A note on entire functions sharing a finite set with applications to difference equations
aa r X i v : . [ m a t h . C V ] J u l A note on entire functions sharing a finite set withapplications to difference equations
Molla Basir Ahamed
Abstract.
Value distribution and uniqueness problems of difference operator of anentire function have been investigated in this article. This research shows that a finiteordered entire function f when sharing a set S = { α ( z ) , β ( z ) } of two entire functions α and β with max { ρ ( α ) , ρ ( β ) } < ρ ( f ) with its difference L nc ( f ) = P nj =0 a j f ( z + jc ),then L nc ( f ) ≡ f , and more importantly certain form of the function f has beenfound. The results in this paper improve those given by k. Liu , X. M. Li , J. Qi,Y. Wang and Y. Gu etc. Some examples have been exhibited to show the conditionmax { ρ ( α ) , ρ ( β ) } < ρ ( f ) is sharp in our main result. Examples have been also exhib-ited to show that if CM sharing is replaced by IM sharing, then conclusion of themain results ceases to hold. Introduction
In this paper, a meromorphic function will always be non-constant and meromorphicin the complex plane C, unless specifically stated otherwise. In what follows, we assumethat the reader is familiar with the elementary Nevanlinna theory,(see [
16, 19, 30 ]). Inparticular, for a meromorphic function f , S ( f ) denotes the family of all meromorphicfunction ζ for which T ( r, ζ ) = S ( r, f ) = o ( T ( r, f )), where r → ∞ outside of a possible setof finite logarithmic measure.For convenience, we agree that S ( f ) includes all constantfunctions and S ( f ) := S ( f ) ∪ {∞} . A set S is called a unique range set (URSE) for a certain class of entire functions ifeach inverse image of the set uniquely determines a function from the given class. Let S be a finite set of some entire functions and f an entire function. Then, a set E f ( S ) isdefined as E f ( S ) = { ( z, m ) ∈ C × Z : f ( z ) − a ( z ) = 0 with multiplicity m, a ∈ S} . If we do not the count multiplicities, then we denote the set as E f ( S ).Assume that g isanother entire funct ion. We say that f and g share S CM if E f ( S ) = E g ( S ), and we say f and g share S IM if E f ( S ) = E g ( S ). Thus, a set S is called U RSE if E f ( S ) = E g ( S ) , where f and g are two entire functions, then f ≡ g. Gross and
Yang [ ], first found an example of a U RSE which is S = { z : e z + z = 0 } ,and as this an infinite set, so it is very natural to investigate whether there exists a finite U RSE or not. In 1995 , Yi [ ] who found a U RSE S = { z : z n + az m + b = 0 } ,where n > m + 4 and a, b chosen in a such a way that the equations z n + az m + b = 0has no repeated roots. Since then there have been many efforts to study the problemof constructing U RSE time to time (see [
11, 12, 23 ]). There is another study on the
U RSE of entire functions, which is to seek a set S for which if E f ( S ) = E f ′ ( S ), then f ≡ f ′ . One can verify that the form of the function will be f ( z ) = ce z . where c is Mathematics Subject Classification.
Primary 30D35 Secondary 30D30.
Key words and phrases.
Uniqueness, difference operator, entire function, shared set. a non-zero complex number. Li and Yang [ ] also deduced that if E f ( S ) = E f ′ ( S ),where S = { a, b } with a + b = 0, then either f ( z ) = A e z or f ( z ) = A e z + a + b , where A is a non-zero complex number. Later, Fang and
Zalcman [ ], using the theory ofnormal families, proved that there exists a set S = { a, b, c } such that E f ( S ) = E f ′ ( S ),then f ≡ f ′ . A special topic widely studied in the uniqueness theory is the case when f ( z ) shares value(s) or set(s) with its derivatives or differential polynomials. We recall aresult of this type from the preceding literature: Theorem A. [ ] Let f be a non-constant entire function and a , a be two distinctcomplex numbers. If E f ( { a , a } ) = E f ′ ( { a , a } ) , then f takes one of the followingconclusions: (1) f = f ′ ; (2) f + f ′ = a + a ; (3) f ( z ) = c e cz + c e − cz , with a + a = 0 , where c , c , c are all non-zero complexnumbers satisfying c = 1 and c c = a (cid:0) − c (cid:1) . Recently, with a more general setting, namely k -th derivative f ( k ) or differentialmonomial M [ f ] or a more general setting namely differential polynomial P [ f ], the presentauthor, deduced that, when some power of a meromorphic function f and its differentialmonomial (or polynomial) sharing a set [ ] or a small function [ ], then even in this case,the function f also assumes a certain form. From the literature of meromorphic functionson sharing value problems, finding the class of the functions satisfying some differentialor difference equations gained a valuable space.Throughout this paper, we denote by ρ ( f ) and λ ( f ), the order of f and the exponentof convergence of zeros of f respectively (see [ ]): We also need the following notation:Let f be a non-constant meromorphic function, and we define difference operators as∆ c ( f ) = f ( z + c ) − f ( z ), and for n >
2, ∆ nc ( f ) = ∆ n − c (∆ c ( f )).In the past recent years, the Navanlinna characteristic of f ( z + c ), the value distribu-tion theory for difference polynomials, the Nevanlinna theory for the difference operatorand most importantly, the difference analogue of the lemmas on the logarithmic deriva-tive had been established (see [
5, 9, 14, 15, 17, 18, 27, 28, 32 ]). For these theories,since the derivative is a difference counterpart of a function, hence there has been recentstudy of whether the derivative f ′ of a entire (meromorphic) function f can be replacedby the difference ∆ c ( f ) = f ( z + c ) − f ( z ) in the above mentioned results. Number ofresearches have been done with difference operator (see[
2, 6, 7, 20, 24, 25, 26, 27, 29 ])In this direction, in 2009,
Liu [ ] considered the problem of sharing set by an entirefunction and its difference, and obtained the following result. Theorem B. [ ] Suppose that α is is a non-zero complex number, and f is atranscendental entire function with finite order. If E f ( {− α, α } ) = E ∆ c ( f ) ( {− α, α } ) , then ∆ c ( f ( z )) = f ( z ) for all z ∈ C . Since ∆ c ( f ) is a very special form of the setting L c ( f ) := a f ( z + c ) + a f ( z ), where a ( = 0) , a ∈ C (see [ ]). Therefore a natural quarry would be as the following: Question . Does
Theorem B still hold if we replace ∆ c ( f ) by L c ( f ) ?From the following two examples, one can ensure that answer of Question 1.1 is notaffirmative.
