Class of Meromorphic Functions Partially Shared Values with Their Differences or Shifts
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For a value s ∈ C∪ {∞}, 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 ∈ ˆS 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
For a value s ∈ C∪ {∞}, 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 ∈ ˆS 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.
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