Further Results On Uniqueness Of Derivatives Of Meromorphic Functions Sharing Three Sets
aa r X i v : . [ m a t h . C V ] M a y FURTHER RESULTS ON UNIQUENESS OF DERIVATIVES OFMEROMORPHIC FUNCTIONS SHARING THREE SETS
ABHIJIT BANERJEE , SUJOY MAJUMDER AND BIKASH CHAKRABORTY Abstract.
We prove some uniqueness theorems concerning the derivatives of meromorphicfunctions when they share three sets which will improve some recent existing results. Introduction, Definitions and Results
In this paper by meromorphic functions we will always mean meromorphic functions in thecomplex plane. We shall use the standard notations of value distribution theory : T ( r, f ) , m ( r, f ) , N ( r, ∞ ; f ) , N ( r, ∞ ; f ) , . . . (see [7]). It will be convenient to let E denote any set of positive real numbers of finite linearmeasure, not necessarily the same at each occurrence. We denote by T ( r ) the maximum of T ( r, f ( k ) ) and T ( r, g ( k ) ). The notation S ( r ) denotes any quantity satisfying S ( r ) = o ( T ( r )) as r −→ ∞ , r E .If for some a ∈ C ∪ {∞} , f and g have the same set of a -points with same multiplicitiesthen we say that f and g share the value a CM (counting multiplicities). If we do not take themultiplicities into account, f and g are said to share the value a IM (ignoring multiplicities).Let S be a set of distinct elements of C ∪ {∞} and E f ( S ) = S a ∈ S { z : f ( z ) − a = 0 } , whereeach zero is counted according to its multiplicity. If we do not count the multiplicity the set E f ( S ) = S a ∈ S { z : f ( z ) − a = 0 } is denoted by E f ( S ). If E f ( S ) = E g ( S ) we say that f and g share the set S CM. On the other hand if E f ( S ) = E g ( S ), we say that f and g share the set S IM. Evidently if S contains only one element, then it coincides with the usual definition ofCM(respectively, IM) shared values.In 1926, R.Nevanlinna showed that a meromorphic function on the complex plane C isuniquely determined by the pre-images, ignoring multiplicities, of 5 distinct values (includinginfinity). A few years latter, he showed that when multiplicities are taken into consideration,4 points are enough and in this case either the two functions coincides or one is the bilineartransformation of the other.This two theories are the starting point of uniqueness theory. Research became more inter-esting although sophisticated when F.Gross and C.C.Yang transferred the study of uniquenesstheory to a more general setup namely sets of distinct elements instead of values. For instancethey proved that if f and g are two non-constant entire functions and S , S and S are threedistinct finite sets such that f − ( S i ) = g − ( S i ) for i = 1 , , f ≡ g .The following analogous question corresponding to meromorphic functions was asked in[18]. Question A
Can one find three finite sets S j ( j = 1 , , such that any two non-constantmeromorphic functions f and g satisfying E f ( S j ) = E g ( S j ) for j = 1 , , must be identical ?Question A may be considered as the inception of a new horizon in the uniqueness of mero-morphic functions concerning three set sharing problem and so far the quest for affirmativeanswer to Question A under weaker hypothesis has made a great stride { see [1]-[2], [5]-[6], [7],[13], [15], [17]-[20], [21] } . But unfortunately the derivative counterparts of the above results AMS -L A TEX are scanty in number. In 2003, in the direction of
Question A concerning the uniqueness ofderivatives of meromorphic functions Qiu and Fang obtained the following result.
Theorem A. [17]
Let S = { z : z n − z n − − } , S = {∞} and S = { } and n ( ≥ , k be two positive integers. Let f and g be two non-constant meromorphic functions such that E f ( k ) ( S j ) = E g ( k ) ( S j ) for j = 1 , and E f ( S ) = E g ( S ) then f ( k ) ≡ g ( k ) . In 2004 Yi and Lin [21] independently proved the following theorem.
Theorem B. [21]
Let S = { z : z n + az n − + b = 0 } , S = {∞} and S = { } , where a , b are nonzero constants such that z n + az n − + b = 0 has no repeated root and n ( ≥ , k be two positive integers. Let f and g be two non-constant meromorphic functions such that E f ( k ) ( S j ) = E g ( k ) ( S j ) for j = 1 , , then f ( k ) ≡ g ( k ) . The following examples show that in
Theorems A, B a = 0 is necessary. Example 1.1. [4]
Let f ( z ) = e z and g ( z ) = ( − k e − z and S = { z : z − } , S = {∞} , S = { } . Since f ( k ) − ω l = g ( k ) − ω − l , where ω = cos π + isin π , ≤ l ≤ , clearly E f ( k ) ( S j ) = E g ( k ) ( S j ) for j = 1 , , but f ( k ) g ( k ) . We now consider the following examples which establish the sharpness of the lower boundof n in Theorems A, B . Example 1.2. [4]
Let f ( z ) = √ α + β √ αβ e z and g ( z ) = ( − k √ α + β √ αβ e − z and S = { α + β, αβ } , S = {∞} , S = { } , where α + β = − a and αβ = b ; a , b are nonzero complexnumbers. Clearly E f ( k ) ( S j ) = E g ( k ) ( S j ) for j = 1 , , but f ( k ) g ( k ) . Example 1.3.
Let f = α √ βe z , g = ( − k β √ αe − z , where α and β be two non zero com-plex numbers such that q αβ = − . Let S = { β √ α, α √ β } , S = {∞} , S = { } . Clearly E f ( k ) ( S j ) = E g ( k ) ( S j ) for j = 1 , , but f ( k ) g ( k ) . Example 1.4.