Example . Let f ( z ) = e z , and for c = log(log 2) , suppose that L c ( f ) = √ √ f ( z + c ) + (1 − √ − i √ f ( z ) . It is clearly f is a finite order entire function and E f ( S ) = E L c ( f ) ( S ), where S = {− α, α } , α ∈ C r { } but L c ( f ) = f. NTIRE FUNCTION SHARING A FINITE SET ... 3
Example . Let f ( z ) = sin ( z ) or cos ( z ) and L π ( f ) = (1 + i √ f ( z + π ) + i √ f ( z ) . Clearly f is of finite order and E f ( S ) = E L π ( f ) ( S ), where S = {− α, α } , α ∈ C r { } but L π ( f ) = f. To investigate with L c ( f ) to get the similar conclusion as of Theorem B , we requiresome extra conditions. Instead of looking for some general setting of the difference, it istherefore reasonable to concentrate for the generalization of the shared set {− α, α } .In this direction, we recall here a question proposed by Liu in [ ] as follows. Question . Let α and β be two small functions of f with period c . When atranscendental entire function f of finite order and its difference ∆ c ( f ) share the set { α, β } CM , what can we say about the relationship between f and ∆ c ( f ) ?In connection with the Question 1.2 , in 2012, Li [ ] established the following result. Theorem C. [ ] Suppose that α , β are two distinct entire functions, and f is a non-constant entire function with ρ ( f ) = 1 and λ ( f ) < ρ ( f ) < ∞ such that max { ρ ( α ) , ρ ( β ) } <ρ ( f ) . If E f ( { α, β } ) = E ∆ c ( f ) ( { α, β } ) , then ∆ c ( f ( z )) ≡ f ( z ) for all z ∈ C . Remark . Next example confirms that conclusion of
Theorem C still holds if weremove the condition ρ ( f ) = 1. Example . Let f ( z ) = 2 z/ π cos( z ), and S = { α ( z ) , β ( z ) } , where α ( z ) and β ( z )are two non-constant polynomials in z . It is clear that ρ ( f ) = 1, and max { ρ ( α ) , ρ ( β ) } =0 < ρ ( f ), and E f ( { α, β } ) = E ∆ π ( f ) ( { α, β } ), and ∆ c ( f ( z )) = f ( z ) for all z ∈ C .Thus, one natural question is : Can we prove Theorem C by omitting the restriction ρ ( f ) = 1. Recently, Qi, Wang and Gu [ ] answered this question by proving thefollowing result. Theorem D. [ ] Suppose that α , β are two distinct entire functions, and f is anon-constant entire function with λ ( f ) < ρ ( f ) < ∞ such that max { ρ ( α ) , ρ ( β ) } < ρ ( f ) .If E f ( { α, β } ) = E ∆ c ( f ) ( { α, β } ) , then f ( z ) = A e λz , where A , λ are two non-zero complexnumbers satisfying e λc = 2 . Furthermore ∆ c ( f ( z )) ≡ f ( z ) for all z ∈ C . Note . In view of
Examples 1.1 and , we have seen that in
Theorem B , it isnot possible to replace ∆ c ( f ) by L c ( f ) in general, but, the next two examples show thatin case of Theorem D , one can do it.
