We study the problems of uniqueness of some types of differential-difference polynomials sharing a small function. We not only solve an open problem formulated in [A. Banerjee and S. Majumder, Anal. Math., 43, 415–444 (2017)] but also present our main results in a more generalized way.
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References
A. Banerjee and S. Majumder, “On the uniqueness of certain types of differential-difference polynomials,” Anal. Math., 43, No. 3, 415–444 (2017).
A. Banerjee and S. Mukherjee, “Uniqueness of meromorphic functions concerning differential monomials sharing the same value,” Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 50, 191–206 (2007).
W. Bergweiler and A. Eremenko, “On the singularities of the inverse to a meromorphic function of finite order,” Rev. Mat. Iberoamer., 11, 355–373 (1995).
Y. M. Chiang and S. J. Feng, “On the Nevanlinna characteristic f (z + 𝜂) and difference equations in complex plane,” Ramanujan J., 16, 105–129 (2008).
M. R. Chen and Z. X. Chen, “Properties of difference polynomials of entire functions with finite order,” Chinese Ann. Math. Ser. A, 33, 359–374 (2012).
J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, and J. L. Zhang, “Value sharing results for shifts of meromorphic function, and sufficient conditions for periodicity,” J. Math. Anal. Appl., 355, 352–363 (2009).
W. K. Hayman, “Picard values of meromorphic functions and their derivatives,” Ann. of Math. (2), 70, 9–42 (1959).
W. K. Hayman, Meromorphic Functions, The Clarendon Press, Oxford (1964).
I. Lahiri, “Value distribution of certain differential polynomials,” Int. J. Math. Math. Sci., 28, 83–91 (2001).
I. Lahiri, “Weighted value sharing and uniqueness of meromorphic functions,” Complex Variables Theory Appl., 46, 241–253 (2001).
I. Laine and C. C. Yang, “Value distribution of difference polynomials,” Proc. Jap. Acad., Ser. A, 83, 148–151 (2007).
S. Lin and W. C. Lin, “Uniqueness of meromorphic functions concerning weakly weighted sharing,” Kodai Math. J., 29, 269–280 (2006).
E. Mues, “Über ein Problem von Hayman,” Math. Z., 164, 239–259 (1979).
P. Sahoo and H. Karmakar, “Results on uniqueness of entire functions whose certain difference polynomials share a small function,” Anal. Math., 41, 257–272 (2015).
P. Sahoo and H. Karmakar, “Erratum to: Results on uniqueness of entire functions whose certain difference polynomials share a small function,” Anal. Math., 43, No. 4, 629–630 (2017).
C. C. Yang, “On deficiencies of differential polynomials II,” Math. Z., 125, 107–112 (1972).
K. Yamanoi, “The second main theorem for small functions and related problems,” Acta Math., 192, 225–294 (2004).
K. Zhang and H. X. Yi, “The value distribution and uniqueness of one certain type of differential-difference polynomials,” Acta Math. Sci. Ser. B Engl. Ed., 34, No. 3, 719–728 (2014).
J. L. Zhang, “Value distribution and shared sets of differences of meromorphic functions,” J. Math. Anal. Appl., 367, 401–408 (2010).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, No. 5, pp. 679–694, May, 2021. Ukrainian DOI: 10.37863/umzh.v73i5.379.
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Majumder, S., Saha, S. A Note on the Uniqueness of Certain Types of Differential-Difference Polynomials. Ukr Math J 73, 791–810 (2021). https://doi.org/10.1007/s11253-021-01960-9
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DOI: https://doi.org/10.1007/s11253-021-01960-9