Let f = √ e z , g = ( − k √ e − z . Let S = { i, − i } , S = {∞} , S = { } .Clearly E f ( k ) ( S j ) = E g ( k ) ( S j ) for j = 1 , , but f ( k ) g ( k ) . Above example assures the fact that in
Theorems A - B , the cardinality of the set S can notbe further reduced. Rather on the basis of above examples one may concentrate to relax thenature of sharing the range sets. For the purpose of relaxation of the nature of sharing the setsthe notion of weighted sharing of values and sets which appeared in [11, 12] has become verymuch effective. We now give the definition. Definition 1.1. [11, 12]
Let k be a nonnegative integer or infinity. For a ∈ C ∪ {∞} we denoteby E k ( a ; f ) the set of all a -points of f , where an a -point of multiplicity m is counted m timesif m ≤ k and k + 1 times if m > k . If E k ( a ; f ) = E k ( a ; g ) , we say that f , g share the value a with weight k . The definition implies that if f , g share a value a with weight k then z is an a -point of f with multiplicity m ( ≤ k ) if and only if it is an a -point of g with multiplicity m ( ≤ k ) and z is an a -point of f with multiplicity m ( > k ) if and only if it is an a -point of g with multiplicity n ( > k ), where m is not necessarily equal to n .We write f , g share ( a, k ) to mean that f, g share the value a with weight k . Clearly if f , g share ( a, k ) then f , g share ( a, p ) for any integer p , 0 ≤ p < k . Also we note that f , g share avalue a IM or CM if and only if f , g share ( a,
0) or ( a, ∞ ) respectively. Definition 1.2. [11]
Let S be a set of distinct elements of C ∪ {∞} and k be a nonnegativeinteger or ∞ . We denote by E f ( S, k ) the set E f ( S, k ) = S a ∈ S E k ( a ; f ) .Clearly E f ( S ) = E f ( S, ∞ ) and E f ( S ) = E f ( S, . In 2009 Banerjee and Bhattacharjee [3] subtly use the concept of weighted sharing of sets toimprove
Theorems A and B as follows : URTHER RESULTS ON UNIQUENESS OF DERIVATIVES OF MEROMORPHIC FUNCTIONS 3
Theorem C. [3]
Let S i , i = 1 , , be defined as in Theorem B and k be a positive integer.If f and g are two non-constant meromorphic functions such that E f ( k ) ( S ,
4) = E g ( k ) ( S , , E f ( S , ∞ ) = E g ( S , ∞ ) and E f ( k ) ( S ,
7) = E g ( k ) ( S , then f ( k ) ≡ g ( k ) . Theorem D. [3]
Let S i , i = 1 , , be defined as in Theorem B and k be a positive integer.If f and g are two non-constant meromorphic functions such that E f ( k ) ( S ,
5) = E g ( k ) ( S , , E f ( S , ∞ ) = E g ( S , ∞ ) and E f ( k ) ( S ,
1) = E g ( k ) ( S , then f ( k ) ≡ g ( k ) . Theorem E. [3]
Let S i , i = 1 , , be defined as in Theorem B and k be a positive integer.If f and g are two non-constant meromorphic functions such that E f ( k ) ( S ,
6) = E g ( k ) ( S , , E f ( S , ∞ ) = E g ( S , ∞ ) and E f ( k ) ( S ,
0) = E g ( k ) ( S , then f ( k ) ≡ g ( k ) . A few years latter in 2011 Banerjee and Bhattacharjee [4] further improved the above resultsin the following manner.
Theorem F. [4]
Let S i , i = 1 , , be defined as in Theorem B and k be a positive integer.If f and g are two non-constant meromorphic functions such that E f ( k ) ( S ,
5) = E g ( k ) ( S , , E f ( S , ∞ ) = E g ( S , ∞ ) and E f ( k ) ( S ,
0) = E g ( k ) ( S , then f ( k ) ≡ g ( k ) . Theorem G. [4]
Let S i , i = 1 , , be defined as in Theorem B and k be a positive integer.If f and g are two non-constant meromorphic functions such that E f ( k ) ( S ,
4) = E g ( k ) ( S , , E f ( S , ∞ ) = E g ( S , ∞ ) and E f ( k ) ( S ,
1) = E g ( k ) ( S , then f ( k ) ≡ g ( k ) . Theorem H. [4]
Let S i , i = 1 , , be defined as in Theorem B and k be a positive integer.If f and g are two non-constant meromorphic functions such that E f ( k ) ( S ,
5) = E g ( k ) ( S , , E f ( S ,
9) = E g ( S , and E f ( k ) ( S , ∞ ) = E g ( k ) ( S , ∞ ) then f ( k ) ≡ g ( k ) . In the present paper we we significantly reduce the weight of the range sets in all the abovetheorems. The following theorems are the main results of the paper:
Theorem 1.1.
Let S i , i = 1 , , be defined as in Theorem B and k be a positive integer. If f and g are two non-constant meromorphic functions such that E f ( k ) ( S , k ) = E g ( k ) ( S , k ) , E f ( S , k ) = E g ( S , k ) and E f ( k ) ( S , k ) = E g ( k ) ( S , k ) , where k ≥ , k ≥ , k ≥ areintegers satisfying k k k > k + k + 2 k + k − kk k − k k − kk + 3 , then f ( k ) ≡ g ( k ) . Remark 1.1.
