Entire function sharing two values partially with its derivative and a conjecture of Li and Yang
-
Junfeng Xu
and
Pradip Das
Abstract
In the paper, using the idea of normal family, we investigate the uniqueness problem of entire functions that share two values partially with their k-th derivatives. The obtained results improve the results of Lü, Xu and Yi (Ann. Polon. Math. 95(1) (2009), 67–75) in a large scale. Also, as an application of our results, we have settled the conjecture posed by Li and Yang (Illinois J. Math. 44(2) (2000), 349–362).
Acknowledgement
The authors are thankful to the referee for his/her valuable suggestions for the improvement of the paper.
(Communicated by Michal Fečkan)
References
[1] Chang, J. M.—Zalcman, L.: Meromorphic functions that share a set with their derivatives, J. Math. Anal. Appl. 338 (2008), 1191–1205.10.1016/j.jmaa.2007.05.079Search in Google Scholar
[2] Frank, G.: Lecture notes on sharing values of entire and meromorphic functions, Workshop in Complex Analysis at Tianjing, China, 1991.Search in Google Scholar
[3] Gundersen, G. G.: Meromorphic functions that share finite values with their derivative, J. Math. Anal. Appl. 75 (1980), 441–446; Correction 86 (1982), 307.10.1016/0022-247X(80)90092-XSearch in Google Scholar
[4] Hayman, W. K.: Meromorphic Functions, Clarendon Press, Oxford, 1964.Search in Google Scholar
[5] Heittokangas, J.—Korhonen, R. J.—Rättyä, J.: Generalized logarithmic derivative estimates of Gol’dberg-Grinshtein type, Bull. London Math. Soc. 36 (2004), 105–114.10.1112/S0024609303002649Search in Google Scholar
[6] Li. P.—Yang, C. C.: When an entire function and its linear differential polynomial share two values, Illinois J. Math. 44 (2) (2000), 349–362.10.1215/ijm/1255984845Search in Google Scholar
[7] Li, J. T.—Yi, H. X.: Normal families and uniqueness of entire functions and their derivatives, Arch. Math. (Basel) 87(1) (2006), 52–59.10.1007/s00013-005-1619-0Search in Google Scholar
[8] Lü, F.—Xu, J. F.—Yi, H. X.: Uniqueness theorems and normal families of entire functions and their derivatives, Ann. Polon. Math. 95(1) (2009), 67–75.10.4064/ap95-1-5Search in Google Scholar
[9] Lü, F.—Yi, H. X.: The Brück conjecture and entire functions sharing polynomials with their k-th derivatives, J. Korean Math. Soc. 48(3) (2011), 499–512.10.4134/JKMS.2011.48.3.499Search in Google Scholar
[10] Mues, E.—Steinmetz, N.: Meromorphe Funktionen, die mit ihrer Ableitung Werte teilen, Manuscripta Math. 29 (1979), 195–206.10.1007/BF01303627Search in Google Scholar
[11] Rubel, L. A.—Yang, C. C.: Values shared by an entire function and its derivative. Lecture Notes in Math., Vol. 599, Springer-Verlag, Berlin, 1977, pp. 101–103.10.1007/BFb0096830Search in Google Scholar
[12] Schiff, J.: Normal Families, Berlin, 1993.10.1007/978-1-4612-0907-2Search in Google Scholar
[13] Yang, C. C.—YI, H. X.: Uniqueness Theory of Meromorphic Functions, Kluwer Academic Publishers, Dordrecht/Boston/London, 2003.10.1007/978-94-017-3626-8Search in Google Scholar
[14] Zalcman, L.: Normal families, new perspectives, Bull. Amer. Math. Soc. 35 (1998), 215–230.10.1090/S0273-0979-98-00755-1Search in Google Scholar
© 2025 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- A new categorical equivalence for stone algebras
- On special classes of prime filters in BL-algebras
- A note on characterized and statistically characterized subgroups of 𝕋 = ℝ/ℤ
- New Young-type integral inequalities using composition schemes
- The structure of pseudo-n-uninorms with continuous underlying functions
- Jensen-type inequalities for a second-order differential inequality condition
- A direct proof of the characterization of the convexity of the discrete Choquet integral
- Envelope of plurifinely plurisubharmonic functions and complex Monge-Ampère type equation
- Fekete-Szegö inequalities for Φ-parametric and β-spirllike mappings of complex order in ℂn
- Entire function sharing two values partially with its derivative and a conjecture of Li and Yang
- Oscillatory properties of third-order semi-canonical dynamic equations on time scales via canonical transformation
- Weighted B-summability and positive linear operators
- Some properties and applications of convolution algebras
- On measures of σ-noncompactess in F-spaces
- On the kolmogorov–feller–gut weak law of large numbers for triangular arrays of rowwise and pairwise negatively dependent random variables
- Intermediately trimmed sums of oppenheim expansions: A strong law
- Novel weighted distribution: Properties, applications and web-tool
- On the q-Gamma distribution: Properties and inference
- Finiteorthoatomistic effect algebras and regular algebraic E-test spaces
- Prof. RNDr. Anatolij Dvurečenskij, DrSc. 75th anniversary
Articles in the same Issue
- A new categorical equivalence for stone algebras
- On special classes of prime filters in BL-algebras
- A note on characterized and statistically characterized subgroups of 𝕋 = ℝ/ℤ
- New Young-type integral inequalities using composition schemes
- The structure of pseudo-n-uninorms with continuous underlying functions
- Jensen-type inequalities for a second-order differential inequality condition
- A direct proof of the characterization of the convexity of the discrete Choquet integral
- Envelope of plurifinely plurisubharmonic functions and complex Monge-Ampère type equation
- Fekete-Szegö inequalities for Φ-parametric and β-spirllike mappings of complex order in ℂn
- Entire function sharing two values partially with its derivative and a conjecture of Li and Yang
- Oscillatory properties of third-order semi-canonical dynamic equations on time scales via canonical transformation
- Weighted B-summability and positive linear operators
- Some properties and applications of convolution algebras
- On measures of σ-noncompactess in F-spaces
- On the kolmogorov–feller–gut weak law of large numbers for triangular arrays of rowwise and pairwise negatively dependent random variables
- Intermediately trimmed sums of oppenheim expansions: A strong law
- Novel weighted distribution: Properties, applications and web-tool
- On the q-Gamma distribution: Properties and inference
- Finiteorthoatomistic effect algebras and regular algebraic E-test spaces
- Prof. RNDr. Anatolij Dvurečenskij, DrSc. 75th anniversary
