A note on the uniqueness of certain types of differential-difference polynomials

S. Majumder; S. Saha

doi:10.37863/umzh.v73i5.379

S. Majumder Raiganj Univ., West Bengal, India
S. Saha Mehendipara Jr. High School, Dist.-Dakshin Dinajpur, West Bengal, India

DOI: https://doi.org/10.37863/umzh.v73i5.379

Ключові слова: Meromorphic function, difference polynomia, uniqueness

Анотація

УДК 517.9

Про унiкальнiсть деяких типiв диференцiально-рiзницевих полiномiв

Вивчається можливість розв'язання задач єдиності для деяких типів диференціально-різницевих поліномів, які мають спільну малу функцію.
У цій роботі не лише наведено розв'язок відкритої задачі з [A. Banerjee, S. Majumder, On the uniqueness of certain types of differential-difference polynomials, Anal. Math., 43, № 3, 415-444 (2017)], а й запропоновано більш загальний вигляд отриманого основного результату.

Біографічна довідка автора

S. Majumder, Raiganj Univ., West Bengal, India

Посилання

A. Banerjee, S. Majumder, On the uniqueness of certain types of differential-difference polynomials, Anal. Math., 43, № 3, 415 – 444 (2017), https://doi.org/10.1007/s10476-017-0402-3 DOI: https://doi.org/10.1007/s10476-017-0402-3

A. Banerjee, 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, A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoam, 11, 355 – 373 (1995), https://doi.org/10.4171/RMI/176 DOI: https://doi.org/10.4171/RMI/176

Y. M. Chiang, S. J. Feng, On the Nevanlinna characteristic $f(z+eta)$ and difference equations in complex plane, Ramanujan J., 16, 105 – 129 (2008), https://doi.org/10.1007/s11139-007-9101-1 DOI: https://doi.org/10.1007/s11139-007-9101-1

M. R. Chen, 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, J. L. Zhang, Value sharing results for shifts of meromorphic function, and sufficient conditions for periodicity, J. Math. Anal. and Appl., 355, no. 1,

– 363 (2009), https://doi.org/10.1016/j.jmaa.2009.01.053 DOI: https://doi.org/10.1016/j.jmaa.2009.01.053

W. K. Hayman, Picard values of meromorphic functions and their derivatives, Ann. Math., 70, 9 – 42 (1959), https://doi.org/10.2307/1969890 DOI: https://doi.org/10.2307/1969890

W. K. Hayman, Meromorphic functions, The Clarendon Press, Oxford (1964).

I. Lahiri, Value distribution of certain differential polynomials, Int. J. Math. and Math. Sci., 28, no. 2, 83 – 91 (2001), https://doi.org/10.1155/S0161171201011036 DOI: https://doi.org/10.1155/S0161171201011036

I. Lahiri, Weighted value sharing and uniqueness of meromorphic functions, Complex Var., Theory and Appl., 46, no. 3, 241 – 253 (2001), https://doi.org/10.1080/17476930108815411 DOI: https://doi.org/10.1080/17476930108815411

I. Laine, C. C. Yang, Value distribution of difference polynomials, Proc. Japan Acad. Ser. A, 83, 148 – 151 (2007). DOI: https://doi.org/10.3792/pjaa.83.148

S. Lin, W. C. Lin, Uniqueness of meromorphic functions concerning weakly weighted sharing, Kodai Math. J., 29, 269 – 280 (2006), https://doi.org/10.2996/kmj/1151936441 DOI: https://doi.org/10.2996/kmj/1151936441

E. Mues, Über ein Problem von Hayman, Math. Z., 164, 239 – 259 (1979), https://doi.org/10.1007/BF01182271 DOI: https://doi.org/10.1007/BF01182271

P. Sahoo, H. Karmakar, Results on uniqueness of entire functions whose certain difference polynomials share a small function, Anal. Math., 41, no. 4, 257 – 272 (2015), https://doi.org/10.1007/s10476-015-0302-3 DOI: https://doi.org/10.1007/s10476-015-0302-3

P. Sahoo, H. Karmakar, Erratum to: Results on uniqueness of entire functions whose certain difference polynomials share a small function, Anal. Math., 43, № 4, 629 – 630 (2017), https://doi.org/10.1007/s10476-017-0506-9 DOI: https://doi.org/10.1007/s10476-017-0506-9

C. C. Yang, On deficiencies of differential polynomials II, Math. Z., 125, 107 – 112 (1972), https://doi.org/10.1007/BF01110921 DOI: https://doi.org/10.1007/BF01110921

K. Yamanoi, The second main theorem for small functions and related problems, Acta Math., 192, 225 – 294 (2004), https://doi.org/10.1007/BF02392741 DOI: https://doi.org/10.1007/BF02392741

K. Zhang, H. X. Yi, The value distribution and uniqueness of one certain type of differential-difference polynomials, Acta Math. Sci. Ser. B, 34, № 3, 719 – 728 (2014), https://doi.org/10.1016/S0252-9602(14)60043-6 DOI: https://doi.org/10.1016/S0252-9602(14)60043-6

J. L. Zhang, Value distribution and shared sets of differences of meromorphic functions, J. Math. Anal. and Appl., 367, no. 2, 401 – 408 (2010), https://doi.org/10.1016/j.jmaa.2010.01.038 DOI: https://doi.org/10.1016/j.jmaa.2010.01.038