Example . Let f ( z ) = (cid:16) π (cid:17) z/c and L c ( f ) = 2 f ( z + c ) + (1 − π ) f ( z ) . Let S = { α, β } , where α and β are two entire functions with ρ ( α ) < ρ ( β ) <
1. Evidently, E f ( S ) = E L c ( f ) ( S ) with max { ρ ( α ) , ρ ( β ) } < ρ ( f ), and also L c ( f ( z )) = f ( z ). Example . Let f ( z ) = (1 + i ) z/c and L c ( f ) = (cid:16) i √ (cid:17) f ( z + c ) + (cid:16) √ − i (cid:16) √ (cid:17)(cid:17) f ( z ) . Let S = { α, β } , where α and β are two entire functions with ρ ( α ) < ρ ( β ) < E f ( S ) = E L c ( S ) with max { ρ ( α ) , ρ ( β ) } < ρ ( f ), and also L c ( f ( z )) = f ( z ). Note . It is not hard to check that the functional form is f ( z ) = (cid:18) − a a (cid:19) z/c g ( z ),where g is a c -periodic function, when f satisfies the relation L c ( f ) = f .In this paper, we are mainly concerned for a more generalization of ∆ c ( f ), and L c ( f ),hence we define L nc ( f ) = a n f ( z + nc ) + a n − f ( z + ( n − c ) + . . . + a f ( z + c ) + a f ( z ) , MOLLA BASIR AHAMED where a n ( = 0) , a i ∈ C for ( i = 0 , , , . . . , n − a j =( − ) j (cid:18) nj (cid:19) , j = 0 , , , . . . , n , then L nc ( f ) = ∆ nc ( f ). With this generalization, our aim isto study Theorems C and D further. So it is interesting to ask the following questionsregarding Theorems C and D . Question . In Theorems C and D , what happen if we replace ∆ c ( f ) by L nc ( f ) ? Question . Can we get a corresponding result like
Theorems C and D , in whichthe condition max { ρ ( α ) , ρ ( β ) } < ρ ( f ) is sharp ? Question . What can be say about the specific form of the function f when∆ c ( f ) is replaced by L c ( f ) ? Main result
In this article, we dealt with the above questions and answered them all affirmatively.Following is the main result of this paper.
Theorem . Suppose that α , β are two distinct entire functions, and f is a non-constant entire function with λ ( f ) < ρ ( f ) < ∞ such that max { ρ ( α ) , ρ ( β ) } < ρ ( f ) and L nc ( f )( . If E f ( { α, β } ) = E L nc ( f ) ( { α, β } ) , then f ( z ) = A λ z/c + A λ z/c + . . . + A n λ z/cn , where A i , λ i ∈ C r { } for i = 1 , , . . . , n , and λ i are the roots of the equation a n w n + a n − w ( n − c + . . . + a w + a − . (2.1) Furthermore L nc ( f ( z )) ≡ f ( z ) for all z ∈ C . Remark . Entire functions satisfying
Theorem 2.1 do exist, and it is shown herefor the case n = 1 and n = 3 only, and we discussed the case n = 2 later in a corollary. Example . Let f ( z ) = (1 + i ) z/ and L ( f ) = 2 f ( z + 3) − (1 + 2 i ) f ( z ). Let S = { z − , z − z +1 } . Evidently, max { ρ ( α ) , ρ ( β ) } = 0 < ρ ( f ) and E f ( S ) = E L ( f ) ( S ),and f has the specific form and also satisfying the relation L ( f ) ≡ f. Example . Let f ( z ) = i √ ! z/ √ and L √ ( f ) = − (cid:16) √ i √ (cid:17) f ( z + √
2) + (cid:16) i √ (cid:16) √ i √ (cid:17)(cid:17) f ( z ) . Let S = { z − z + √ , √ z − √ z − i √ β } . Clearly, max { ρ ( α ) , ρ ( β ) } = 0 < ρ ( f )and E f ( S ) = E L √ ( f ) ( S ), and f has the specific form and also satisfying the relation L √ ( f ) ≡ f. Example . Let f ( z ) = 6 z/c + (cid:0) ω + 3 ω (cid:1) z/c + (cid:0) ω + 3 ω (cid:1) z/c and S = { α, β } , where α and β are any two polynomials in z . Then, we see that max { ρ ( α ) , ρ ( β ) } =0 < ρ ( f ) and E f ( S ) = E L c ( f ) ( S ), where L c ( f ) = f ( z + 3 c ) − f ( z + 2 c ) − f ( z + c ) − f ( z ) . Clearly f has the specific form and also satisfying the relation L c ( f ) ≡ f . Example . Let f ( z ) = A z/c + B z/c + C z/c and S = { α, β } , where α and β areany two polynomials in z . Then, we see that max { ρ ( α ) , ρ ( β ) } = 0 < ρ ( f ) and E f ( S ) = E L c ( f ) ( S ), where L c ( f ) = f ( z + 3 c ) − ( A + B + C ) f ( z + 2 c ) + ( AB + BC + CA ) f ( z + c ) − ( ABC − f ( z ) . Clearly f has the specific form and also satisfying the relation L c ( f ) ≡ f NTIRE FUNCTION SHARING A FINITE SET ... 5
Remark . In Theorem 2.1 , the condition max { ρ ( α ) , ρ ( β ) } < ρ ( f ) can not bereplaced by max { ρ ( α ) , ρ ( β ) } = ρ ( f ) i.e., the condition is sharp, which can be seen in thenext examples for the case n = 1 , Example . Let f ( z ) = e z + e z , and L c ( f ) = √ f ( z + c ) + −√ − i p √ − f ( z ) , where c = log √ i p √ − √ ! . Let S = { α ( z ) , β ( z ) } , where α ( z ) = A e z and β ( z ) = (1 − A ) e z , A ∈
