Abstract
PurposeThe paper aims to build the relationship between an entire function of restricted hyper-order with its linear c-shift operator.Design/methodology/approachStandard methodology for papers in difference and shift operators and value distribution theory have been used.FindingsThe relation between an entire function of restricted hyper-order with its linear c-shift operator was found under the periphery of sharing a set of two small functions IM (ignoring multiplicities) when exponent of convergence of zeros is strictly less than its order. This research work is an improvement and extension of two previous papers.Originality/valueThis is an original research work.
Highlights
By a meromorphic function f, we always mean that it is defined on C. For such a meromorphic function, we recall some basic terminologies of value distribution theory such as the Nevanlinna characteristic function T(r, f), the proximity function m(r, f) and the counting function of a-points of f
Δcf ðzÞ 1⁄4 f ðz þ cÞ À f ðzÞ; Δkc f ðzÞ 1⁄4 ΔcðΔkc−1f ðzÞÞ; k ∈ N; k ≥ 2: We introduce the more generalized linear c-shift operator Lcf by Xk
The uniqueness problem of entire functions sharing set with their derivatives, shifts, different types of difference operators has been developed as an interesting direction of research in the realm of value distribution theory
Summary
By a meromorphic function f, we always mean that it is defined on C. For such a meromorphic function, we recall some basic terminologies of value distribution theory such as the Nevanlinna characteristic function T(r, f), the proximity function m(r, f) and the counting function (reduced counting function) of a-points of f. With the help of the standard notations, we would like to recall the following useful terms, namely exponent of convergence of zeros, order and hyper-order of f respectively defined as follows: log N λðf Þ 1⁄4 lim sup r;.
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