An Entire Function Weakly Sharing a Doubleton with its Derivative
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An Entire Function Weakly Sharing a Doubleton with its Derivative Indrajit Lahiri1 Received: 25 January 2020 / Revised: 24 August 2020 / Accepted: 27 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In the paper we consider the uniqueness problem of an entire function f when it shares a doubleton with its derivative f (k) , k ≥ 1. Our result extends a result of Li and Yang (J Math Soc Japan 51(4):781–799, 1999). Keywords Entire function · Derivative · Set sharing · Uniqueness Mathematics Subject Classification 30D35
1 Introduction, Definitions and Results Let f be a non-constant meromorphic function in the open complex plane C. For a ∈ C∪{∞} we denote by E(a; f ) the set of a-points of f counted with multiplicities and E(a; f ) the set of distinct a-points of f . If for two non-constant meromorphic functions f and g in C, E(a; f ) = E(a; g), then we say that f and g share the value a CM (counting multiplicities). If E(a; f ) = E(a; g), then we say that f and g share the value a IM (ignoring multiplicities). Rubel and Yang [7] considered the uniqueness of an entire function when it shares two values CM with its first derivative. Their following result is the first of this kind. Theorem A [7] If a non-constant entire function f shares two values CM with f (1) , then f = f (1) . In Theorem z A two shared values cannot be reduced to one as is evident from t ez f (z) = e e−e (1 − et )dt, see [1, p. 386]. However, Mues and Steinmetz [6] 0
Communicated by Ilpo Laine.
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Indrajit Lahiri [email protected] Department of Mathematics, University of Kalyani, Kalyani, West Bengal 741235, India
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improved Theorem A by considering two IM shared values. Yang [8] also extended Theorem A by replacing f (1) with a higher order derivative f (k) . In the year 2000 Li and Yang [4] improved the result of Yang [8] and proved the following theorem. Theorem B [4] Let f be a non-constant entire function, k be a positive integer and a, b be distinct finite numbers. If f and f (k) share a and b IM, then f = f (k) . Let S ⊂ C ∪ {∞} and f be a non-constant meromorphic function. We denote by E(S; f ) the set E(S; f ) = a∈S E(a; f ). Similarly E(S; f ) = a∈S E(a; f ). Two meromorphic functions f and g are said to share a set S with counting multiplicities (CM) or with ignoring multiplicities (IM) if E(S; f ) = E(S; g) or E(S; f ) = E(S; g) respectively. In 1999 Li and Yang [3] considered the problem of improving Theorem A using the notion of set sharing in stead of value sharing. They prove the following theorem. Theorem C [3] Let f be a non-constant entire function and a1 , a2 be two distinct finite complex numbers. If f and f (1) share the set {a1 , a2 } CM, then one and only one of the following holds: (i) f = f (1) , (ii) f + f (1) = a1 + a2 , (iii) f = c1 ecz + c2 e−cz with a1 + a2 = 0, where c, c1 and c2 are non-zero constants which satisfy c2 = 1 and 4c2 c1 c2 = a1 (c2 − 1). We now explain a weaker notion of set sharing. Let f and g be two non-constant meromorphic functions in C
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