The Uniqueness of Differential-Difference Polynomials of Meromorphic Functions Sharing A Small Function

The Uniqueness of Differential-Difference Polynomials of Meromorphic Functions Sharing A Small Function

Somalatha M.T (1) , Shilpa N (2) and Y. Veeranna (3)

(1) Department of Mathematics, Government Science College, Bangalore – 560 001, Karnataka, India.

(2) Department of Mathematics, Presidency University, Itgalpura, Bangalore, 560 064, Karnataka, India.

(3) Department of Mathematics, Government Science College, Bangalore – 560 001, Karnataka, India.

Abstract – In this article, we establish the uniqueness outcomes of a meromorphic function in which a small function is shared through Differential Difference polynomials. We use second order difference operator 2 () =

and Utilizing a non-constant differential polynomial 

we aim to establish the uniqueness of two transcendental meromorphic functions with zero order. To achieve this, we will apply Leibnitz’s theorem to substantiate the obtained result.

Keywords: Entire function, Meromorphic function, Uniqueness, Shift operator, Difference polynomial, Differential polynomial.

Subject Code: MSC 30D35

1 INTRODUCTION

The Nevanlinna theory of value distribution focuses on the distribution density of points where a meromorphic function attains a specific value in the complex plane. For standard definitions and concepts in Nevanlinna theory, one can refer to works by W. K. Hayman [9] and C. C. Yang [15]. Let () and () be two meromorphic functions in the complex plane that are not constant. For {}, we say that () and () are if and have the same zeros and and share if multiplicity is not counted.

The amalgamation of complex derivatives and complex differences results in a complex differential-difference polynomial. Nevanlinna extended classical results on the distribution of a-points from entire functions to meromorphic functions. A differential-difference polynomial comprises terms involving the function f(z), its shifts, and the derivatives of these shifts.

Numerous research articles have delved into the study of entire and meromorphic functions, particularly focusing on scenarios where differential polynomials exhibit sharing of specific values, small functions, or fixed points. Mathematicians worldwide have contributed to this body of work, as evidenced by publications such as [2, 4, 12, 13].

Contemporary research efforts have witnessed a significant number of scholars exploring and attaining uniqueness results pertaining to difference polynomials. In 2006, R. G. Halburd and R. J. Korhonen [6] introduced a version of Nevanlinna theory grounded in difference operators, accompanied by the formulation of the difference logarithmic derivative lemma [7], thus advancing the development of the difference analogue of Nevanlinna theory. Chiang and Feng [3] have independently played a crucial role in contemplating the difference analogues of this theory. Consequently, mathematicians worldwide have directed their focus towards scrutinizing the distribution of zeros in various types of difference polynomials.

In 2007, I. Laine and C.C Yang [11] gave the result that If n2, then expression ()( + ) for a finite-order transcendental entire function f(z) and a non-zero complex constant takes on any non-zero value infinitely often.

In 2010, J. Zhang [16] explored the zeros of difference polynomials and derived the result that for every integer n2, the expression ()(() 1)( + ) () has infinitely many zeros if () is a transcendental entire function with finite order, ()( 0) is a small function with respect to (), and is a non-zero complex constant.

In the same paper, the author also established the following result regarding uniqueness.

Let () be the small function with regard to both () and () whose are transcendental entire functions with finite order. Assuming c to be a non-zero complex constant with 7, () () if ()(() 1)( + ) and ()(() 1)( + ) share () CM.

Recently, the uniqueness results on difference polynomial of entire functions of the form ()(() 1)()( + ) and ()(() 1)()( + ) as well as shift difference polynomial of meromorphic functions of the form ()(( + )() 1 and ()( + )() 1 have been investigated by R.S Dyavanal [4] and Q. Zhao et.al [18] respectively.

To improve the above result, the author N.V Thin[14] in 2017, gave a new conception(idea) for the unicity

theorems by considering qdifference polynomial of meromorphic function. In this paper, we prove the result.

Inspired by the extensive research in this field, we have demonstrated the following results by extending the findings of N.V. Thin and Renukadevi S. Dyavanal, among others, focusing on the differential-difference polynomial.

