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1.
LetM(z)=z n +…,N(z)=z n +… be analytic in the unit disc Δ and let λ(z)=N(z)/zN′(z). The classical result of Sakaguchi-Libera shows that Re(M′(z)/N′(z))<0 implies Re(M(z)/N(z))>0 in Δ whenever Re(λ(z))>0 in Δ. This can be expressed in terms of differential subordination as follows: for anyp analytic in Δ, withp(0)=1,p(z)+λ(z)zp′(z)<1+z/1−z impliesp(z)<1+z/1−z, for Reλ(z)>0,z∈Δ. In this paper we determine different type of general conditions on λ(z),h(z) and ϕ(z) for which one hasp(z)+λ(z)zp′(z)<h(z) impliesp(z)<ϕ(z)<h(z) z∈Δ. Then we apply the above implication to obtain new theorems for some classes of normalized analytic funotions. In particular we give a sufficient condition for an analytic function to be starlike in Δ.  相似文献   

2.
The paper studies the region of values of the system {c 2, c 3, f(z 1), f′(z 1)},where z 1 is an arbitrary fixed point of the disk |z| < 1; fT,and the class T consists of all the functions f(z) = z + c 2 z 2 + c 3z3 + ⋯ regular in the disk |z| < 1 that satisfy the condition Im z · Im f(z) > 0 for Im z ≠ 0. The region of values of f′(z 1) in the subclass of functions fT with prescribed values c 2, c 3, and f(z 1) is determined. Bibliography: 10 titles.  相似文献   

3.
A criterion of normality based on a single holomorphic function   总被引:1,自引:0,他引:1  
Let F be a family of functions holomorphic on a domain D ⊂ ℂ Let k ≥ 2 be an integer and let h be a holomorphic function on D, all of whose zeros have multiplicity at most k −1, such that h(z) has no common zeros with any fF. Assume also that the following two conditions hold for every fF: (a) f(z) = 0 ⇒ f′(z) = h(z); and (b) f′(z) = h(z) ⇒ |f (k)(z)| ≤ c, where c is a constant. Then F is normal on D.  相似文献   

4.
On Homogeneous Differential Polynomials of Meromorphic Functions   总被引:2,自引:0,他引:2  
In this paper, we study one conjecture proposed by W. Bergweiler and show that any transcendental meromorphic functions f(z) have the form exp(αz+β) if f(z)f″(z)–a(f′ (z))2≠0, where . Moreover, an analogous normality criterion is obtained. Supported by National Natural Science Foundation and Science Technology Promotion Foundation of Fujian Province (2003)  相似文献   

5.
For an analytic function f (z) on the unit disk |z| < 1 with f (0) = f′(0) − 1 = 0 and f (z) ≠ 0, 0 < |z| < 1, we consider the power deformation f c (z) = z(f (z)/z) c for a complex number c. We determine those values c for which the operator maps a specified class of univalent functions into the class of univalent functions. A little surprisingly, we will see that the set is described by the variability region of the quantity zf′(z)/ f (z), |z| < 1, for most of the classes that we consider in the present paper. As an unexpected by-product, we show boundedness of strongly spirallike functions.  相似文献   

6.
Let T be the family of all typically real functions, i.e. functions that are analytic in the unit disk Δ:= {z ∈ ℂ: |z| < 1}, normalized by f(0) = f′(0) − 1 = 0 and such that Imz Im f(z) ⩾ 0 for z ∈ Δ.  相似文献   

7.
In this paper, we obtain a version of subordination lemma for hyperbolic disk relative to hyperbolic geometry on the unit disk D. This subordination lemma yields the distortion theorem for Bloch mappings f ∈ H(B^n) satisfying ||f||0 = 1 and det f'(0) = α ∈ (0, 1], where||f||0 = sup{(1 - |z|^2 )n+1/2n det(f'(z))[1/n : z ∈ B^n}. Here we establish the distortion theorem from a unified perspective and generalize some known results. This distortion theorem enables us to obtain a lower bound for the radius of the largest univalent ball in the image of f centered at f(0). When a = 1, the lower bound reduces to that of Bloch constant found by Liu. When n = 1, our distortion theorem coincides with that of Bonk, Minda and Yanagihara.  相似文献   

8.
LetS denote the usual class of functionsf holomorphic and univalent in the unit diskU such thatf(0)=f′(0)−1=0. The main result of the paper is that area (f(U) ≥27π/7)(2-α)−2 for allfS such that |f″(0)|=2α, 1/2<α<2. This solves a long-standing extremal problem for the class of functions considered.  相似文献   

9.
Let f(z) = e2πiθz(1 z/d)d,θ∈R\Q be a polynomial. Ifθis an irrational number of bounded type, it is easy to see that f(z) has a Siegel disk centered at 0. In this paper, we will show that the Hausdorff dimension of the Julia set of f(z) satisfies Dim(J(f))<2.  相似文献   

10.
Let k be a positive integer, let M be a positive number, let F be a family of meromorphic functions in a domain D, all of whose zeros are of multiplicity at least k, and let h be a holomorphic function in D, h ≢ 0. If, for every fF, f and f (k) share 0, and |f(z)| ≥ M whenever f (k)(z) = h(z), then F is normal in D. The condition that f and f (k) share 0 cannot be weakened, and the condition that |f(z)| ≥ M whenever f (k)(z) = h(z) cannot be replaced by the condition that |f(z)| ≥ 0 whenever f (k)(z) = h(z). This improves some results due to Fang and Zalcman [2] etc.  相似文献   