Note that
Theorem 1.1 holds for k = 4 , k = 2 and k = 0 . So Theorem 1.1 improves
Theorems A-H . Remark 1.2.
Examples 1.2-1.4 assures the fact that in
Theorem 1.1 , n ≥ is the best possible. Though we follow the standard definitions and notations of the value distribution theoryavailable in [9], we explain some notations which are used in the paper.
Definition 1.3. [10]
For a ∈ C ∪ {∞} we denote by N ( r, a ; f | = 1) the counting function ofsimple a points of f . For a positive integer m we denote by N ( r, a ; f |≤ m )( N ( r, a ; f |≥ m )) the counting function of those a points of f whose multiplicities are not greater(less) than m where each a point is counted according to its multiplicity. N ( r, a ; f |≤ m ) ( N ( r, a ; f |≥ m )) are defined similarly, where in counting the a -points of f we ignore the multiplicities.Also N ( r, a ; f | < m ) , N ( r, a ; f | > m ) , N ( r, a ; f | < m ) and N ( r, a ; f | > m ) are definedanalogously. Definition 1.4.
We denote by N ( r, a ; f | = k ) the reduced counting function of those a -pointsof f whose multiplicities is exactly k , where k ≥ is an integer. A.BANERJEE, S.MAJUMDER AND B. CHAKRABORTY
Definition 1.5. [2]
Let f and g be two non-constant meromorphic functions such that f and g share ( a, k ) where a ∈ C ∪ {∞} . Let z be an a -point of f with multiplicity p , a a -point of g with multiplicity q . We denote by N L ( r, a ; f ) the counting function of those a -points of f and g where p > q ; each point in this counting functions is counted only once. In the same way wecan define N L ( r, a ; g ) . Definition 1.6. [12]
We denote N ( r, a ; f ) = N ( r, a ; f ) + N ( r, a ; f |≥ Definition 1.7. [11, 12]
Let f , g share a value a IM. We denote by N ∗ ( r, a ; f, g ) the reducedcounting function of those a -points of f whose multiplicities differ from the multiplicities of thecorresponding a -points of g .Clearly N ∗ ( r, a ; f, g ) ≡ N ∗ ( r, a ; g, f ) and N ∗ ( r, a ; f, g ) = N L ( r, a ; f ) + N L ( r, a ; g ) . Definition 1.8. [14]
Let a, b ∈ C ∪{∞} . We denote by N ( r, a ; f | g = b ) the counting functionof those a -points of f , counted according to multiplicity, which are b -points of g . Definition 1.9. [14]
Let a, b , b , . . . , b q ∈ C ∪{∞} . We denote by N ( r, a ; f | g = b , b , . . . , b q ) the counting function of those a -points of f , counted according to multiplicity, which are notthe b i -points of g for i = 1 , , . . . , q . Definition 1.10.
Let f and g be two non-constant meromorphic functions such that E f ( S, k ) = E g ( S, k ) . Let a and b be any two elements of S . We denote by N ∗ ( r, a ; f | g = b ) the reducedcounting function of those a -points of f whose multiplicities differ from the multiplicities of thecorresponding b -points of g .Clearly N ∗ ( r, a ; f | g = b ) = N ∗ ( r, b ; g | f = a ) . Also if a = b , then N ∗ ( r, a ; f | g = b ) = N ∗ ( r, a ; f, g ) . Lemmas
In this section we present some lemmas which will be needed in the sequel. Let F and G betwo non-constant meromorphic functions defined as follows.(2.1) F = (cid:0) f ( k ) (cid:1) n − ( f ( k ) + a ) − b , G = (cid:0) g ( k ) (cid:1) n − ( g ( k ) + a ) − b , where n ( ≥
2) and k are two positive integers.Henceforth we shall denote by H , Φ ,Φ and Φ the following three functions H = ( F ′′ F ′ − F ′ F − − ( G ′′ G ′ − G ′ G − , Φ = F ′ F − − G ′ G − , Φ = ( f ( k ) ) ′ f ( k ) − ( g ( k ) ) ′ g ( k ) and Φ = ( ( f ( k ) ) ′ f ( k ) − ω i − ( f ( k ) ) ′ f ( k ) ) − ( ( g ( k ) ) ′ g ( k ) − ω j − ( g ( k ) ) ′ g ( k ) ) , where ω i and ω j be any two roots of the equation z n + az n − + b = 0 Lemma 2.1. ( [12] , Lemma 1) Let F , G share (1 , and H . Then N ( r, F | = 1) = N ( r, G | = 1) ≤ N ( r, H ) + S ( r, F ) + S ( r, G ) . URTHER RESULTS ON UNIQUENESS OF DERIVATIVES OF MEROMORPHIC FUNCTIONS 5
Lemma 2.2.