C r (cid:26) , , (cid:27) . We see that max { ρ ( α ) , ρ ( β ) } = 1 = ρ ( f ),and E f ( S ) = E L c ( f ) ( S ) but neither f has the specific form nor satisfying the relation L c ( f ) = f. Example . Let f ( z ) = e z + e z + e − z , and L πi ( f ) = − f ( z + πi ) − f ( z ) . Let S = { α ( z ) , β ( z ) } , where α ( z ) = e z and β ( z ) = e − z . Clearly max { ρ ( α ) , ρ ( β ) } = 1 = ρ ( f ),and E f ( S ) = E L πi ( f ) ( S ) but note that f is neither in the form nor satisfying the relation L πi ( f ) = f. Example . Let f ( z ) = sin z + e z and L π ( f ) = (cid:18) e π − e π (cid:19) f ( z + 2 π ) + 2 f ( z + π ) + (cid:18) ( e π + 2) e π e π − (cid:19) f ( z ) . Let S = { α, β } , where α ( z ) = B e z and β ( z ) = (1 − B ) e z , B ∈
C r (cid:26) , , (cid:27) . We checkthat E f ( S ) = E L π ( f ) ( S ) and max { ρ ( α ) , ρ ( β ) } = 1 = ρ ( f ) but f is neither in the specificform nor satisfying L π ( f ) = f . Example . Let f ( z ) = cos z + e − z and L π ( f ) = (cid:18) e π + e π − e − π + 1 (cid:19) f ( z + 3 π ) + e π f ( z + π ) − e π f ( z + π )+ (cid:18) e π + e π − e − π + 1 − e π − e π (cid:19) f ( z ) . Let S = { α, β } , where α ( z ) = C cos z and β ( z ) = (1 − C ) cos z , C ∈
C r (cid:26) , , (cid:27) . Wecheck that E f ( S ) = E L π ( f ) ( S ) and max { ρ ( α ) , ρ ( β ) } = 1 = ρ ( f ) but f is not in thespecific form and also not satisfying L π ( f ) = f . Remark . The next examples show that, conclusion of
Theorem 2.1 ceases tohold if one replace the CM sharing by IM sharing. The examples have been exhibitedfor n = 1 and n = 2 only. Example . Let f ( z ) = − (cid:0) e z + a e − z (cid:1) for a ∈ Cr { } . We choose c ∈ Cr { kπi } , for k ∈ Z be such that L c ( f ) = 2 e c − e c f ( z + c ) − − e c f ( z ) . Evidently E f ( S ) = E L c ( f ) ( S ) where S = { α, β } = {− a, a } and max { ρ ( α ) , ρ ( β ) } =0 < ρ ( f ) but f has neither the specific form as in Theorem 1.1 nor satisfies therelation L c ( f ) ≡ f . Example . Let f ( z ) = 12 (cid:0) e z + e − z (cid:1) and for c ∈ C r (cid:26) kπi k ∈ Z (cid:27) , L c ( f ) = 21 − e c f ( z + 2 c ) − e − c f ( z + c ) + 2 e c e c − f ( z ) . MOLLA BASIR AHAMED
Evidently E f ( S ) = E L c ( f ) ( S ), where S = {− , } with max ρ ( α ) , ρ ( β ) = 0 < ρ ( f )but f has neither the specific form nor satisfying L c ( f ) = f. For a ( = 0) , a , a ∈ C , we define two constants D and D as follows D = − a + p a − a ( a − a and D = − a − p a − a ( a − a . We have a corollary of the main result.
Corollary . Suppose that α , β are two distinct entire functions, and f is anon-constant entire function with λ ( f ) < ρ ( f ) < ∞ such that max { ρ ( α ) , ρ ( β ) } < ρ ( f ) and L c ( f )( . If E f ( { α, β } ) = E L c ( f ) ( { α, β } ) , then f ( z ) = A (cid:18) − a a (cid:19) z/c when a + 4 a = 4 a a A D z/c + A D z/c , otherwisewhere A , A , A ∈ C r { } . Furthermore L c ( f ( z )) ≡ f ( z ) for all z ∈ C . Remark . From the next examples, one can observe that entire functions insupport of
Corollary 2.1 when (i) a + 4 a = 4 a a and (ii) a + 4 a = 4 a a . Here S = { α, β } , where α and β are two polynomials in z . Example . Let f ( z ) = 2 √ √ √ √ ! z/c , and L c ( f ) = √ f ( z + 2 c ) − √ f ( z + c ) + 3 + 4 √ √ f ( z ) . One can check that a + 4 a = 4 a a , and all the conditions of Corollary 2.1 are satisfiedand f has the specific form and also L c ( f ) = f . Example . Let f ( z ) = √ p
16 + √
17 + p √ √ ! z/c + √ p
16 + √ − p √ √ ! z/c and L c ( f ) = 2 √ f ( z + 2 c ) − q
16 + √ f ( z + c ) + (1 + √ f ( z ) . Verify that a + 4 a = 4 a a , and all the conditions of Corollary 2.1 are satisfied and f is in the form satisfying L c ( f ) = f . Some useful lemmas
As discussed in section 1,
