory of normal families, proved that there exists a set S = {a, b, c} such that E f (S) = E f ′ (S), then f ≡ f ′ . ...
A note on entire functions sharing a finite set with applications to difference equations
Preprint
Full-text available
  • Jul 2020
Value distribution and uniqueness problems of difference operator of an entire function have been investigated in this article. This research shows that a finite ordered entire function f when sharing a set S={α(z),β(z)} \mathcal{S}=\{\alpha(z), \beta(z)\} of two entire functions α \alpha and β \beta with max⁡{ρ(α),ρ(β)}<ρ(f) \max\{\rho(\alpha), \rho(\beta)\}<\rho(f) with its difference Lcn(f)=∑j=0najf(z+jc) \mathcal{L}^n_c(f)=\sum_{j=0}^{n}a_jf(z+jc) , then Lcn(f)≡f \mathcal{L}^n_c(f)\equiv f , and more importantly certain form of the function f has been found. The results in this paper improve those given by \emph{k. Liu}, \emph{X. M. Li}, \emph{J. Qi, Y. Wang and Y. Gu} etc. Some constructive examples have been exhibited to show the condition max⁡{ρ(α),ρ(β)}<ρ(f) \max\{\rho(\alpha), \rho(\beta)\}<\rho(f) is sharp in our main result. Examples have been also exhibited to show that if CM sharing is replaced by IM sharing, then conclusion of the main results ceases to hold.
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... There is another study direction on the URSE of entire functions, which is to seek a set S such that if E(f , S) = E(f , S), then f = f for an entire function f . Li and Yang [8] deduced that if E(f , S) = E(f , S) with S consisting of two distinct constants, then f has specific forms. Later, based on the theory of the normal family, Fang and Zalcman [9] answered the question by proving that there exists a finite set S including three elements ...
A note on entire functions sharing a finite set with their difference operators
Article
Full-text available
  • Mar 2019
  • ADV DIFFER EQU-NY
In this note, we will show that an entire function is equal to its difference operator if it has a growth property and shares a set, where the set consists of two entire functions of smaller orders. This result generalizes a result of Li (Comput. Methods Funct. Theory 12:307–328, 2012 and partially answers Liu’s (J. Math. Anal. Appl. 359:384–393, 2009) question.
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... 1977, Rubel and Yang [10] proved the following result. Li and Yang [8] gave an example to show that the "CM" in Theorem C cannot be replaced by "IM". However, if 0 is a Picard exceptional value of f and f (k) , Zheng and Wang [13] proved the following result. ...
Meromorphic Functions Sharing One Value with Their Derivatives Concerning the Difference Operator
Article
  • Nov 2017
Let f(z) be a non-constant meromorphic function of finite order, c∈C\{0} and k∈N. Suppose f(z) and f(k)(z+c) share 1 CM (IM), f(z) and f(z+c) share ∞ CM. If N(r,0;f)=S(r,f)Nr,0;f(z)+Nr,0;f(k)(z+c)=S(r,f), then either f(z)≡f(k)(z+c) or f(z) is a solution of the following equation: f′(z+c)-1=a(z)f(z)-1f(z)+1a(z),andNr,0;f(z)+1a(z)=S(r,f)f′(z+c)-1=a(z)f(z)-1f(z)+1a(z)where a(z)≢-1,0,∞a(z)≢0,∞ is a meromorphic function satisfying T(r,a)=S(r,f). Also we exhibit some examples to show that the conditions of our results are the best possible.
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... If f is entire and shares two finite, distinct values with its linear di¤erential polynomials of similar forms to P½ f but with small function coefficients, say, b l A M f ðCÞ ðl ¼ 0; 1; . . . ; nÞ with b n 2 0, many results have been obtained (see [1], [14] and [16]). ...
On the uniqueness problems of entire functions and their linear differential polynomials
Article
  • Mar 2007
  • KODAI MATH J
https://projecteuclid.org/journals/kodai-mathematical-journal/volume-30/issue-1/On-the-uniqueness-problems-of-entire-functions-and-their-linear/10.2996/kmj/1175287622.full
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... From Case 1 in the proof of Theorem 1.3, we can easily get the following corollary which has greatly extended Theorem C. The following example shows that it is necessary that the complex number a is finite. In the end, we use the theory of normal families to prove a uniqueness theorem which was discussed by Li and Yang [6] with Nevanlinna theory. Theorem 1.6. ...
