Partial sharing of small functions with exact difference

Audrija Choudhury, Rupa Pal

Abstract


We have proven certain uniqueness results related owing to the partial sharing of small functions of $f$ with difference polynomials of the form $L(f)=b_k(z)f(z+kc)+\ldots +b_0(z)f(z)$, where $b_i$ are small functions of $f$.  As a consequence of this, we have deduced the conditions under which, $f$ is identically equal to its exact difference operator $\Delta_c^k f$.

Keywords


Meromorphic functions, difference polynomials, partial sharing

Full Text:

PDF

References


K.S. Charak, R. J. Korhonen, and G. Kumar, A note on partial sharing of values of meromorphic functions with their shifts, J. Math.Anal.Appl. 435 (2016), 1241-1248.

S. Chen, On Uniqueness of Meromorphic Functions and Their Difference Operators with Partially Shared Values, Comput. Methods Funct. Theory 18 (2018), 529-536.

S. Chen, and A. Xu, Uniqueness on entire functions and their nth order exact differences with two shared values, De Gruyter Open Mathematics 18 (2020), 211--215.

P. Chern, On meromorphic functions with finite logarithmic order, Transactions of the American Mathematical Society, 358 (2006), 473--489.

Y. M. Chiang, and S. J. Feng, On the Nevanlinna Characteristic of f(z+eta) and difference equations in the complex plane, Ramanujan J., 16 (2008), 105--129.

Y. M. Chiang, and S. J. Feng, On the growth of logarithmic difference, difference equations and logarithmic derivatives of meromorphic functions, Trans. Am. Math. Soc., 361 (2009), 3767--3791.

Z. Gao, R. Korhonen, J. Zhang, and Y. Zhang, Uniqueness of meromorphic functions sharing values with their nth order exact differences, Analysis Mathematica 45 (2019), 321-334.

G. G. Gundersen, Meromorphic functions that share two finite values with their derivative, Pacific J. Math. 105 (1983), 299--309.

R. G. Halburd, and R. J. Korhonen, Difference analogue of the Lemma on the Logarithmic Derivative with applications to difference equations, J. Math. Anal. Appl. 314 (2006), 477--487.

R. G. Halburd, and R. J. Korhonen, Nevanlinna Theory for the difference operator, Annales Academiae Scientiarum Fennicae Mathematica 31 (2006), 463--478.

R. Halburd, R. Korhonen, and K. Tohge, Holomorphic curves with shift-invariant hyperplane preimages, Trans. Am. Math. Soc. 366 (2014), 4267-4298.

W. K. Hayman, Meromorphic Functions, 195. The Clarendon Press, Oxford, 1964.

J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and J. Zhang, Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity, J. Math. Anal. Appl. 355 (2009) 352-363.

J. Heittokangas, R. Korhonen, I. Laine, and J. Rieppo, Uniqueness of meromorphic functions sharing values with their shifts, Complex Var. Elliptic Equ. 56 (2011) 81-92.

I. Laine, Nevanlinna Theory and Complex Differential Equations, 341. Walter de Gruyter & Co., Berlin, 1993.

X. M. Li, H. X. Yi and C. Y. Kang, Results on meromorphic functions sharing three values with their difference operators, Bull. Korean Math. Soc., 52 (2015), 1401--1422.

S. Li, and Z. S. Gao, Entire functions sharing one or two finite values CM with their shifts or difference operators, Arch. Math. 97 (2011), 475-483.

Y. Lo, Value Distribution theory, 268. Springer-Verlag Berlin Heidelberg, 1993.

K. Yamanoi, The second main theorem for small functions and related problems, Acta Math. 192 (2004), 225--294.

C. C. Yang, and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Kluwer Academic Publishers, Dordrecht, 2003.




DOI: http://dx.doi.org/10.23755/rm.v51i0.1458

Refbacks

  • There are currently no refbacks.


Copyright (c) 2024 Audrija Choudhury, Rupa Pal

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Ratio Mathematica - Journal of Mathematics, Statistics, and Applications. ISSN 1592-7415; e-ISSN 2282-8214.