Halburd - Korhonen [ ] and Chiang - Feng [ ] inspectedthe value distribution theory of difference expressions, including the difference analogueof the logarithmic derivative lemma, independently, we recall here their results. Lemma . [ ] Let f ( z ) be a non-constant meromorphic function of finite order,and c ∈ C r { } , δ < . Then m (cid:18) r, f ( z + c ) f ( z ) (cid:19) = o (cid:18) T ( r + | z | , f ) r δ (cid:19) , for all r outside of a possible exceptional set with finite logarithmic measure. By [ , Lemma 2.1], we have T ( r + | c | , f ( z )) = (1 + o (1)) T ( r, f ) for all r outside ofa set with finite logarithmic measure, when f is of finite order. NTIRE FUNCTION SHARING A FINITE SET ... 7
Lemma . [ ] Let f be a meromorphic function of finite order, and η , η be twodistinct arbitrary complex numbers. Assume that σ is the order of f , then for each ǫ > ,we have m (cid:18) r, f ( z + η ) f ( z + η ) (cid:19) = O (cid:0) r σ − ǫ (cid:1) . Lemma . [ ] Let g be a function transcendental and meromorphic in the plane oforder less than . Set h > . Then there exists an ǫ -set E such that g ( z + η ) g ( z ) → , when z → ∞ in C r E uniformly in η for | η | h . Lemma . Let α , β are two distinct entire functions, and f is a non-constantentire function such that max { ρ ( α ) , ρ ( β ) } < ρ ( f ) and E f ( { α, β } ) = E L c ( f ) ( { α, β } ) . Let f ( z ) = G ( z ) e P ( z ) , where G ( is an entire function and P be a polynomial in z . If W ( z ) = n X j =0 a j G ( z + jc ) e P ( z + jc ) − P ( z ) , then W ( z ) . Proof.
By the
Hadamard factorization theorem , since f is an entire function, wecan write f ( z ) = G ( z ) e P ( z ) , where G (
0) is an entire function and P be a polynomial in z . Since E f ( { α, β } ) = E L nc ( f ) ( { α, β } ), so we can write( L nc ( f ) − α )(( L nc ( f ) − β ))( f − α )( f − β ) = e Q , (3.1)where Q is an entire function. Furthermore, it follows from (3.1) and the conditionmax { ρ ( α ) , ρ ( β ) } < ρ ( f ) < ∞ , that Q is a polynomial in z . Since f ( z ) = G ( z ) e P ( z ) , thenwe can write L nc ( f ) as L nc ( f ) = n X j =0 a j f ( z + jc ) = (cid:20) n X j =0 a j G ( z + jc ) e P ( z + jc ) − P ( z ) (cid:21) e P ( z ) . (3.2)Substituting the forms of the functions f and L nc ( f ) in (3.1), we have (cid:26)(cid:20) n X j =0 a j G ( z + jc ) e P ( z + jc ) − P ( z ) (cid:21) e P ( z ) − α ( z ) (cid:27) (3.3) × (cid:26)(cid:20) n X j =0 a j G ( z + jc ) e P ( z + jc ) − P ( z ) (cid:21) e P ( z ) − β ( z ) (cid:27) = (cid:26) H ( z ) e P ( z ) − α ( z ) (cid:27)(cid:26) H ( z ) e P ( z ) − β ( z ) (cid:27) e Q ( z ) . On contrary, let if possible
W ≡
0. This shows that n X j =0 a j G ( z + jc ) e P ( z + jc ) − P ( z ) = 0 . (3.4)In view of (3.2) and (3.4), we see that L nc ( f ) ≡ , which contradicts L nc ( f ) W ( z ) . (cid:3) Lemma . Let f and G be two entire functions and P ( z ) is a polynomial in zj ,and a j , j = 0 , , . . . , n be all complex constants with a n = 0 , such that n X j =0 a j G ( z + jc ) e P ( z + jc ) − P ( z ) = G ( z ) , MOLLA BASIR AHAMED then for each j = 1 , , . . . , n and ǫ > , m (cid:16) r, e P ( z + jc ) − P ( z ) (cid:17) = [ A + o (1)] r ρ ( f ) − ǫ , for a complex number A . Proof.
We prove this lemma by inductive way. Let n = 1.Then, we have a G ( z ) + a G ( z + c ) e P ( z + c ) − P ( z ) = G ( z )which we can rewrite as e P ( z + c ) − P ( z ) = 1 − a a G ( z ) G ( z + c ) . (3.5)In view of Lemma 3.3 , we get from (3.5) that m (cid:16) r, e P ( z + c ) − P ( z ) (cid:17) = m (cid:18) r, − a a G ( z ) G ( z + c ) (cid:19) = m (cid:18) r, G ( z ) G ( z + c ) (cid:19) + O (1)= O (cid:16) r ρ ( G ) − ǫ (cid:17) + O (1) . On the other hand we have m (cid:0) r, e P ( z + c ) − P ( z ) (cid:1) = [ A + o (1)] r ρ ( f ) − ǫ , where A is acomplex number.Let n = 2, then we have e P ( z +2 c ) − P ( z ) = 1 − a a G ( z ) G ( z + 2 c ) − a a G ( z + c ) G ( z + 2 c ) . (3.6)By Lemma 3.3 , we get from (3.6) m (cid:16) r, e P ( z +2 c ) − P ( z ) (cid:17) = m (cid:18) r, G ( z ) G ( z + 2 c ) (cid:19) + m (cid:18) r, G ( z + c ) G ( z + 2 c ) (cid:19) + m (cid:16) r, e P ( z + c ) − P ( z ) (cid:17) + O (1)= O (cid:16) r ρ ( G ) − ǫ (cid:17) + O (1) . On the other hand we have m (cid:0) r, e P ( z +2 c ) − P ( z ) (cid:1) = [ A + o (1)] r ρ ( f ) − ǫ . So, continuing in this way, one can prove that m (cid:16) r, e P ( z + jc ) − P ( z ) (cid:17) = [ A + o (1)] r ρ ( f ) − ǫ , for j = 3 , , . . . , n. This completes the proof. (cid:3) Proof of Theorem
In this section, we give the proof of the main result of this article.