Theorem 1.1: Consider () and () as two transcendental meromorphic (or entire) functions of zero order, such that 2 () 0 and 2 () 0 where is nonzero complex constant. , , are positive integers. () be a small function and let () = () + 1(1) + + 1(1) + 0be a nonconstant differential polynomial with constant coefficients 0, 1, 1, ( 0) and be the distinct zeros of (). If 

14. and [() 2 ()]() and [() 2 ()]() share (), CM. Then, one of the two cases listed below is

true.

1. for a constant with = 1 when = {; = 0,1, ( 1)} and

2. and satisfy the algebraic equations (, ) = 0 when

Theorem 1.2: Let () and () be transcendental meromorphic (entire) functions of zero order, such that 2 () 0 and 2 () 0 where is nonzero complex constant. ,

, +. () be a small function and let () = () + 1(1) + + 1(1) + 0f

be a nonconstant differential polynomial with constant coefficients 0, 1, 1, ( 0) and

be the distinct zeros of () . If > 5 + 7 + 8 + 38 and [() 2 ()]() and

[() 2 ()]() share() . Then, one of the three cases listed below is true.

1. [() 2 ()]()[() 2 ()]() = 2

2. for a constant with = 1 when = {; = 0,1, ( 1)} and

3. and satisfy the algebraic equations (, ) = 0 when

2 LEMMA SECTION

Lemma 2.1 [3] Assuming () is a transcendental meromorphic function of finite order, the Nevanlinna characteristic function equality holds:

.

Lemma 2.2 [3] By using Lemma 2.1 and properties of (, )

.

Lemma 2.3 [8] Given () as a transcendental meromorphic function of finite order and as a non-zero complex constant, the relation

holds.

Lemma 2.4. [3] Considering () as a meromorphic function of finite order and as a non-zero complex constant, the expression

holds true.

Lemma 2.5 [15] If and are non-constant meromorphic functions sharing a common value, then one of the following three cases must occur:

Lemma 2.6 [12] Assume is a non-constant meromorphic function, and consider two positive integers and . Then,

Lemma 2.7 [17] Consider two non-constant meromorphic functions, and , and let

()( 0, ) be a small function of both and . If and share () , then one of the following three cases must be true:

and similar equality for (, )

Lemma 2.8 Let () be a transcendental meromorphic function of zero order and let =

() 2 (()) , where is positive integer, then

Proof. Derived from the first fundamental theorem, we acquire:

Lemma 2.9 If () be a transcendental entire function of zero order and () = () +

1(1) + + 0. Let = () 2 () , where is positive integer. Then

Proof.

3 MAIN RESULTS

Proof of Theorem 1.1:

Proof. Let

and

Since, () and () share (), , CM, then there exists a nonzero constant such that

(3.1)

we get,

Now, we will prove that = 1

On the contrary, if 1. UsingSecondFundamental Theorem and Lemma2.6

This implies,

Combining this with Lemma 2.7, we get

Combining (3.2) and (3.3) we get

which is contradiction to > 2 + 2 + 2 + 14. Thus, we get = 1, Hence from (3.1) we have

We get

where () is a polynomial. Suppose () 0, then we get

By utilizing Lemma 2.7 and Nevanlinna’s Second Fundamental Theorem, we obtain:

Similarly, we have

Combining (3.5) and (3.6) we get

Which contradicts to > 2 + 2 + 2 + 14 > 2 + 14. Therefore, () = 0. Therefore, deriving from equation (3.4), we obtain:

That is

which implies

Substitute = or = , and we examine the following scenarios:

Case 1: If () is a constant, then substitute = in (3.7), we have

Applying Leibnitz theorem,

Put

which implies = 1 where = {

Thus, = , where is constantwith = 1, where = { = 0,1, ( 1)} and

Case 2: Assume that, () is notaconstant, In such a case, and fulfill the algebraic equation

(, ) = 0, (1, 2) = (1)[1( + 2) 21( + ) + 1()] (2)[2( + 2) 22( + ) + 2()].

