Suppose that f（ z ） is a transcendental entire function and that F（f） contains unbounded Fatou components. In this article, we obtained some links between the lowor bounds of the lower order of f and the angle of ...Suppose that f（ z ） is a transcendental entire function and that F（f） contains unbounded Fatou components. In this article, we obtained some links between the lowor bounds of the lower order of f and the angle of an angular sector which is completely contained in an unbounded Fatou component of F（f）. Then, we investigate the bounded components for the Julia set J（f） of a transcendental entire function f（z ） and obtain a sufficient and necessary condition.展开更多
Let f be a transcendental entire function with order ρ 〈 1/2 and let a be a sufficiently large constant. We prove that if there exists r0 〉 1 such that, for all r 〉 r0 and any small ε 〉0,M（r^σ,f）≥M（r,f）σ...Let f be a transcendental entire function with order ρ 〈 1/2 and let a be a sufficiently large constant. We prove that if there exists r0 〉 1 such that, for all r 〉 r0 and any small ε 〉0,M（r^σ,f）≥M（r,f）σ＋ε.then every component of the Fatou set F（f） is bounded.展开更多
In this paper, we discuss the rational maps Fλ（z）=z^n＋λ/z^n,n≥2with the positive real parameter ）λ. It is shown that the immediately attracting basin Bλ of ∞ for Fλ is always a Jordan domain if the Julia se...In this paper, we discuss the rational maps Fλ（z）=z^n＋λ/z^n,n≥2with the positive real parameter ）λ. It is shown that the immediately attracting basin Bλ of ∞ for Fλ is always a Jordan domain if the Julia set of Fλ is not a Cantor set. Fuhermore, Bλ is a quasidisk if there is no parabolic fixed point on the boundary of Bλ. It is also shown that if the Julia set of Fλ is connected, then it is locally connected and all Fatou components are Jordan domains. Finally, a complete description to the problem when the Julia set is a Sierpirlski curve is given.展开更多
Let f and g be two permutable transcendental entire functions. In this paper, we first prove that J(fg)=J(f n g m) for any positive integers n and m . Then we prove that the function h(p(z))+az ∈/ B , where h(z) is...Let f and g be two permutable transcendental entire functions. In this paper, we first prove that J(fg)=J(f n g m) for any positive integers n and m . Then we prove that the function h(p(z))+az ∈/ B , where h(z) is any transcendental entire function with h′(z)=0 having infinitely many solutions, p(z) is a polynomial with deg p ≥2 and a(≠0) ∈ C .展开更多
This article studies the inverse image of rational functions. Several theorems are obtained on the Julia set expressed by the inverse image, and a mistake is pointed out in H.Brolin＇ theorem incidentally.
Suppose that f and g are two transcendental entire functions, and h is a non-constant periodic entire function. We denote the Julia set and Fatou set off by J（f） and F（f）, respectively, lffand g are semiconjugated...Suppose that f and g are two transcendental entire functions, and h is a non-constant periodic entire function. We denote the Julia set and Fatou set off by J（f） and F（f）, respectively, lffand g are semiconjugated, that is, h · f = g · h, in this paper, we will show that z ∈ J（f） if and only if h（z） ∈ J（g） （ similarly, z F（f） if and only ifh（z） ∈ F（g））, and this extends a result of Bergweiler.展开更多
文摘Suppose that f（ z ） is a transcendental entire function and that F（f） contains unbounded Fatou components. In this article, we obtained some links between the lowor bounds of the lower order of f and the angle of an angular sector which is completely contained in an unbounded Fatou component of F（f）. Then, we investigate the bounded components for the Julia set J（f） of a transcendental entire function f（z ） and obtain a sufficient and necessary condition.
基金Supported by National Natural Science Foundation of China(Grant Nos.11261002 and 11261069)Natural Science Foundation of Yunnan Province of China(Grant No.2012FZ167)Educational Commission of Yunnan Province of China(Grant No.2012Z121)
文摘Let f be a transcendental entire function with order ρ 〈 1/2 and let a be a sufficiently large constant. We prove that if there exists r0 〉 1 such that, for all r 〉 r0 and any small ε 〉0,M（r^σ,f）≥M（r,f）σ＋ε.then every component of the Fatou set F（f） is bounded.
基金Supported by National Natural Science Foundation of China (Grant Nos. 10831004, 10871047)Science and Technology Commission of Shanghai Municipality (NSF Grant 10ZR1403700)
文摘In this paper, we discuss the rational maps Fλ（z）=z^n＋λ/z^n,n≥2with the positive real parameter ）λ. It is shown that the immediately attracting basin Bλ of ∞ for Fλ is always a Jordan domain if the Julia set of Fλ is not a Cantor set. Fuhermore, Bλ is a quasidisk if there is no parabolic fixed point on the boundary of Bλ. It is also shown that if the Julia set of Fλ is connected, then it is locally connected and all Fatou components are Jordan domains. Finally, a complete description to the problem when the Julia set is a Sierpirlski curve is given.
文摘Let f and g be two permutable transcendental entire functions. In this paper, we first prove that J(fg)=J(f n g m) for any positive integers n and m . Then we prove that the function h(p(z))+az ∈/ B , where h(z) is any transcendental entire function with h′(z)=0 having infinitely many solutions, p(z) is a polynomial with deg p ≥2 and a(≠0) ∈ C .
基金Project Supported by the National Natural Science Foundation of China(10471048)the Research Fund for the Doctoral Program of Higher Education(20050574002)
文摘This article studies the inverse image of rational functions. Several theorems are obtained on the Julia set expressed by the inverse image, and a mistake is pointed out in H.Brolin＇ theorem incidentally.
文摘Suppose that f and g are two transcendental entire functions, and h is a non-constant periodic entire function. We denote the Julia set and Fatou set off by J（f） and F（f）, respectively, lffand g are semiconjugated, that is, h · f = g · h, in this paper, we will show that z ∈ J（f） if and only if h（z） ∈ J（g） （ similarly, z F（f） if and only ifh（z） ∈ F（g））, and this extends a result of Bergweiler.