Proof of Theorem 2.1.
From the conditions of
Theorem 2.1 , since we have E f ( { α, β } ) = E L nc ( f ) ( { α, β } ), there must exists an entire function Q such that( L nc ( f ) − α )(( L nc ( f ) − β ))( f − α )( f − β ) = e Q . (4.1)Again, it follows from (4.1) and the condition max { ρ ( α ) , ρ ( β ) } < ρ ( f ) < ∞ , that Q is a polynomial in z . By the Hadamard factorization theorem , we may suppose that
NTIRE FUNCTION SHARING A FINITE SET ... 9 f ( z ) = G ( z ) e P ( z ) , where G (
0) is an entire function, and P is a polynomial satisfying λ ( f ) = ρ ( G ) < ρ ( f ) = deg( P ) . We set W ( z ) = n X j =0 a j G ( z + jc ) e P ( z + jc ) − P ( z ) . By Lemma 3.4 , we must have W ( z )
0. It is not hard to check that W is a smallfunction of e P ( z ) . Rewriting (3.3), we have e Q ( z ) = W ( z ) (cid:20) e P ( z ) − α ( z ) W ( z ) (cid:21)(cid:20) e P ( z ) − β ( z ) W ( z ) (cid:21) G ( z ) (cid:20) e P ( z ) − α ( z ) G ( z ) (cid:21)(cid:20) e P ( z ) − β ( z ) G ( z ) (cid:21) . (4.2)By our assumption, since α ( z ) β ( z ), without any loss of generality, we may supposethat α ( z ) . Let z be a zero of e P ( z ) − α ( z ) H ( z ) such that W ( z ) = 0, then it follows from (4.2)that z must be a zero of e P ( z ) − α ( z ) W ( z ) or e P ( z ) − β ( z ) W ( z ) . We denote now two countingfunctions here, one is N (cid:0) r, e P (cid:1) of the common zeros of e P ( z ) − α ( z ) G ( z ) and e P ( z ) − α ( z ) W ( z ) , and the second one is N (cid:0) r, e P (cid:1) be the reduced counting of those zeros of e P ( z ) − α ( z ) G ( z )and e P ( z ) − β ( z ) W ( z ) . Since G ( z ) is a small function of e P ( z ) , then by applying First and
Second Fundamental Theorem , we deduce that T (cid:0) r, e P (cid:1) = N r, e P ( z ) − α ( z ) H ( z ) + S (cid:16) r, e P ( z ) (cid:17) (4.3) = N (cid:16) r, e P ( z ) (cid:17) + N (cid:16) r, e P ( z ) (cid:17) + S (cid:16) r, e P ( z ) (cid:17) . It is clear from (4.3) thateither N (cid:16) r, e P ( z ) (cid:17) = S (cid:16) r, e P ( z ) (cid:17) or N (cid:16) r, e P ( z ) (cid:17) = S (cid:16) r, e P ( z ) (cid:17) . In our next discussions, we are going to appraise the following two cases.Case 1. Suppose N (cid:0) r, e P ( z ) (cid:1) = S (cid:0) r, e P ( z ) (cid:1) . Let z be a common zero of e P − α ( z ) G ( z ) and e P − α ( z ) W ( z ) . Therefore, z must be a zeroof (cid:18) e P − α ( z ) W ( z ) (cid:19) − (cid:18) e P − α ( z ) G ( z ) (cid:19) = α ( z ) G ( z ) − α ( z ) W ( z ) . Subcase 1.1. Let is possible, G ( z )
6≡ W ( z ). Then it follows that α ( z ) G ( z ) − α ( z ) W ( z ) . Wenext deduce that S (cid:0) r, e P (cid:1) = N (cid:0) r, e P (cid:1) N r, α ( z ) H ( z ) − α ( z ) W ( z ) T (cid:18) r, α ( z ) H ( z ) − α ( z ) W ( z ) (cid:19) = S ( r, e P ) , which is a contradiction.Subcase 1.2. Suppose G ( z ) ≡ W ( z ). Which in turn shows that n X j =0 a j G ( z + jc ) e P ( z + jc ) − P ( z ) = G ( z ) , (4.4)In view of Lemma 3.5 and (4.4), for each j = 1 , , . . . , n and ǫ >
0, we have m (cid:16) r, e P ( z + jc ) − P ( z ) (cid:17) = [ A + o (1)] r ρ ( f ) − ǫ , for a complex number A . Let if possible ρ ( f ) > ρ ( f ) > ρ ( G ) and the estimates of m (cid:0) r, e P ( z + jc ) − P ( z ) (cid:1) , we can easily get acontradiction.Thus, we see that ρ ( f )
1, and which in turn shows that e P ( z + jc ) − P ( z ) is a non-zeroconstant, say, η j for j = 0 , , , . . . , n . We have e P ( z + c ) = η j e P ( z ) . Therefore, it followsfrom (4.4) that, n X j =0 a j η j G ( z + jc ) G ( z ) = 1 . (4.5)Since ρ ( G ) < ρ ( f )
1, then in view of
Lemma 3.3 , there exists an ǫ -set E , as z E and | z | → ∞ , such that for each j = 1 , , . . . , n we can obtained G ( z + jc ) G ( z ) → n X j =0 a j η j = 1 . If we approach inductively, it follows from (4.5) that G ( z + c ) = G ( z ) for all z ∈ C i.e., G is a periodic entire function of period c . Let ifpossible G is non-constant, then G must be transcendental, and hence ρ ( G ) >
