Uniqueness of L function with special class of meromorphic function in the light of two shared sets
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Uniqueness of L function with special class of meromorphic function in the light of two shared sets Arpita Kundu1 · Abhijit Banerjee1 Received: 14 July 2020 / Accepted: 3 August 2020 © Springer-Verlag Italia S.r.l., part of Springer Nature 2020
Abstract In this article we study the uniqueness problem of an L function in the Selberg class with an arbitrary meromorphic function having finite poles sharing two sets. One of our result will extend a result of Yuan et al. (Lith Math J 58(2):249–262, 2018). Most importantly, we have pointed out a big gap in the analysis of a recent result of Sahoo and Halder (Comput Methods Funct Theory 19:601–612, 2019) which makes the validity as well as existance of most of the part of the paper under question. Ultimately we have obtained the accurate form of the result, which together with another theorem established in the paper provide an answer to a question raised by Lin and Lin (Filomat 30:3795–3806, 2016) in some sense. Keywords Meromorphic function · L function · Uniqueness · Shared sets Mathematics Subject Classification Primary 11M36 · Secondary 30D35
1 Introduction and results By a meromorphic function we shall always mean a meromorphic function in the complex plane. We adopt the standard notations of Nevanilinna theory of meromorphic functions as explained in [4]. Let ℂ = ℂ ∪ {∞} , ℂ∗ = ℂ⧵{0} and ℕ = ℕ ∪ {0} , where ℂ and ℕ denote the set of all complex numbers and natural numbers respectively and by ℤ we denote the set of all integers. For any non-constant meromorphic function h(z) we define S(r, h) = o(T(r, h)), (r ⟶ ∞, r ∉ E) where E denotes any set of positive real numbers having finite linear measure.
Definition 1.1 For a non-constant meromorphic function f and a ∈) ℂ , let (
Ef (a) = {(z, p) ∈ ℂ × ℕ ∶ f (z) = a with multiplicity p} Ef (a) = {(z, 1) ∈ ℂ × ℕ ∶ f (z) = a} . Then
* Arpita Kundu [email protected]
Abhijit Banerjee [email protected]; [email protected]
1
Department of Mathematics, University of Kalyani, Kalyani, West Bengal 741235, India
13
Vol.:(0123456789)
A. Kundu, A. Banerjee
( ) we say f, g share the value a CM(IM) if Ef (a) = Eg (a) Ef (a) = Eg (a) . For a = ∞ , we ( ) define Ef (∞) ∶= E1∕f (0) Ef (∞) ∶= E1∕f (0) .
Definition 1.2 For a non-constant meromorphic function f and S ⊂ ℂ , let ⋃ �Ef (S) = ⋃a∈S {(z, p) ∈ ℂ × ℕ ∶ f (z) = a with � multiplicity p} Ef (S) = a∈S {(z, 1) ∈ ℂ × ℕ ∶ f (z) = a} . Then we say f, g share the set S CM(IM) if ( ) Ef (S) = Eg (S) Ef (S) = Eg (S) . When S contains only one element the definition coincides with the classical definition of value sharing. This paper deals with the uniqueness problems of set sharing related to L-functions and an arbitrary meromorphic function in ℂ. In 1989, Selberg [13] introduced a new class of Dirichlet series, called the Selberg class, which later became an important field of research in analytic number theory. In this paper, by an L-function we mean a Selberg class function with the Riemann zeta function ∑ 1 𝜁 (s) = ∞ prototype. The Selberg
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