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In this paper taking a question in [1] into background we investigate the uniqueness of a non-constant polynomial with the differential monomial generated by a non-constant mermorphic function f. Our result will also extend a result of Banerjee-Majumder[2] given earlier. An open question is also posed, in the paper, for future investigation.

With the help of weighted sharing of sets, we find out the class of meromorphic functions f, when P(f) and [P(f)](k) share a set Sm of small functions. Our results improve and extend the results of Zhang and Yang [Ann. Acad. Sci. Fenn. Math. 34 (2009), 249-260] and Xu et al. [Rev. Mat. Teor. Apl. 23(1) (2016), 291-308]. A number of examples are exhibited to validate certain claims of the main results. [ABSTRACT FROM AUTHOR]

For a value s 2 C[f1g, two meromorphic functions f and g are said to share the value s, CM, (or IM), provided that f(z)s and g(z)s have the same set of zeros, counting multiplicities, (respectively, ignoring multiplicities). We say that a meromorphic function f shares s 2 partially with a meromorphic function g if E(s; f) E(s; g). It is easy to see that "partially shared values CM" are more general than "shared values CM". With the help of partially shared values, in this paper, we prove some uniqueness results between a non-constant meromorphic function and its generalized differences or shifts. We exhibit some examples to show that the result of Charak et al. [8] is not true for k = 2 or k = 3. We find some gaps in proof of the result of Lin et al. [24]. We not only correct these resuts, but also generalize them in a more convenient way. We give a number of examples to validate certain claims of the main results of this paper and also to show that some of conditions are sharp. Finally, we pose some open questions for further investigation. [ABSTRACT FROM AUTHOR]

In this paper, we have considered two sets sharing problems, and investigated on some sufficient conditions for the periodicity of meromorphic functions and obtained two results improving the result of Bhoosnurmath-Kabbur [6], Qi- Dou-Yang [17] and Zhang [20]. The results are: Let S1 = {z: ∫0 z-a (t- a)n (t- b) 4dt + 1 = 0} and S2 = {a,b}, where n ≥ 4 (n ≥ 2) be an integer. Let f (z) be a non-constant meromorphic (entire) function satisfying ... f (z) (Sj) = ... f(z+c)(Sj), (j = 1, 2) then f (z) = f (z + c). Some examples have been exhibited to show that, the meromorphic functions, we have considered may be of infinite order, and also to show that the sets considered in the main results, can't be replace by some arbitrary sets. At the last section, we have posed a question for the future research in this direction. [ABSTRACT FROM AUTHOR]

Taking two and three shared set problems into background, the uniqueness problem of a meromorphic function together with its shift operator have been studied. Our results will improve a number of recent results in the literature. Some examples have been provided in the last section to show that certain conditions used in the paper, is the best possible. [ABSTRACT FROM AUTHOR]

The paper deals with the problem of determining some sufficient conditions under which a polynomial generated by a meromorphic function and its k-th derivative sharing a set become identical. In this context, we have somehow been able to present an unique range set for a special class of meromorphic functions with smallest cardinalities. We have exhibited a number of examples relevant to the content of the paper. [ABSTRACT FROM AUTHOR]

In this paper we have mainly dealt with the relation between a generalized di erential polynomial and a rational function R(f) of a non-constant meromorphic function f sharing a small function a ≡ a(z)(≢ 0, ∞). Our results will extend recent results in [4], [5] and [9] in the direction of Brück Conjecture. We have exhibited some examples which show that the result of this paper may or may not be true because non-constant entire functions and conditions obtained in the theorems cannot be removed. Other examples have also substantiated our certain claims. [ABSTRACT FROM AUTHOR]