Sharing set and normal families of entire functions and their derivatives
Article
Full-text available
  • Jan 2008
  • HOUSTON J MATH
In this paper, we use the idea of sharing set to prove: Let F be a family of functions holomorphic in a domain, let a and b be two distinct complex numbers with a + b ≠ 0. If for all f C. F, f and f′ share S = {a, b} CM, then F is normal in D. As an application, we prove a uniqueness theorem.
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... For a set S ⊂Ŝ(f ), we define that Ì ÓÖ Ñ Aº ( [14]) Let f be a non-constant entire function and a 1 , a 2 be two distinct complex numbers. If f and f share the set {a 1 , a 2 } CM, then f takes one of the following conclusions: ...
Entire functions Sharing sets of small functions with their difference operators or Shifts
Article
Full-text available
  • Dec 2013
  • MATH SLOVACA
We show some interesting results concerning entire functions sharing two sets of small functions CM with their difference operators or shifts. (C) 2013 Mathematical Institute Slovak Academy of Sciences
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... This result is due to Frank and Ohlenroth [3] for the case that the shared values are nonzero and Frank and Weissenborn [4] for the general case. In addition, Li [7] gave an example which shows that condition f and f have two shared CM is essential. In [5], Gundersen gave the following example. ...
Meromorphic functions that share one value and the solution of Riccati differential equation
Article
Full-text available
  • Jun 2013
In this paper, we shall give new examples on meromorphic functions that share one value with their first derivative and also give the solution for Riccati differential equation.
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... A special topic widely studied in the uniqueness theory is the case when f (z) shares values with its derivatives or differential polynomials. We recall a result of this type from the preceding literature: [13,Theorem 3].) Let f be a non-constant entire function and a 1 , a 2 be two distinct complex numbers. ...
Meromorphic functions sharing a set with applications to difference equations
Article
  • Nov 2009
This paper is devoted to proving some uniqueness type results for an entire function f(z) that shares a common set with its shift f(z+c) or its difference operator Δcf. We also give some applications to solutions of non-linear difference equations related to a conjecture proposed by C.C. Yang.
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... This result has been generalized to the case that f share two values with a linear differential polynomial in f by several mathematicians, see, e.g., [1] and [9,10]. Li–Yang [7] considered the case that f share two values CM jointly with its derivative, and proved that if f and f share the set {a 1 , a 2 } CM, then f assume one of the following cases: ...
Value sharing and differential equations
Article
  • Oct 2005
Suppose that f is a nonconstant entire function and L[f] a linear differential polynomial in f with constant coefficients. In this paper, by considering the existence of the solutions of some differential equations, we find all the forms of entire functions f in most cases when f and L[f] share two values counting multiplicities jointly. This result generalize some known results due to Rubel–Yang and Li–Yang.
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... Lemma 4 [5] ...
Unicity of meromorphic functions and their derivatives
Article
  • Sep 2003
In this paper, we prove that if a transcendental meromorphic function f shares two distinct small functions CM with its kth derivative f(k) (k>1), then f=f(k). We also resolve the same question for the case k=1. These results generalize a result due to Frank and Weissenborn.
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... Moreover, if f is transcendental, then r = O(T (r, f )). 2.14 (See [16]). Let f be a nonconstant entire function and g = b 1 + n j=0 b j f (j) , where b j (j = −1, 0, 1, 2, . . . ...
Some results on the regular solutions of a linear differential equation
Article
  • Nov 2008
  • COMPUT MATH APPL
In this paper, we prove a theorem on the regular growth of the solutions of a linear differential equation, as an application of which we obtain some uniqueness theorems of an entire function sharing a finite nonzero complex number with its nth derivative and a linear differential polynomial of its nth derivative under the restriction of finite lower order or of finite lower hyper-order that is not a positive integer. The results in this paper improve many known results. Some examples are provided to show that the results in this paper are the best possible.
