Some further difference results on Hayman conjecture and value-sharing
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Some further difference results on Hayman conjecture and value-sharing Sujoy Majumder1 · Arup Dam2 Received: 13 May 2019 / Accepted: 6 August 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract In this paper we investigate the zeros distributions of difference polynomials of entire functions of finite order, which can be viewed as the Hayman Conjecture for difference. We also study the uniqueness of difference polynomials of entire functions of finite order sharing a common value and obtain uniqueness theorems for difference. Keywords Entire functions · Difference · Finite order · Uniqueness · Value sharing Mathematics Subject Classification Primary 30D35; Secondary 39A05
1 Introduction, definitions and results In this paper by meromorphic functions we shall always mean meromorphic functions in the complex plane. We adopt the standard notations of value distribution theory (see [3]). For a non-constant meromorphic function f , we denote by T (r , f ) the Nevanlinna characteristic of f and by S(r , f ) any quantity satisfying S(r , f ) = o{T (r , f )} as r → ∞ possibly outside a set of finite linear measure. We denote by T (r ) the maximum of T (r , f ) and T (r , g). The notation S(r ) denotes any quantity satisfying S(r ) = o(T (r )) as r −→ ∞, outside of a possible exceptional set of finite linear measure. A meromorphic function a(z) is called a small function with respect to f , if T (r , a) = S(r , f ). We denote by S( f ) the set of all small functions of f . We use the symbol σ ( f ) to denote the order of f . Define the difference of f (z) by c f (z) = f (z + c) − f (z). Let a ∈ S( f ) ∩ S(g). We say that f (z) and g(z) share a(z) CM (counting multiplicities) if f (z) − a(z) and g(z) − a(z) have the same zeros with the same multiplicities and we say that f (z), g(z) share a(z) IM (ignoring multiplicities) if we do not consider the multiplicities.
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Sujoy Majumder [email protected] ; [email protected] ; [email protected] Arup Dam [email protected]
1
Department of Mathematics, Raiganj University, Raiganj, West Bengal 733134, India
2
Department of Mathematics, North Bengal St. Xavier’s College, Balai Gachh, West Bengal 735135, India
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S. Majumder, A. Dam
Let k ∈ N and a ∈ C ∪ {∞}. We use the notations Nk) (r , a; f ) and N(k+1 (r , a; f ) to denote the counting function of a-points of f with multiplicity not greater than k and the counting function of a-points of f with multiplicity greater than k respectively. Similarly N k) (r , a; f ) and N (k+1 (r , a; f ) are their reduced functions respectively. For p ∈ N we p denote by N p (r , a; f ) the sum i=1 N (i (r , a; f ). Let f and g share a IM. We denote by N ∗ (r , a; f , g) the reduced counting function of those a-points of f whose multiplicities differ from the multiplicities of the corresponding a-points of g. Let k ∈ N ∪ {0} ∪ {∞}. For a ∈ C ∪ {∞} we denote by E k (a; f ) the set of all a-points of f where an a-point of multiplicity m is counted m times if m ≤ k and k+1 times if m > k. If E k (a; f ) = E
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