Let S , S and S be defined as in Theorem 1.1 and F , G be given by (2.1). Iffor two non-constant meromorphic functions f and g E f ( k ) ( S ,
0) = E g ( k ) ( S , , E f ( k ) ( S ,
0) = E g ( k ) ( S , , E f ( S ,
0) = E g ( S , and H then N ( r, H ) ≤ N ∗ ( r, f ( k ) , g ( k ) ) + N ∗ ( r, F, G ) + N ( r, − a n − n ; f ( k ) ) + N ( r, − a n − n ; g ( k ) )+ N ∗ ( r, ∞ ; f, g ) + N ( r,
0; ( f ( k ) ) ′ ) + N ( r,
0; ( g ( k ) ) ′ ) , where N ( r,
0; ( f ( k ) ) ′ ) is the reduced counting function of those zeros of ( f ( k ) ) ′ which are notthe zeros of f ( k ) ( F − and N ( r,
0; ( g ( k ) ) ′ ) is similarly defined.Proof. Since E f ( k ) ( S ,
0) = E g ( k ) ( S ,
0) it follows that F and G share (1 , F ′ = [ nf ( k ) + ( n − a ]( f ( k ) ) n − ( f ( k ) ) ′ / ( − b )and G ′ = [ ng ( k ) + ( n − a ]( g ( k ) ) n − ( g ( k ) ) ′ / ( − b ) . We can easily verify that possible poles of H occur at (i) those zeros of f ( k ) and g ( k ) whosemultiplicities are distinct from the multiplicities of the corresponding zeros of g ( k ) and f ( k ) respectively, (ii)zeros of nf ( k ) + a ( n −
1) and ng ( k ) + a ( n − f and g whose multiplicities are distinct from the multiplicities of the corresponding poles of g and f respectively, (iv) those 1-points of F and G with different multiplicities, (v) zeros of ( f ( k ) ) ′ which are not the zeros of f ( k ) ( F − g ( k ) ) ′ which are not zeros of g ( k ) ( G − (cid:3) Lemma 2.3. [16]
Let f be a non-constant meromorphic function and let R ( f ) = n P k =0 a k f km P j =0 b j f j be an irreducible rational function in f with constant coefficients { a k } and { b j } where a n = 0 and b m = 0 Then T ( r, R ( f )) = dT ( r, f ) + S ( r, f ) , where d = max { n, m } . Lemma 2.4. [4]
Let F and G be given by (2.1). If f ( k ) , g ( k ) share (0 , and is not a Picardexceptional value of f ( k ) and g ( k ) . Then Φ ≡ implies F ≡ G . Lemma 2.5. [4]
Let F and G be given by (2.1), n ≥ be an integer and Φ . If F , G share (1 , k ) ; f , g share ( ∞ , k ) , and f ( k ) , g ( k ) share (0 , k ) , where ≤ k < ∞ then [( n − k + n − N ( r, f ( k ) |≥ k + 1) ≤ N ∗ ( r, F, G ) + N ∗ ( r, ∞ ; F, G ) + S ( r ) . Lemma 2.6.
Let f , g be two non-constant meromorphic functions, F , G be given by (2.1), n ≥ be an integer and Φ . If F , G share (1 , k ) and f ( k ) , g ( k ) share (0 , k ) ; f , g share ( ∞ , k ) , where ≤ k ≤ ∞ then k N ( r, F | ≥ k + 1) ≤ N ∗ ( r, f ( k ) , g ( k ) ) + N ∗ ( r, ∞ ; f, g ) + S ( r, f ( k ) ) + S ( r, g ( k ) ) . Proof.
Note that k N ( r, F | ≥ k + 1) ≤ N ( r,
0; Φ ) ≤ N ( r, Φ ) + S ( r, f ( k ) ) + S ( r, g ( k ) ) ≤ N ∗ ( r, f ( k ) , g ( k ) ) + N ∗ ( r, ∞ ; f, g ) + S ( r, f ( k ) ) + S ( r, g ( k ) ) . (cid:3) A.BANERJEE, S.MAJUMDER AND B. CHAKRABORTY
Lemma 2.7.
Let f , g be two non-constant meromorphic functions. Also let F , G be given by(2.1), n ≥ an integer and Φ , Φ . If F , G share (1 , k ) , where k ≥ , f ( k ) , g ( k ) share (0 , k ) and f , g share ( ∞ , k ) , ≤ k ≤ ∞ then N ( r, f ( k ) | ≥ k + 1) ≤ k + 1 k [( n − k + ( n − − N ∗ ( r, ∞ ; f, g ) + S ( r, f ( k ) ) + S ( r, g ( k ) ) , Similar result holds for g ( k ) .Proof. Using
Lemma 2.5 and
Lemma 2.6 and noting that N ∗ ( r, f ( k ) , g ( k ) ) ≤ N ( r, f ( k ) | ≥ k + 1) we see that[( n − k + ( n − N ( r, f ( k ) | ≥ k + 1) ≤ N ∗ ( r, F, G ) + N ∗ ( r, ∞ ; f, g ) + S ( r, f ( k ) ) + S ( r, g ( k ) ) ≤ k N ( r, f ( k ) | ≥ k + 1) + k + 1 k N ∗ ( r, ∞ ; f, g )+ S ( r, f ( k ) ) + S ( r, g ( k ) ) , from which the lemma follows. (cid:3) Lemma 2.8.
Let f and g be two non-constant meromorphic functions. Suppose f , g share ( ∞ , and ∞ is not an Picard exceptional value of f and g . Then Φ ≡ implies f ( k ) ≡ ω i ω j g ( k ) .Proof. Suppose Φ ≡
0. Then by integration we obtain1 − ω i f ( k ) ≡ A (1 − ω j g ( k ) ) , where A = 0. Since f , g share ( ∞ ,
0) it follows that A = 1 and hence f ( k ) ≡ ω i ω j g ( k ) . (cid:3) Lemma 2.9.
