Unique Range Set for Meromorphic Functions with Deficient Values
We employ the idea of weighted sharing of sets to find a unique range set for meromorphic functions with deficient values. Our result improves, generalises, and extends the result of Lahiri. Examples are exhibited that a condition in one of our results is the best possible one.
1. Introduction Definitions and Results
In this paper by meromorphic functions we will always mean meromorphic functions in the complex plane. It will be convenient to let denote any set of positive real numbers of finite linear measure, not necessarily the same at each occurrence. For any nonconstant meromorphic function we denote by any quantity satisfying
We denote by the maximum of and . The notation denotes any quantity satisfying as , .
We adopt the standard notations of the Nevanlinna theory of meromorphic functions as explained in . For , we define Let and be two nonconstant meromorphic functions, and let be a finite complex number. We say that and share CM, provided that and have the same zeros with the same multiplicities. Similarly, we say that and share IM, provided that and have the same zeros ignoring multiplicities. In addition we say that and share CM, if and share CM, and we say that and share IM, if and share IM.
Let be a set of distinct elements of and , where each point is counted according to its multiplicity. Denote by the reduced form of . If , we say that and share the set CM. On the other hand if , we say that and share the set IM. Evidently, if contains only one element, then it coincides with the usual definition of CM (resp., IM) shared values.
Let a set and and be two nonconstant meromorphic (entire) functions. If implies , then is called a unique range set for meromorphic (entire) functions or in brief URSM (URSE). We will call any set a unique range set for meromorphic functions ignoring multiplicity (URSM-IM) for which implies for any pair of nonconstant meromorphic functions.
Inspired by Nevanlinna’s 5 and 4 value theorem, in [2, 3], Gross raised the problem of finding out a finite set so that an entire function in the complex plane is determined by the preimage of , where each pre-image of related to some entire function is counted according to its multiplicity.
In 1982 Gross and Yang  proved the following theorem.
Theorem A. Let . If two entire functions and satisfy , then .
Noting that the set in Theorem A is an infinite set, we know that Theorem A does not give a solution to the problem of Gross.
In 1994 Yi  exhibited a URSE with 15 elements, and in 1995 Li and Yang  exhibited a URSM with 15 elements and a URSE with 7 elements. Since then, the quest for minimum cardinality for a URSM created an increasing interest among the researchers.
By now the URSM with 11 elements is the smallest available URSM obtained by Frank and Reinders . It is also seen that the finite URSMs are the set of distinct zeros of some suitable polynomials.
A polynomial in is called a strong uniqueness polynomial for meromorphic (entire) functions if, for any nonconstant meromorphic (entire) functions and , implies , where is a suitable nonzero constant. We say that is SUPM (SUPE) in brief. On the other hand for a polynomial in , if the condition implies for any nonconstant meromorphic (entire)s function and , then is called a uniqueness polynomial for meromorphic (entire) functions. We say that is a UPM (UPE) in brief.
Suppose that is a polynomial of degree in having only simple zeros and that is the set of all zeros of . If is a URSM (URSE), then from the definition it follows that is UPM (UPE). However the converse is not, in general, true. For () is clearly a UPM, but for and we see that , where is the set of zeros of .
To find under which condition the converse is true Fujimoto  first invented a special property of a polynomial, which he called the property (). Fujimoto’s property () may be stated as follows: a polynomial is said to satisfy the property () if for any two distinct zeros , of the derivative .
Fujimoto found a sufficient condition for a set of zeros of a SUPM (SUPE) to be a URSM (URSE) as follows.
Theorem B (see ). Let be polynomial of degree in having only simple zeros satisfying the condition (). Let have distinct zeros, and let either or and have no simple zero. Further suppose that is a SUPM (SUPE). If is the set of zeros of and (), then is a URSM (URSE).
Answers to the question of Gross [2, 3] and the analogous question for meromorphic functions on were given by Yi  and Li and Yang [6, 10] who investigated the zero sets of polynomials of the form where and and are so chosen so that has distinct roots. When , then there is no problem regarding property as in  it has been shown that the zero set of is a URSM, and hence is a UPM. But whenever , is not satisfying property as here has two distinct zeros and one of them is simple. So natural question would be whether, for , the zero set of can be a URSM or even be a URSE.