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... E. Mues and M. Reinders, G. Frank and X. H. Hua, and P. Li continuously obtained that a meromorphic function f in C is equal to a linear differential polynomial P (f ) of f if f and P (f ) share three distinct finite complex numbers IM. In particular, when f in C is entire, C. A. Bernstein, C. D. Chang and B. Q. Li, and P. Li and C. C. Yang also obtained the relationship f = P (f ) if and only if f and P (f ) share two distinct finite complex numbers CM (see, e.g., [1], [8], [10], [11] or [12]). Let κ be an algebraically closed field of characteristic zero, complete for a nontrivial non-Archimedean absolute value | · |. ...
Unicity of meromorphic functions related to their derivatives
Article
  • Dec 2007
  • B BELG MATH SOC-SIM
https://projecteuclid.org/journals/bulletin-of-the-belgian-mathematical-society-simon-stevin/volume-14/issue-5/Unicity-of-meromorphic-functions-related-to-their-derivatives/10.36045/bbms/1197908902.full
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Entire function with its linear differential polynomial normally or weakly shares a set and some related topics
Article
  • Jan 2024
We strive to find different forms of an entire function sharing a doubleton of small functions CM with its linear differential polynomial to extend a result of Li [J Math Anal Appl 310:412–423, 2005]. In our another attempt, we have conveniently first time answered an open question raised in Li (2005) under weakly sharing environment. We have also improved a very recent result of Lahiri [Comput Methods Funct Theo 21:379–397, 2021] by responding an open question of the same. Finally, in the concluding section we have exhibited a two fold improvement of a recent result of Qi et al. [Comput Methods Funct Theo 18:567–582, 2018] which can somehow been interlinked to one of our theorem in the previous section. We also put two open questions and a handful number of examples relevant to the content of the paper.
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Unicity on Entire Function Concerning Its Differential-Difference Operators
Article
Full-text available
  • Aug 2021
  • RESULTS MATH
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 ηη\eta be a non-zero complex number, n≥1,k≥0n≥1,k≥0n\ge 1, k\ge 0 two integers and let a and b be two distinct finite complex numbers. If f and (Δηnf)(k)(Δηnf)(k)(\Delta _{\eta }^{n}f)^{(k)} share a CM and share b IM, then f≡(Δηnf)(k)f≡(Δηnf)(k)f\equiv (\Delta _{\eta }^{n}f)^{(k)}.
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A proof of conjecture of Li-Yang
Preprint
  • May 2021
In this paper, we prove a conjecture posed by Li-Yang in \cite{ly3}. We prove the following result: Let f(z) be a nonconstant entire function, and let a(z)≢∞,b(z)≢∞a(z)\not\equiv\infty, b(z)\not\equiv\infty be two distinct small meromorphic functions of f(z). If f(z) and f(k)(z)f^{(k)}(z) share a(z) and b(z) IM. Then f(z)≡f(k)(z)f(z)\equiv f^{(k)}(z), which confirms a conjecture due to Li and Yang (in Illinois J. Math. 44:349-362, 2000).
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Entire functions that share two pairs of small functions
Article
Full-text available
  • May 2021
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].
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Uniqueness of Entire Functions Sharing Certain Values with Derivatives
Chapter
Full-text available
  • Dec 2014
The problem of entire functions sharing values with their derivatives is a special case of uniqueness of entire functions. In 1977 L. A. Rubel and C. C. Yang initiated this kind of investigation. Afterwards, a number of researchers like Mues, Steinmetz, Zhang, Zheng, Wang, Yang, Yi etc. proceeded further with this problem. Now-a-days the uniqueness problem of entire functions sharing values with derivatives has become one of the most well explored branches of the value distribution theory. In the short survey we discuss the gradual development of this branch starting from the result of Rubel and Yang. For the purpose, sometimes we shall mention some results for meromorphic functions also.
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Unicity of meromorphic functions and their linear differential polynomials.
Article
  • Jan 2014
  • J INEQUAL APPL
This paper studies the unicity of meromorphic functions that share an arbitrary nonzero small function with their general linear differential polynomials. The result derived here extends one by Brück in 1996 and others.
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Unicity of meromorphic functions and their linear differential polynomials
Article
  • Oct 2014
  • J INEQUAL APPL
https://journalofinequalitiesandapplications.springeropen.com/articles/10.1186/1029-242X-2014-388
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Unicity of meromorphic functions and their linear differential polynomials.