Let f and g be two non-constant meromorphic functions and Φ . Also let F and G be given by (2.1). If f ( k ) , g ( k ) share (0 , k ) ; f and g share ( ∞ , k ) , where ≤ k ≤ ∞ and E f ( k ) ( S , k ) = E g ( k ) ( S , k ) , where S is the same set as used in the Theorem 1.1 and ≤ k ≤ ∞ then ( k + k ) N ( r, ∞ ; f |≥ k + 1) ≤ N ∗ (cid:16) r, f ( k ) , g ( k ) (cid:17) + N ∗ ( r, F, G ) + S ( r ) . Similar expressions hold for g ( k ) also.Proof. If ∞ is an e.v.P of f ( k ) and g ( k ) then the assertion follows immediately.Next suppose ∞ is not an e.v.P of f ( k ) and g ( k ) . Since E f ( k ) ( S , k ) = E g ( k ) ( S , k ), it followsthat N ∗ ( r, ω i ; f ( k ) | g ( k ) = ω j ) ≤ N ∗ ( r, F, G ). Note that( k + k ) N ( r, ∞ ; f | ≥ k + 1)= ( k + k ) N ( r, ∞ ; g | ≥ k + 1) ≤ N ( r,
0; Φ ) ≤ N ( r, Φ ) + S ( r, f ( k ) ) + S ( r, g ( k ) ) ≤ N ∗ ( r, f ( k ) , g ( k ) ) + N ∗ ( r, ω i ; f ( k ) | g ( k ) = ω j ) + S ( r, f ( k ) ) + S ( r, g ( k ) ) ≤ N ∗ ( r, f ( k ) , g ( k ) ) + N ∗ ( r, F, G ) + S ( r ) . (cid:3) Lemma 2.10.
Let f , g be two non-constant meromorphic functions and Φ , Φ .Also let F and G be given by (2.1).If f ( k ) , g ( k ) share (0 , k ) ; f and g share ( ∞ , k ) , where ≤ k < ∞ and F , G share (1 , k ) , where k > then N ( r, ∞ ; f |≥ k + 1) ≤ k + 1 k ( k + k ) − N ∗ (cid:16) r, f ( k ) , g ( k ) (cid:17) + S ( r ) . URTHER RESULTS ON UNIQUENESS OF DERIVATIVES OF MEROMORPHIC FUNCTIONS 7
Similar expressions hold for g ( k ) also.Proof. Using
Lemma 2.6 and
Lemma 2.9 and noting that N ∗ ( r, ∞ ; f, g ) ≤ N ( r, ∞ ; f | ≥ k + 1)we see that( k + k ) N ( r, ∞ ; f | ≥ k + 1) ≤ N ∗ ( r, F, G ) + N ∗ ( r, f ( k ) , g ( k ) ) + S ( r ) ≤ k N ( r, ∞ ; f | ≥ k + 1) + k + 1 k N ∗ ( r, f ( k ) , g ( k ) )+ S ( r, f ( k ) ) + S ( r, g ( k ) ) , from which the lemma follows. (cid:3) Lemma 2.11. [4]
Let F , G be given by (2.1) and H . If f ( k ) , g ( k ) share (0 , k ) ; f and g share ( ∞ , k ) , where ≤ k < ∞ and F , G share (1 , k ) , where ≤ k ≤ ∞ then { ( nk + nk + n ) − } N ( r, ∞ ; f |≥ k + 1) ≤ N ∗ (cid:16) r, f ( k ) , g ( k ) (cid:17) + N (cid:16) r, f ( k ) + a (cid:17) + N (cid:16) r, g ( k ) + a (cid:17) + N ∗ ( r, F, G ) + S ( r ) . Similar expressions hold for g also. Lemma 2.12.
Let f , g be two non-constant meromorphic functions. Also let F , G be givenby (2.1), n ≥ an integer and Φ , Φ and Φ . If F , G share (1 , k ) ; f ( k ) , g ( k ) share (0 , k ) and f , g share ( ∞ , k ) , where k > , k ≥ and k ≥ are integers satisfying k k k > k + k + 2 k + k − kk k − k k − kk + 3 , then N ( r, F | ≥ k + 1) + N ( r, ∞ ; f | ≥ k + 1) + N ( r, f ( k ) | ≥ k + 1) = S ( r ) . Proof.
Since Φ Lemma 2.5 we get(2 k + 1) N ( r, f ( k ) | ≥ k + 1) ≤ N ( r, F |≥ k + 1) + N ( r, ∞ ; f | ≥ k + 1) + S ( r ) . Again since Φ Lemmas 2.6 , 2.9 respectively k N ( r, F | ≥ k + 1) ≤ N ( r, f ( k ) |≥ k + 1) + N ( r, ∞ ; f | ≥ k + 1) + S ( r ) , and( k + k ) N ( r, ∞ ; f | ≥ k + 1) ≤ N ( r, F |≥ k + 1) + N ( r, f ( k ) | ≥ k + 1) + S ( r ) . Using the above inequalities and following the same procedure as done in
Lemma 2.6 [19]the rest of the lemma can be proved. So we omit the details. (cid:3)
Lemma 2.13. [12] If N ( r, f ( k ) | f = 0) denotes the counting function of those zeros of f ( k ) which are not the zeros of f , where a zero of f ( k ) is counted according to its multiplicity then N ( r, f ( k ) | f = 0) ≤ kN ( r, ∞ ; f ) + N ( r, f | < k ) + kN ( r, f |≥ k ) + S ( r, f ) . Lemma 2.14.
Let F , G be given by (2.1), F , G share (1 , k ) , ≤ k ≤ ∞ and Φ and n ≥ . Also f ( k ) , g ( k ) share (0 , k ) and f , g share ( ∞ , ∞ ) . Then N ( r, f ( k ) ) ≤ k ( n − − N ( r, ∞ ; f ) + S ( r, f ( k ) ) . A.BANERJEE, S.MAJUMDER AND B. CHAKRABORTY
Proof.