Theorem C. Let . If and are two nonconstant entire functions satisfying , then .
Clearly is a UPE. So it will be interesting to investigate the analogous results for meromorphic functions for more general polynomial, namely, . In this perspective in 1996 Yi proved the following theorem.
Theorem D (see ). Let , where is an integer and and are two nonzero constants such that the algebraic equation has no multiple roots. If and are nonconstant meromorphic functions satisfying , then either or , , where .
Theorem D is not much significant as it do not commensurate with the aforesaid discussion. That is to say under the supposition of Theorem D, cannot be a URSM.
In the meantime Fang and Hua  extended Theorem C to meromorphic functions with the help of some additional conditions on the ramification indexes of and . Fang and Hua  proved the following theorem.
Theorem E (see ). Let be defined as in Theorem C. If two meromorphic functions and are such that , , and , then .
We now require the following definition known as weighted sharing of sets and values which renders a useful tool for the purpose of relaxation of the nature of sharing the sets.
Definition 1 (see [13, 14]). Let be a nonnegative integer or infinity. For one denotes by the set of all -points of , where an -point of multiplicity is counted times if and times if . If , one says that and share the value with weight .
We write that and share to mean that and share the value with weight . Clearly if and share , then and share for any integer , . Also we note that , share a value IM or CM if and only if and share or , respectively.
Definition 2 (see ). Let be a set of distinct elements of and a nonnegative integer or . One denotes by the set . Clearly and .
Definition 3 (see ). For one denotes by the counting function of simple -points of . For a positive integer one denotes by the counting function of those -points of whose multiplicities are not greater (less) than , where each -point is counted according to its multiplicity.
are defined similarly, where in counting the -points of we ignore the multiplicities.
Also , , , and are defined analogously.
We define where .
Lahiri  improved Theorem E in the following direction.
Theorem F (see ). Let be defined as in Theorem C. If, for two nonconstant meromorphic functions and , and , then .
In 2004 Lahiri and Banerjee  further improved Theorem C in a more compact and convenient way and obtained the following result.
Theorem G (see ). Let , where is an integer and , are two nonzero constants such that has no multiple root. If and , then .
So we observe that deficiencies of poles play a vital role in order to find the sufficient condition for which the conclusion of Theorems D and F-G holds well. The first motivation of the paper is to show that even the deficiencies of other values contribute significantly for the conclusion of Theorems F and G. We now give the following example which establishes the fact that the set in Theorems F-G can not be replaced by any arbitrary set containing six distinct elements.
Example 4. Let , , and , where , , and are three nonzero distinct complex numbers. Clearly , but .
So it remains an open problem for investigations whether the degree of the equation defining in Theorem G can be reduced to six. This is the second motivation of writing the paper. In short we will further improve, generalize, and extend Theorems F and G.
The following theorem is the main result of the paper.
Theorem 5. Let , where is an integer and , are two nonzero constants such that has no multiple roots. Suppose that and are two nonconstant meromorphic functions satisfying . If (i) and (ii)or and (iii)or and , then , where and can be similarly defined.
The following examples show that the condition is sharp in Theorem 5 when and .
Example 6 (Example 2, ). Let and , where , , and is an integer.
Then , , and . Further we see that and so for any complex number , . We also note that a root of is neither a pole nor a zero of and . Hence . On the other hand and , where . Also and . Therefore . Clearly because , but .
Example 7. Let and be given as in Example 6, where , , and is an integer.
We now explain some definitions and notations which are used in the paper.
Definition 8 (see ). Let and be two nonconstant meromorphic functions such that and share . Let be an -point of with multiplicity , an -point of with multiplicity . One denotes by the reduced counting function of those -points of and , where , by the counting function of those -points of and , where , and by the reduced counting function of those -points of and , where . In the same way one can define , , and . In a similar manner one can define and for .
When and share , , then .
Definition 9. One denotes by the reduced counting function of those -points of whose multiplicities are exactly , where is an integer.
