# Uniqueness of Entire Functions Sharing Polynomials with Their Derivatives

• Sahoo, Pulak (Department of Mathematics, University of Kalyani) ;
• Biswas, Gurudas (Department of Mathematics, Hooghly Women's College)
• Accepted : 2018.06.28
• Published : 2018.09.23
• 122 7

#### Abstract

In this paper, we investigate the uniqueness problem of entire functions sharing two polynomials with their k-th derivatives. We look into the conjecture given by $L{\ddot{u}}$, Li and Yang [Bull. Korean Math. Soc., 51(2014), 1281-1289] for the case $F=f^nP(f)$, where f is a transcendental entire function and $P(z)=a_mz^m+a_{m-1}z^{m-1}+{\ldots}+a_1z+a_0({\not{\equiv}}0)$, m is a nonnegative integer, $a_m,a_{m-1},{\ldots},a_1,a_0$ are complex constants and obtain a result which improves and generalizes many previous results. We also provide some examples to show that the conditions taken in our result are best possible.

#### Keywords

entire function;derivative;uniqueness

#### Acknowledgement

Supported by : UGC-DRS-SAP

#### References

1. A. Al-Khaladi, On meromorphic functions that share one value with their derivatives, Analysis, 25(2005), 131-140.
2. R. Bruck, On entire functions which share one value CM with their first derivative, Results Math., 30(1996), 21-24.
3. Z. X. Chen and K. H. Shon, On conjecture of R. Bruck concerning entire function sharing one value CM with its derivative, Taiwanese J. Math., 8(2004), 235-244. https://doi.org/10.11650/twjm/1500407625
4. J. Clunie, On integral and meromorphic functions, J. London Math. Soc., 37(1962), 17-27.
5. G. G. Gundersen, Meromorphic functions that share finite values with their derivatives, J. Math. Anal. Appl., 75(1980), 441-446. https://doi.org/10.1016/0022-247X(80)90092-X
6. G. G. Gundersen and L. Z. Yang, Entire functions that share one value with one or two of their derivatives, J. Math. Anal. Appl., 223(1998), 88-95. https://doi.org/10.1006/jmaa.1998.5959
7. W. K. Hayman, Meromorphic Functions, The Clarendon Press, Oxford, 1964.
8. G. Jank, E. Mues and L. Volkmann, Meromorphe Funktionen, die mit ihrer ersten und zweiten Ableitung einen endchen Wert teilen, Complex Variables Theory Appl., 6(1986), 51-71. https://doi.org/10.1080/17476938608814158
9. I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin/New York, 1993.
10. W. Lu, Q. Li and C. C. Yang, On the transcendental entire solutions of a class of differential equations, Bull. Korean Math. Soc., 51(2014), 1281-1289. https://doi.org/10.4134/BKMS.2014.51.5.1281
11. F. Lu and H.-X. Yi, The Bruck conjecture and entire functions sharing polynomials with their k-th derivatives, J. Korean Math. Soc., 48(2011), 499-512. https://doi.org/10.4134/JKMS.2011.48.3.499
12. S. Majumder, A result on a conjecture of W. Lu, Q. Li and C. C. Yang, Bull. Korean Math. Soc., 53(2016), 411-421.
13. S. Majumder, Values shared by meromorphic functions and their derivatives, Arab J. Math. Sci., 22(2016), 265-274.
14. L. A. Rubel and C. C. Yang, Values shared by an entire function and its derivative, Complex Analysis (Proc. Conf., Univ. Kentucky, Lexington, Ky.) (1976), 101-103.
15. C. C. Yang, On deficiencies of differential polynomials II, Math. Z., 125(1972), 107-112. https://doi.org/10.1007/BF01110921
16. L. Z. Yang, Entire functions that share finite values with their derivatives, Bull. Aust. Math. Soc., 41(1990), 337-342.
17. C.-C. Yang and H.-X. Yi, Uniqueness theory of meromorphic functions, Science Press, Beijing, 1995.
18. L.-Z. Yang and J.-L. Zhang, Non-existence of meromorphic solutions of Fermat Type functional equation, Aequationes Math., 76(2008), 140-150. https://doi.org/10.1007/s00010-007-2913-7
19. H. X. Yi, On characteristic function of a meromorphic function and its derivative, Indian J. Math., 33(1991), 119-133.
20. Q. C. Zhang, Meromorphic function that shares one small function with its derivative, JIPAM. J. Inequal. Pure. Appl. Math., 6(2005), Article 116, 13 pp.
21. J.-L. Zhang and L.-Z. Yang, Some results related to a conjecture of R. Bruck, JIPAM. Inequal. Pure Appl. Math., 8(2007), Article 18, 11 pp.
22. J.-L. Zhang and L.-Z. Yang, A Power of an entire function sharing one value with its derivative, Comput. Math. Appl., 60(2010), 2153-2160.