Abstract
The generalized Yang’s Conjecture states that if, given an entire function f(z) and positive integers n and k, f(z)^nf^{(k)}(z) is a periodic function, then f(z) is also a periodic function. In this paper, it is shown that the generalized Yang’s conjecture is true for meromorphic functions in the case k=1. When kge 2 the conjecture is shown to be true under certain conditions even if n is allowed to have negative integer values.
Highlights
Introduction and Main ResultsYang’s Conjecture on the periodicity of transcendental entire functions, proposed in [8] and [15, Conj. 1.1], has recently prompted intensive research activity in the field of periodic entire functions, see, e.g., [10–12,15], which include results both on Yang’s Conjecture and its difference version
Remark 1.4 (1) If f is of finite order in Theorem 1.3, f (z) = eAz+B, where A is a non-zero constant
We proceed to consider the case of f (z) admitting a non-zero Picard exceptional value d, i.e. f (z) = eh(z) + d, where h(z) is an entire function
Summary
Introduction and Main ResultsYang’s Conjecture on the periodicity of transcendental entire functions, proposed in [8] and [15, Conj. 1.1], has recently prompted intensive research activity in the field of periodic entire functions, see, e.g., [10–12,15], which include results both on Yang’s Conjecture and its difference version. Keywords Meromorphic functions · Periodicity · Yang’s Conjecture Yang’s Conjecture on the periodicity of transcendental entire functions, proposed in [8] and [15, Conj.
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