Note: In the scenario where () and () are transcendental entire functions, it follows that

By performing calculations analogous to the scenario of meromorphic functions, we straightforwardly deduce the conclusion of Theorem 1.1 when > 2 + 2 + 2 + 11.

Proof of Theorem 1.2:

Proof. Let

and

we see that () and () share () . If the condition stated in equation 1 of Lemma 2.5 is satisfied, then, through the application of Lemma 2.6, we derive:

which implies,

Therefore,

similarly,

We have,

and

Similarly, 

and

Substituting (3.10) to (3.12) in (3.8), we get

Using Lemma 2.8 in L.H.S

Similarly, substituting (3.13) to (3.15) in (3.9), we get

Now, following is derived from equations (3.16) and (3.17),

which is contradiction to 

Therefore, based on Lemma 2.7, we observe either

Case 1: Assume that ()() 2()

[() 2 ()]()[() 2 ()]() = 2(), This corresponds to one of the conclusions drawn from Theorem 1.1.

Case 2: Now, let’s examine () (), implying that, following a reasoning similar to Theorem 1.1, we deduce that and satisfy one of the following two statements.

1. = for a constant with = 1, where = { = 0,1, . . . , ( 1)}

   and

2. and meeting the requirements of the algebraic equation (, ) = 0, where

2()]. Hence the proof.

Conclusion: The uniqueness of a differential-difference polynomial sharing a small function is confirmed through the verification of Lemmas (2.8) and (2.9). To establish the uniqueness of two transcendental meromorphic functions with zero order, we applied Leibniz’s theorem. The utilization of a second-order operator is instrumental in demonstrating the conditional existence of two meromorphic functions with zero order.

REFERENCES

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2. Bhoosnurmath, S. S., Shilpa, N., and Barki, M. (2019). Results on entire and meromorphic functions that share small function with their homogeneous and linear differential polynomials. Tbilisi Mathematical Journal, 12(4), 227-236.

3. Chiang, Y. M., and Feng, S. J. (2008). On the Nevanlinna characteristic of and difference equations in the complex plane. The Ramanujan Journal, 16(1), 105-129.

4. Dyavanal, R. S., and Desai, R. V. (2014). Uniqueness of difference polynomials of entire functions. Appl. Math. Sci, 8(69), 3419-3424.

5. Fang, M. L., and Hua, X. H. (1996). Entire functions that share one value. Nanjing Univ. Math. Biquarterly, 13(1), 44-48.

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About Authors

1st Author, Somalatha M.T (Corresponding Author): Dr. Somalatha M.T. is Associate Professor in the Department of Studies and Research in Mathematics,Government Science College (Autonomous) who has 25 years of teaching experience in UG and 15 years of teaching experience in PG and 8 years of Research Experience. She received her M. Sc. Degree from Bangalore University, Ph.D Degree from Presidency University .

2nd Author, Shilpa N: Dr. Shilpa N. is Associate Professor in the Department of Mathematics, Presidency University, who has more than 18 years of teaching experience and 12 years of Research Experience. She received her M.Phil. Degree from Vinayaka Mission University, B.Ed. and M. Sc. Degree from Bangalore University. She has qualified KSET examination conducted by Mysore University. She has successfully completed a UGC project as a principle investigator. She was also the member of the organizing committee for two days National Conference, International Symposium and several Workshops at M.E.S College, Bengaluru. She is currently guiding five students for the doctoral award and additionally collaborating her research work with Prof. Dr. Subash S. Bhoosnurmath, Honorary Professor and Principle Investigator of DST project, and Dr. Renukadevi S. Dyavanal, Associate Professor, Karnatak University, Dharwad, in the field of Nevanlinna theory.

3rd Author, Y. Veeranna: Dr. Y Veeranna is Associate Professor in the Department of Studies and Research in Mathematics,Government Science College (Autonomous) who has 20 years of teaching experience in both UG and PG and 9 years of Research Experience. He received his M. Sc. Degree from Bangalore University, Ph.D Degree from Bangalore University .