Using
Lemma 2.3 and
Lemma 2.13 we see that N ∗ ( r, F, G ) ≤ N ( r, F |≥ k + 1) ≤ k (cid:0) N ( r, F ) − N ( r, F ) (cid:1) ≤ k [ n X j =1 (cid:16) N ( r, ω j ; f ( k ) ) − N ( r, ω j ; f ( k ) ) (cid:17) ] ≤ k (cid:16) N ( r,
0; ( f ( k ) ) ′ | f ( k ) = 0) (cid:17) ≤ k h N ( r, f ( k ) ) + N ( r, ∞ ; f ) i + S ( r, f ( k ) ) , where ω , ω . . . ω n are the distinct roots of the equation z n + az n − + b = 0. Rest of the prooffollows from the Lemma 2.5 for k = 0.This proves the lemma. (cid:3) Lemma 2.15.
Let F , G be given by (2.1), F , G share (1 , k ) , ≤ k ≤ ∞ and Φ and n ≥ . Also f ( k ) , g ( k ) share (0 , k ) and f , g share ( ∞ , ∞ ) , where ≤ k ≤ ∞ . Then N L ( r, F ) ≤ k ( n − k + 1)[ k ( n − − N ( r, ∞ ; f ) + S ( r, f ( k ) ) . Similar expression holds for G also.Proof. Using
Lemma 2.3 and
Lemma 2.13 we see that N L ( r, F ) ≤ N ( r, F |≥ k + 2) ≤ k + 1 (cid:0) N ( r, F ) − N ( r, F ) (cid:1) ≤ k + 1 h N ( r, f ( k ) ) + N ( r, ∞ ; f ) i + S ( r, f ( k ) ) . Now using
Lemma 2.14 the rest of the lemma can be easily proved. So we omit it. (cid:3)
Lemma 2.16. [1]
Let f and g be two non-constant meromorphic functions sharing (1 , k ) ,where ≤ k ≤ ∞ . Then N ( r, f | = 2) + 2 N ( r, f | = 3) + . . . + ( k − N ( r, f | = k ) + k N L ( r, f )+( k + 1) N L ( r, g ) + k N ( k +1 E ( r, g ) ≤ N ( r, g ) − N ( r, g ) . Lemma 2.17.
Let F , G be given by (2.1) and they share (1 , k ) . If f ( k ) , g ( k ) share (0 , k ) and f , g share ( ∞ , k ) , where ≤ k ≤ ∞ and H . nT ( r, f ( k ) ) ≤ N ( r, ∞ ; f ) + N ( r, − a n − n ; f ( k ) ) + N ( r, ∞ ; g ) + N ( r, − a n − n ; g ( k ) )+ N ( r, f ( k ) ) + N ( r, g ( k ) ) + N ∗ ( r, f ( k ) , g ( k ) ) + N ∗ ( r, ∞ ; f, g ) − ( k − N ∗ ( r, F, G ) + N L ( r, F ) + S ( r, f ( k ) ) + S ( r, g ( k ) ) . Similar result holds for g ( k ) . URTHER RESULTS ON UNIQUENESS OF DERIVATIVES OF MEROMORPHIC FUNCTIONS 9
Proof.
Using
Lemma 2.13 and
Lemma 2.16 we see that N ( r,
0; ( g ( k ) ) ′ ) + N ( r, F |≥
2) + N ∗ ( r, F, G )(2.2) ≤ N ( r,
0; ( g ( k ) ) ′ ) + N ( r, F | = 2) + N ( r, F | = 3) + . . . + N ( r, F | = k )+ N ( k +1 E ( r, F ) + N L ( r, F ) + N L ( r, G ) + N ∗ ( r, F, G ) ≤ N ( r,
0; ( g ( k ) ) ′ ) − N ( r, F | = 3) − . . . − ( k − N ( r, F | = k ) − ( k − N L ( r, F ) − k N L ( r, G ) − ( k − N ( k +1 E ( r, F ) + N ( r, G ) − N ( r, G ) + N ∗ ( r, F, G ) ≤ N ( r,
0; ( g ( k ) ) ′ ) + N ( r, G ) − N ( r, G ) − ( k − N L ( r, F ) − ( k − N L ( r, G ) ≤ N ( r,
0; ( g ( k ) ) ′ ) + N ( r, G ) − N ( r, G ) − ( k − N L ( r, F ) − ( k − N L ( r, G ) ≤ N ( r,
0; ( g ( k ) ) ′ | g ( k ) = 0) − ( k − N L ( r, F ) − ( k − N L ( r, G ) ≤ N ( r, g ( k ) ) + N ( r, ∞ ; g ) − ( k − N L ( r, F ) − ( k − N L ( r, G )= N ( r, g ( k ) ) + N ( r, ∞ ; g ) − ( k − N ∗ ( r, F, G ) + N L ( r, F ) , where N ( r,
0; ( g ( k ) ) ′ ) has the same meaning as in the Lemma 2.2 . Hence using (2.2),
Lemmas2.1 , and we get from second fundamental theorem that n T ( r, f ( k ) )(2.3) ≤ N ( r, f ( k ) ) + N ( r, ∞ ; f ) + N ( r, F | = 1) + N ( r, F |≥ − N ( r,
0; ( f ( k ) ) ′ )+ S ( r, f ( k ) ) ≤ N ( r, f ( k ) ) + N ( r, ∞ ; f ) + N ( r, − a n − n ; f ( k ) ) + N ( r, − a n − n ; g ( k ) )+ N ∗ ( r, f ( k ) , g ( k ) ) + N ∗ ( r, ∞ ; f, g ) + N ∗ ( r, F, G ) + N ( r, F | ≥
2) + N ( r,
0; ( g ( k ) ) ′ )+ S ( r, f ( k ) ) + S ( r, g ( k ) ) ≤ N ( r, ∞ ; f ) + N ( r, − a n − n ; f ( k ) ) + N ( r, ∞ ; g ) + N ( r, − a n − n ; g ( k ) ) + N ( r, f ( k ) )+ N ( r, g ( k ) ) + N ∗ ( r, ∞ ; f, g ) + N ∗ ( r, f ( k ) , g ( k ) ) − ( k − N ∗ ( r, F, G ) + N L ( r, F )+ S ( r, f ( k ) ) + S ( r, g ( k ) ) . This proves the Lemma. (cid:3)
Lemma 2.18. [4]
Let F , G be given by (2.1), n ≥ and they share (1 , k ) . If f ( k ) , g ( k ) share (0 , , and f , g share ( ∞ , k ) and H ≡ . Then f ( k ) ≡ g ( k ) . Proofs of the theorem
Proof of Theorem 1.1.
