Abstract
Let f be a meromorphic function in the complex plane. A value θ ∈ [0, 2π) is called a Julia limiting direction of f if there is an unbounded sequence {zn} in the Julia set J(f) satisfying limn→∞ arg zn = θ (mod 2π). We denote by L(f) the set of all Julia limiting directions of f. Our main result is that, for any non-empty compact set E ⊆ [0, 2π) and ρ ∈ [0, ∞], there are an entire function f of infinite lower order and a transcendental meromorphic function g of order ρ such that L(f) = L(g) = E. In addition, we have also constructed some transcendental entire functions whose lower order is ρ ∈ (1/2, ∞) and whose L(f) coincides with a certain kind of compact set. To prove our results, we have established a criterion for a direction θ to be a Julia limiting direction of a function by utilizing the growth rate of the function in the direction θ. The criterion may be of independent interest.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.