1, whichcontradicts ρ ( G ) ρ ( f ) < G is constant. Since, f is a finite order non-constant entire function withdeg( P ) = ρ ( f )
1, so we must have deg( P ) = 1, and hence P ( z ) will be of the form P ( z ) = d z + d , for some d ( = 0) , d ∈ C . Therefore, we can write the function f as f ( z ) = C e µz , where C and µ are two non-zero complex constants.By our assumption in Case 1, we see that f − α and L nc ( f ) − α have common zeros,which are not the zeros of α ( z ). Let z be a common zero of f − a and L nc ( f ) so that α ( z ) = 0 . Then, we see that z must be a zero f ( z ) − α ( z ) + L nc ( f ) − α ( z ) = n X j =1 a j f ( z + jc ) + ( a + 1) f ( z ) − α ( z ) . Thus we have C e µz − α ( z ) = 0 n X j =1 a j C e µz e µjc + ( a + 1) C e µz − α ( z ) = 0Thus we must have n X j =1 a j e µjc + a = 1 which can written as a n ( e µc − λ ) ( e µc − λ ) . . . ( e µc − λ n ) = 0 , (4.6)where λ , . . . , λ n are distinct roots of the equation a n z n + a n − z n − + . . . + a = 1 . From (4.6) , we obtained that e µc = λ i , for some i = 1 , , . . . , n, NTIRE FUNCTION SHARING A FINITE SET ... 11 and thus, the general form of the function is f ( z ) = C λ z/c + C λ z/c + . . . + C n λ z/cn , where C , C , . . . , C n are all complex constants.Case 2. Let N (cid:0) r, e P (cid:1) = S (cid:0) r, e P (cid:1) . Let z be a common zero of e P ( z ) − α ( z ) G ( z ) and e P ( z ) − β ( z ) W ( z ) . One can check that z must be a zero of α ( z ) H ( z ) − β ( z ) W ( z ) . Let α ( z ) G ( z ) − β ( z ) W ( z )
0. Then we see that S (cid:0) r, e P (cid:1) = N (cid:0) r, e P (cid:1) N r, α ( z ) G ( z ) − β ( z ) W ( z ) T (cid:18) r, α ( z ) G ( z ) − β ( z ) W ( z ) (cid:19) = S (cid:0) r, e P (cid:1) , which is clearly a contradiction.Thus we have α ( z ) G ( z ) ≡ β ( z ) W ( z ) . (4.7)Suppose β ( z ) ≡
0, then α ( z ) G ( z ) ≡ α ( z ) ≡
0, which is absurd.Let z is a zero of e P − β G but not a zero of W . Then it follows from (4.2) that z is azero of e P − α W or e P − β G . Here, we denote by N (cid:0) r, e P (cid:1) the reduced counting functionof those common zeros of e P − β G and e P − α W . Similarly, we denote by N (cid:0) r, e P (cid:1) thereduced counting function of those common zeros of e P − β G and e P − β W . Applying
First and
Second Fundamental Theorem , we have T (cid:0) r, e P (cid:1) = N r, e P − β H ! + S (cid:0) r, e P (cid:1) = N (cid:0) r, e P (cid:1) + N (cid:0) r, e P (cid:1) + S (cid:0) r, e P (cid:1) , which implies that either N (cid:0) r, e P (cid:1) = S (cid:0) r, e P (cid:1) or N (cid:0) r, e P (cid:1) = S (cid:0) r, e P (cid:1) .Subcase 2.1. If N (cid:0) r, e P (cid:1) = S (cid:0) r, e P (cid:1) , then in the same manner as in Case 1, we get ourdesired result.Subcase 2.2. Next we assume that N (cid:0) r, e P (cid:1) = S (cid:0) r, e P (cid:1) .Then similar to the Case 2, one can simply deduce that β G − α W ≡ . (4.8)Combining (4.7) and (4.8) yields that α = β which in turn implies that α = − β as α = β . Thus, we have from (4.8) that W = G . Rest of the proof of the result next followsfrom the Subcase 1.2. References [1] M. B. Ahamed,
Uniqueness of two differential polynomials of a meromorphic function sharing aset , Comm. Korean Math. Soc., 33(4)(2018), 1181–1203.[2] M. B. Ahamed,
On the periodicity of meromorphic functions when sharing two sets IM , Stud. Univ.Babes-Bolyai Math. 64(3)(2019), 497-510.[3] M. B. Ahamed,
An investigation on the conjecture of Chen and Yi , Results. Math., 74(2019): 122.[4] M. B. Ahamed and Santosh Linkha, Br ¨ u ck conjceture and certain solution of some differentialequation , Tamkang. J. Math., (2019)(In Press). [5] A. Banerjee and M. B. Ahamed, Uniqueness of meromorphic function with its shift operator underthe purview of two or three shared sets , Math. Slovaca., 69(3)(2019), 557–572.[6] A. Banerjee and M. B. Ahamed,