Let F , G be given by (2.1). Then F and G share (1 , k ), ( ∞ ; k ). Weconsider the following cases. Case 1.
Let H
0. Clearly F G and so f ( k ) g ( k ) . Subcase 1.1:
Let Φ Subcase 1.1.1:
Suppose Φ f ( k ) and g ( k ) . Then by Lemma 2.4 we get Φ
0. Since f ( k ) and g ( k ) share (0 , k ) it follows that N ∗ ( r, f ( k ) , g ( k ) ) ≤ N ( r, f ( k ) ). Now successively using Lemmas 2.17 , for k = 0, for k = 0 and we obtain nT ( r, f ( k ) )(3.1) ≤ N ( r, ∞ ; f ) + N ( r, − a n − n ; f ( k ) ) + N ( r, ∞ ; g ) + N ( r, − a n − n ; g ( k ) )+2 N ( r, f ( k ) ) + N ∗ ( r, f ( k ) , g ( k ) ) + N ∗ ( r, ∞ ; f, g ) − ( k − N ∗ ( r, F, G )+ N L ( r, F ) + S ( r, f ( k ) ) + S ( r, g ( k ) ) ≤ N ( r, − a n − n ; f ( k ) ) + N ( r, − a n − n ; g ( k ) ) + 3 N ( r, f ( k ) ) + 2 N ( r, ∞ ; f )+ N ∗ ( r, ∞ ; f, g ) + S ( r, f ( k ) ) + S ( r, g ( k ) ) ≤ N ( r, − a n − n ; f ( k ) ) + N ( r, − a n − n ; g ( k ) ) + 3 k + 3( n − k − N ( r, ∞ ; f | ≥ k + 1)+ N ( r, ∞ ; f | ≥ k + 1) + 2 k + 2 k k − N ( r, f ( k ) | ≥ k + 1) + S ( r, f ( k ) ) + S ( r, g ( k ) ) ≤ T ( r, f ( k ) ) + T ( r, g ( k ) ) + S ( r, f ( k ) ) + S ( r, g ( k ) ) ≤ T ( r ) + S ( r ) . Next suppose 0 is an e.v.P. of f ( k ) and g ( k ) . Then N ( r, f ( k ) ) = S ( r, f ( k ) ).Suppose that Φ
0. Then by
Lemma 2.10 for k = 0 we get N ( r, ∞ ; f ) = S ( r ). So N ∗ ( r, ∞ ; f, g ) = S ( r ). Consequently (3.1) holds.Next assume Φ ≡
0. Then ( F − ≡ d ( G − d = 0 ,
1. Since f and g share ( ∞ , k ),it follows that f , g share ( ∞ , ∞ ) which implies N ∗ ( r, ∞ ; f, g ) = S ( r ). Also by Lemma 2.10 for k = 0 we get N ( r, ∞ ; f ) = S ( r ). Clearly in this case also (3.1) holds.In a similar manner as above we can obtain nT ( r, g ( k ) ) ≤ T ( r ) + S ( r ) . (3.2)Combining (3.1) and (3.2) we get ( n − T ( r ) ≤ S ( r ) , (3.3)which leads to a contradiction for n ≥ Subcase 1.1.2:
Suppose Φ ≡
0. Then by integration we obtain1 − ω i f ( k ) ≡ A (1 − ω j g ( k ) ) , where A = 0. If A = 1 then f ( k ) = ω i ω j g ( k ) , which contradicts Φ
0. So A = 0 ,
1. Since f and g share ( ∞ , k ), it follows that N ( r, ∞ ; f ) = S ( r, f ( k ) ) and N ( r, ∞ ; g ) = S ( r, g ( k ) ). Nowproceeding in the same way as done in the Subcase 1.1.1 we can arrive at a contradiction.