Article
  • Jan 2014
  • J INEQUAL APPL
This paper studies the unicity of meromorphic functions that share an arbitrary nonzero small function with their general linear differential polynomials. The result derived here extends one by Brück in 1996 and others.
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When an entire function and its linear differential polynomial share two values
Article
  • Jun 2000
  • Illinois J Math
Let f be a nonconstant entire function and a and b be two distinct complex numbers. Let a 0 ,a 1 ,⋯,a k (k≥1) be constants with a k ¬≡0. Let g=a 0 f+a 1 f (1) +⋯+a k f (k) . If f and g share a and b, ignoring multiplicities, the authors prove φ=(f ' (f-g))/(f-a)(f-b) is a constant satisfying a 0 φ+a 1 φ 2 +⋯+a k φ k+1 ≡0. When a j =0 for 0≤j≤k=1 and a k =1, the result verifies a conjecture of Günter Frank. The paper includes a more general theorem and an examination of solutions for the linear differential equation f ' (f-g)-φ(f-a)(f-b)=0 when g=f ' and φ is a nonconstant entire function. These interesting results follow from expert use of the Nevanlinna theory including judicious application of the well-known Clunie theorem on differential polynomials (cf [J. Lond. Math. Soc. 37, 17-27 (1962; Zbl 0104.29504)]).
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Entire functions sharing simple a-points with their first derivative
Article
  • Apr 2011
  • HOUSTON J MATH
We show that if a complex entire function f and its derivative f′f' share their simple zeroes and their simple a-points for some nonzero constant a, then f≡f′f\equiv f'. We also discuss how far these conditions can be relaxed or generalized. Finally, we determine all entire functions f such that for 3 distinct complex numbers a1,a2,a3a_1,a_2,a_3 every simple aja_j-point of f is an aja_j-point of f′f'.
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Uniqueness of meromorphic functions and a question of C. C. Yang
Article
Full-text available
  • Apr 1990
In this paper the following result is proved Let f and g be nonconstant meromorphic functions in the plane. n be a nonnegative integer If . and , then . The result answers a question posed by C C Yang. An example is provided to show that this result is sharp.
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Meromorphe Funktionen, die mit ihrer Ableitung Werte teilen
Article
Full-text available
  • Jun 1979
In this paper it is shown that the functions f(z)=ez are the only nonconstant entire (meromorphic) functions which share two (three) distinct finite values with their derivative.
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On the unique range set of meromorphic functions
Article
  • Jan 1996
This paper studies the unique range set of meromorphic functions and shows that there exists a finite set S such that for any two nonconstant meromorphic functions f and g the condition E f (S) = Eg(S) implies f ≡ g. As a special case this also answers an open question posed by Gross (1977) about entire functions and improves some results obtained recently by Yi.
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Meromorphe Funktionen, die mit einer Ableitung drei Werte teilen
Article
  • Nov 1992
Herrn Professor Helmut Grunsky gewidmet In an earlier joint paper of the first author, the following theorem was “Suppose f is a nonconstant meromorphic function in the complex plane, that shares two finite nonzero values counting multiplicities with one of its derivatives. Then f is identical with that derivative.” In this paper we settle the case that one of the shared values is allowed to be zero. The claim is then the same.
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die mit ibres erzeugenden Funktion zwei Werte teilen, Dissertation
  • Jan 1993
  • P Russmann
P. Russmann, U ¨ ber Di¤erentialpolynome, die mit ibres erzeugenden Funktion zwei Werte teilen, Dissertation, Technischen Universitit Berlin, 1993.
A unicity theorem of slowly growing functions
  • Nov 1993
  • 826-833
  • Q.-D Chang
Q.-D. Chang, A unicity theorem of slowly growing functions, Acta Math. Sinica, Vol. 36, No. 6, Nov., 1993, 826-833.
On uniqueness of entire functions in C n and their partial di¤erential polynomials
  • C A Bernstein
  • D C Chang
  • B Q Li
C. A. Bernstein, D. C. Chang and B. Q. Li, On uniqueness of entire functions in C n and their partial di¤erential polynomials, To appear in Forum Math..