On some sufficient conditions for periodicity of meromorphicfunctions under new shared sets , Filomat, 33(18)(2019), 6055-6072.[7] A. Banerjee and M. B. Ahamed,
Results on meromorphic function sharing two sets with its linearc-difference operator , J. Contem. Math. Anal., 55(3)(2020), 143-155.[8] W. Bergweiler and J. K. Langly,
Zeros of differences of meromorphic functions , Math. Proc. Camb.Philos. Soc., 142(2007), 133–147.[9] Y. M. Chiang and S. J. Feng,
On the Nevanlinna characteristic of f ( z + η ) and difference equationsin the complex plane , Ramanujan J. 16(2008), 105129.[10] M. L. Fang and LL. Zalcman, Normal families and uniqueness theorems for entire functions , J.Math. Anal. Appl., 280(2003), 273–283.[11] G. Frank and M. Reinders,
A unique range set for meromorphic functions with 11 elements , Com-plex Var. Theory Appl., 37(1998), 185-193.[12] H. Fujimoto,
On uniqueness polynomials for meromorphic functions , Nagoya Math. J., 170(2003),33-46.[13] F. Gross and C. C. Yang,
On pre-image range sets of meromorphic functions , Proc. Jpn. Acad.,Ser. A 58(1982), 1720.[14] R. G. Halburd and R. J. Korhonen,
Difference analogue of the lemma on the logarithmic derivativewith applications to difference equations , J. Math. Anal. Appl., 314(2006), 477487.[15] R. G. Halburd and R. J. Korhonen,
Finite-order meromorphic solutions and the discrete Painlev ´ e equations , Proc. London Math. Soc., 94(3)(2007), 443474.[16] W. K. Hayman, Meromorphic Functions , Clarendon, Oxford (1964)[17] J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, J. Zhang,
Value sharing results for shifts ofmeromorphic function, and sufficient conditions for periodicity , J. Math. Anal. Appl. 355 (2009)352363.[18] J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo,
Uniqueness of meromorphic functions sharingvalues with their shifts , Complex Var. Elliptic Equ., 55(1-4)(2011), 81-92.[19] I. Laine,
Nevanlinna Theory and Complex Differential Equations , de Gruyter, Berlin (1993)[20] S. Li and Z. H. Gao,
Entire functions sharing one or two finite values CM with their shifts ordifference operators , Archiv. der Mathemtik., 97(2011), 475–483.[21] P. Li and C. C. Yang,
Value sharing of an entire function and its derivative , J. Math. Soc. Jpn.,51(1999), 781-799.[22] P. Li and C. C. Yang,
Meromorphic solutions of functional equations with nonconstant coefficients ,Proc. Japan Acad. Ser. A 82(2) (2006), 183186.[23] P. Li and C. C. Yang,
Some further results on the unique range sets of meromorphic functions ,Kodai Math. J., 18(1995), 437450.[24] X. M. Li,
Entire functions sharing a finite set with their difference operators , Comput. MethodsFunct. Theory, 12(2012), 307-328.[25] S. Li and B. Q. Chen,
Results on meromorphic solutions of linear difference equations , Adv. Differ.Equ. 2012(2012), 203.[26] S. Li, D. Mei and B. Q. Chen,
Uniqueness of entire functions sharing two values with their differenceoperators , Adv. Differ. Equ. 2017(2017), 390.[27] K. Liu,
Meromorphic functions sharing a set with applications to difference equation , J. Math. Anal.Appl. 359(2009), 384393.[28] X. Luo and W. C. Lin,
Value sharing results for shifts of meromorphic functions , J. Math. Anal.Appl., 377(2011), 441–449.[29] J. Qi, Y. Wang and Y. Gu,
A note on entire functions sharing a finite set with their differenceoperators , Adv. Diff. Equ., 2019(2019), 114.[30] C. C. Yang and H. X. Yi,
Uniqueness Theory of Meromorphic Functions , (Kluwer, Dordrecht, 2003).[31] H. X. Yi,
A question of Gross and the uniqueness of entire function , Nagoya Math. J. 138(1995),169177.[32] J. Zhang,
Value distribution and shared sets of differences of meromorphic functions , J. Math. Anal.Appl., 367(2010), 401-408.
Department of Mathematics, Kalipada Ghosh Tarai Mahavidyalya, West Bengal, 734014,India.
E-mail address ::