Definition 10 (see [13, 14]). Let , share a value IM. One denotes by the reduced counting function of those -points of whose multiplicities differ from the multiplicities of the corresponding -points of .
Clearly and .
In this section we present some lemmas which will be needed in the sequel. Let and be two nonconstant meromorphic functions defined as follows:
Henceforth we will denote by and the following two functions:
Lemma 11 (see ). Let be a nonconstant meromorphic function, and let be an irreducible rational function in with constant coefficients and , where and . Then where .
Lemma 12 (see ). If and are two nonconstant meromorphic functions such that they share and , then
Lemma 13. Let , where and are nonzero constants such that has no repeated root, is an integer, and and are given by (5). If for two nonconstant meromorphic functions and and , then where is the reduced counting function of those zeros of which are not the zeros of , and is similarly defined.
Proof. Since , it follows that and share . We can easily verify that possible poles of occur at (i) zeros of and , (ii) multiple zeros of and , (iii) poles of and , (iv) those 1-points of and with different multiplicities, (v) zeros of which are not the zeros of , and (vi) zeros of which are not zeros of . Since has only simple poles, the lemma follows from above calculations. This proves the lemma.
Lemma 14 (see ). Let , be two nonconstant meromorphic functions. Then , where and are nonzero finite constants, and is an integer.
Lemma 15. Let , be two nonconstant meromorphic functions such that ; then implies , where is an integer and is a nonzero finite constant.
And suppose . We consider two cases.
Case I. Let be a constant. Then from (11) it follows that , , , and , a constant, which is impossible.
Case II. Let be nonconstant. Then From (2) we see by Lemma 11 that
We first note that the zeros of contribute to the zeros of both and . In addition to this the poles of contribute to the zeros of , and, since the zeros of contribute to the zeros of . So from (2) we see that where for and for .
Also since , from (2) we can find that .
By the second fundamental theorem we get That is, where .
Again putting , noting that , and proceeding in the same way as done in the above explanation we get that Adding (16) and (17) we get which is a contradiction.
Hence , and this proves the lemma.
Lemma 16 (see ). If denotes the counting function of those zeros of which are not the zeros of , where a zero of is counted according to its multiplicity, then
3. Proofs of the Theorems
Proof of Theorem 5. We know from the assumption that the zeros of are simple, and we denote them by , . Let , be given by (5) and (6). Since , it follows that and share .
Case 1. If possible let us suppose that .
Subcase 1.1. Consider . While , using Lemma 16 we note that Hence using (20) and Lemmas 11, 12 and 13 we get from the second fundamental theorem for that where and are defined in Theorem 5.
In a similar way we can obtain Combining (21) and (3) we see that Since , (23) leads to a contradiction.
While , using Lemma 16, (20) changes to So using (24) and Lemmas 12 and 13 and proceeding as in (21) we get from the second fundamental theorem for that Similarly we can obtain Combining (25) and (3) we see that Since , (27) leads to a contradiction.
Subcase 1.2. Consider . Using Lemma 16 we note that Hence using (28) and Lemmas 12 and 13 we get from the second fundamental theorem for that In a similar manner we can obtain Combining (29) and (3) we see that Since , (31) leads to a contradiction.
Case 2. Take . On integration we get from (6) where and are constants and . From (32) we obtain Clearly (33) together with Lemma 11 yields
Subcase 2.1. Suppose that .
If , from (33) we obtain From above calculations, Lemma 11, and the second fundamental theorem we obtain which in view of (34) implies a contradiction as . Thus , and hence (33) reduces to From this we have Again by Lemma 11 and the second fundamental theorem we have which in view of (34) leads to a contradiction since .
Subcase 2.2. Suppose that .
From (33) we obtain If , from (40) we obtain So using the same argument as used in the above subcase we can again obtain a contradiction. Hence , and we have from (40) that that means , which is impossible by Lemma 14.
Subcase 2.3. Suppose that .
From (33) we obtain If , from (42) we obtain So in the same manner as above calculations we again get a contradiction. So , and hence ; that is, . Now the theorem follows from Lemma 15.
The authors wish to thank the referee for his/her valuable remarks and suggestions. Abhijit Banerjee is thankful to DST-PURSE programme for financial assistance.
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