Geodesic
Generalizations on the results of Cao and Zhang
Mathematica Bohemica, Tome 147 (2022) no. 1, pp. 65-94
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We establish some uniqueness results for meromorphic functions when two nonlinear differential polynomials $P(f)\prod _{i=1}^{k}(f^{(i)})^{n_{i}}$ and $P(g)\prod _{i=1}^{k}(g^{(i)})^{n_{i}}$ share a nonzero polynomial with certain degree and our results improve and generalize some recent results in Y.-H. Cao, X.-B. Zhang (2012). Also we exhibit two examples to show that the conditions used in the results are sharp.
We establish some uniqueness results for meromorphic functions when two nonlinear differential polynomials $P(f)\prod _{i=1}^{k}(f^{(i)})^{n_{i}}$ and $P(g)\prod _{i=1}^{k}(g^{(i)})^{n_{i}}$ share a nonzero polynomial with certain degree and our results improve and generalize some recent results in Y.-H. Cao, X.-B. Zhang (2012). Also we exhibit two examples to show that the conditions used in the results are sharp.
DOI : 10.21136/MB.2021.0072-19
Classification : 30D35
Keywords: meromorphic function; uniqueness; weighted sharing; differential polynomial 
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Majumder, Sujoy; Mandal, Rajib. Generalizations on the results of Cao and Zhang. Mathematica Bohemica, Tome 147 (2022) no. 1, pp. 65-94. doi: 10.21136/MB.2021.0072-19

[1] Alzahary, T. C., Yi, H. X.: Weighted value sharing and a question of I. Lahiri. Complex Variables, Theory Appl. 49 (2004), 1063-1078. | DOI | MR | JFM

[2] Banerjee, A.: On a question of Gross. J. Math. Anal. Appl. 327 (2007), 1273-1283. | DOI | MR | JFM

[3] Bergweiler, W., Eremenko, A.: On the singularities of the inverse to a meromorphic function of finite order. Rev. Mat. Iberoam. 11 (1995), 355-373. | DOI | MR | JFM

[4] Cao, Y.-H., Zhang, X.-B.: Uniqueness of meromorphic functions sharing two values. J. Inequal. Appl. 2012 (2012), Article ID 100, 10 pages. | DOI | MR | JFM

[5] Chen, H., Fang, M.: The value distribution of $f^{n}f'$. Sci. China, Ser. A. 38 (1995), 789-798 \99999ZBL99999 0839. 30026. | MR

[6] Fang, M., Hua, X.: Entire functions that share one value. J. Nanjing Univ., Math. Biq. 13 (1996), 44-48. | MR | JFM

[7] Fang, M., Qiu, H.: Meromorphic functions that share fixed-points. J. Math. Anal. Appl. 268 (2002), 426-439. | DOI | MR | JFM

[8] Hayman, W. K.: Meromorphic Functions. Oxford Mathematical Monographs. Clarendon Press, Oxford (1964). | MR | JFM

[9] Köhler, I.: Meromorphic functions sharing zeros and poles and also some of their derivatives sharing zeros. Complex Variables, Theory Appl. 11 (1989), 39-48. | DOI | MR | JFM

[10] Lahiri, I.: Weighted value sharing and uniqueness of meromorphic functions. Complex Variables, Theory Appl. 46 (2001), 241-253. | DOI | MR | JFM

[11] Lahiri, I., Dewan, S.: Value distribution of the product of a meromorphic function and its derivative. Kodai Math. J. 26 (2003), 95-100. | DOI | MR | JFM

[12] Lahiri, I., Sarkar, A.: Nonlinear differential polynomials sharing 1-points with weight two. Chin. J. Contemp. Math. 25 (2004), 325-334. | MR | JFM

[13] Xu, J., Yi, H., Zhang, Z.: Some inequalities of differential polynomials. Math. Inequal. Appl. 12 (2009), 99-113. | DOI | MR | JFM

[14] Yamanoi, K.: The second main theorem for small functions and related problems. Acta Math. 192 (2004), 225-294. | DOI | MR | JFM

[15] Yang, C.-C.: On deficiencies of differential polynomials II. Math. Z. 125 (1972), 107-112. | DOI | MR | JFM

[16] Yang, C.-C., Hua, X.: Uniqueness and value-sharing of meromorphic functions. Ann. Acad. Sci. Fenn., Math. 22 (1997), 395-406. | MR | JFM

[17] Yang, C.-C., Yi, H.-X.: Uniqueness Theory of Meromorphic Functions. Mathematics and its Applications 557. Kluwer Academic Publishers, Dordrecht (2003). | DOI | MR | JFM

[18] Yi, H.-X.: On characteristic function of a meromorphic function and its derivative. Indian J. Math. 33 (1991), 119-133. | MR | JFM

[19] Zhang, Q.: Meromorphic function that shares one small function with its derivative. JIPAN, J. Inequal. Pure Appl. Math. 6 (2005), Article ID 116, 13 pages. | MR | JFM

[20] Zhang, Z., Li, W.: Picard exceptional values for two differential polynomials. Acta Math. Sin. 37 (1994), 828-835 Chinese. | MR | JFM

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