Subcase 1.2:
Let Φ ≡ f ( k ) ≡ cg ( k ) , where c = 0 ,
1. Since f ( k ) and g ( k ) share (0 , k ) and f , g share ( ∞ , k ), it follows that N ∗ ( r, f ( k ) , g ( k ) ) = 0 and N ∗ ( r, ∞ ; f, g ) = 0. Subcase 1.2.1
Suppose Φ f ( k ) and g ( k ) then by Lemma 2.4 we get Φ
0. Now consecutively
URTHER RESULTS ON UNIQUENESS OF DERIVATIVES OF MEROMORPHIC FUNCTIONS 11 using
Lemmas 2.17, 2.14, 2.9 for k = 0, and we obtain nT ( r, f ( k ) )(3.4) ≤ N ( r, ∞ ; f ) + N ( r, − a n − n ; f ( k ) ) + N ( r, ∞ ; g ) + N ( r, − a n − n ; g ( k ) )+2 N ( r, f ( k ) ) + N ∗ ( r, f ( k ) , g ( k ) ) + N ∗ ( r, ∞ ; f, g ) − ( k − N ∗ ( r, F, G )+ N L ( r, F ) + S ( r, f ( k ) ) + S ( r, g ( k ) ) ≤ N ( r, − a n − n ; f ( k ) ) + N ( r, − a n − n ; g ( k ) ) + 2 N ( r, ∞ ; f ) + 2 k ( n − − N ( r, ∞ ; f ) − ( k − N ∗ ( r, F, G ) + N L ( r, F ) + S ( r, f ( k ) ) + S ( r, g ( k ) ) ≤ N ( r, − a n − n ; f ( k ) ) + N ( r, − a n − n ; g ( k ) ) + 3 N ( r, ∞ ; f ) − ( k − N ∗ ( r, F, G )+ N L ( r, F ) + S ( r, f ( k ) ) + S ( r, g ( k ) ) ≤ T ( r ) + 3 k N ∗ ( r, F, G ) − ( k − N ∗ ( r, F, G ) + N L ( r, F ) + S ( r, f ( k ) ) + S ( r, g ( k ) ) ≤ T ( r ) + k ( n − k + 1)[ k ( n − − N ( r, ∞ ; g ) + S ( r, f ( k ) ) + S ( r, g ( k ) ) ≤ (cid:18) k ( n − k + 1)( k + 1)[ k ( n − − (cid:19) T ( r ) + S ( r ) . That is (cid:18) n − − k ( n − k + 1)( k + 1)[ k ( n − − (cid:19) T ( r ) ≤ S ( r ) . (3.5)Since n ≥
3, (3.5) leads to a contradiction.Suppose 0 is an e.v.P. of f ( k ) and g ( k ) . Then Lemma 2.9 for k = 0 we get N ( r, ∞ ; f ) = k N ∗ ( r, F, G ). Proceeding as above in this case also we arrive at a contradiction.
Subcase 1.2.2:
Suppose Φ ≡ ∞ is not an e.v.P. of f and g . Since f ( k ) and g ( k ) share (0 , k ) and f , g share ( ∞ , k ),from Lemma 2.8 it follows that N ∗ ( r, f ( k ) , g ( k ) ) = 0 and N ∗ ( r, ∞ ; f, g ) = 0.Suppose 0 is not an e.v.P of f ( k ) and g ( k ) then by Lemma 2.4 we get Φ
0. Nowconsecutively using
Lemmas 2.17, 2.5 for k = 0, for k = 0 we obtain nT ( r, f ( k ) )(3.6) ≤ N ( r, ∞ ; f ) + N ( r, − a n − n ; f ( k ) ) + N ( r, ∞ ; g ) + N ( r, − a n − n ; g ( k ) )+2 N ( r, f ( k ) ) + N ∗ ( r, f ( k ) , g ( k ) ) + N ∗ ( r, ∞ ; f, g ) − ( k − N ∗ ( r, F, G )+ N L ( r, F ) + S ( r, f ( k ) ) + S ( r, g ( k ) ) ≤ N ( r, − a n − n ; f ( k ) ) + N ( r, − a n − n ; g ( k ) ) + 2 N ( r, ∞ ; f ) + 2 N ∗ ( r, F, G ) − ( k − N ∗ ( r, F, G ) + N L ( r, F ) + S ( r, f ( k ) ) + S ( r, g ( k ) ) ≤ N ( r, − a n − n ; f ( k ) ) + N ( r, − a n − n ; g ( k ) ) − ( k − N ∗ ( r, F, G ) + N L ( r, F )+ 2 nk + n − { N ( r, f ( k ) + a ) + N ( r, g ( k ) + a ) + N ∗ ( r, F, G ) } + S ( r, f ( k ) ) + S ( r, g ( k ) ) ≤ T ( r ) + 4 nk + n − T ( r ) + 25 N L ( r, F ) + S ( r, f ( k ) ) + S ( r, g ( k ) ) ≤ (cid:18) nk + n − k ( n − k + 1)[ k ( n − − (cid:19) T ( r ) + S ( r ) . That is (cid:18) n − − nk + n − − k ( n − k + 1)[ k ( n − − (cid:19) T ( r ) ≤ S ( r ) . (3.7)Since n ≥
3, (3.7) leads to a contradiction.If 0 is an e.v.P. of f ( k ) and g ( k ) then with the help of Lemmas 2.17 and 2.11 for k = 0 andproceeding as above we arrive at a contradiction.If ∞ is an e.v.P. of f and g then proceeding as in th Subcase 1.2.1 we can arrive at a contra-diction.
Case 2.
Let H ≡
0. Then the theorem follows from
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E-mail address : abanerjee [email protected], abanerjee [email protected] Department of Mathematics, Katwa College, Burdwan, India. Department of Mathematics, Raiganj University, Raiganj, India
E-mail address : [email protected], [email protected] Department of Mathematics, University of Kalyani, West Bengal, India. Department of Mathematics, Ramakrishna Mission Vivekananda Centenary College, Rahara,India
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