11.
Assume thatf is an integer transcendental solution of the differential equationP n (z, f, f′)=P n−1(z, f, f′, ... f (p)), whereP n andP n−1 are polynomials in all variables, the degree ofP n with respect tof andf′ is equal ton, and the degree ofP n−1 with respect tof, f′, ... f (p) is at mostn−1. We prove that the order ρ of growth off satisfies the relation 1/2≤ρ<∞. We also prove that if ρ=1/2, then, for a certain real ν, in the domain {z: ν<argz<ν+2π}/E *, whereE * is a certain set of disks with finite sum of radii, the estimate lnf(z)=z 1/2 (β+o(1)), β∈C, holds forz=re iϕ,rr(ϕ)≥0. Furthermore, on the ray {z: argz=ν}, the following relation is true: ln‖f(re iν)‖=o(r 1/2),r→+∞,r>0, , where Δ is a certain set on the semiaxisr>0 with mes Δ<∞. “L'vivs'ka Politekhnika” University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 69–77, January, 1999.  相似文献   

12.
The authors discuss the normality concerning holomorphic functions and get the following result. Let F be a family of holomorphic functions on a domain D ⊂ ℂ, all of whose zeros have multiplicity at least k, where k ≥ 2 is an integer. And let h(z) ≢ 0 be a holomorphic function on D. Assume also that the following two conditions hold for every fF: (a) f(z) = 0 ⇒ |f (k)(z)| < |h(z)|; (b) f (k)(z) ≠ h(z). Then F is normal on D.  相似文献   

13.
Oscillation of Solutions of Linear Differential Equations   总被引:1,自引:0,他引:1  
This paper is devoted to studying the growth problem, the zeros and fixed points distribution of the solutions of linear differential equations f″+e^-zf′+Q(z)f=F(z),whereQ(z)≡h(z)e^cz and c∈R.  相似文献   

14.
Let Co(α) denote the class of concave univalent functions in the unit disk \mathbbD{\mathbb{D}}. Each function f ? Co(a){f\in Co(\alpha)} maps the unit disk \mathbbD{\mathbb{D}} onto the complement of an unbounded convex set. In this paper we find the exact disk of variability for the functional (1-|z|2)( f¢¢(z)/f(z)), f ? Co(a){(1-|z|^2)\left ( f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)}. In particular, this gives sharp upper and lower estimates for the pre-Schwarzian norm of concave univalent functions. Next we obtain the set of variability of the functional (1-|z|2)(f¢¢(z)/f(z)), f ? Co(a){(1-|z|^2)\left(f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)} whenever f′′(0) is fixed. We also give a characterization for concave functions in terms of Hadamard convolution. In addition to sharp coefficient inequalities, we prove that functions in Co(α) belong to the H p space for p < 1/α.  相似文献   

15.
Letf be a holomorphic self-map of the punctured plane ℂ*=ℂ\{0} with essentially singular points 0 and ∞. In this note, we discuss the setsI 0(f)={z ∈ ℂ*:f n (z) → 0,n → ∞} andI (f)={z ∈ ℂ*:f n (z) → 0,n → ∞}. We try to find the relation betweenI 0(f),I (t) andJ(f). It is proved that both the boundary ofI 0(f) and the boundary ofI )f) equal toJ(f),I 0(f) ∩J(f) ≠ θ andI (f) ∩J(f) ≠ θ. As a consequence of these results, we find bothI 0(f) andI (f) are not doubly-bounded. Supported by the National Natural Science Foundation of China  相似文献   

16.
Denote byS * (⌕), (0≤⌕<1), the family consisting of functionsf(z)=z+a 2z2+...+anzn+... that are analytic and starlike of order ⌕, in the unit disc ⋎z⋎<1. In the present article among other things, with very simple conditions on μ, ⌕ andh(z) we prove the f’(z) (f(z)/z)μ−1<h(z) implies f∈S*(⌕). Our results in this direction then admit new applications in the study of univalent functions. In many cases these results considerably extend the earlier works of Miller and Mocanu [6] and others.  相似文献   

17.
Let p be a prime and f(z)=∑ n a(n)q n be a weakly holomorphic modular function for \varGamma 0*(p2)\varGamma _{0}^{*}(p^{2}) with a(0)=0. We use Bruinier and Funke’s work to find the generating series of modular traces of f(z) as Jacobi forms. And as an application we construct Borcherds products related to the Hauptmoduln jp2*j_{p^{2}}^{*} for genus zero groups \varGamma 0*(p2)\varGamma _{0}^{*}(p^{2}).  相似文献   

18.
We consider the differential operators Ψ k , defined by Ψ1(y) =y and Ψ k+1(y)=yΨ k y+d/dz k (y)) fork ∈ ℕ fork∈ ℕ. We show that ifF is meromorphic in ℂ and Ψ k F has no zeros for somek≥3, and if the residues at the simple poles ofF are not positive integers, thenF has the formF(z)=((k-1)z+a)/(z 2+β z+γ) orF(z)=1/(az+β) where α, β, γ ∈ ℂ. If the residues at the simple poles ofF are bounded away from zero, then this also holds fork=2. We further show that, under suitable additional conditions, a family of meromorphic functionsF is normal if each Ψ k (F) has no zeros. These conditions are satisfied, in particular, if there exists δ>0 such that Re (Res(F, a)) <−δ for all polea of eachF in the family. Using the fact that Ψ k (f /f) =f (k)/f, we deduce in particular that iff andf (k) have no zeros for allf in some familyF of meromorphic functions, wherek≥2, then {f /f :fF} is normal. The first author is supported by the German-Israeli Foundation for Scientific Research and Development G.I.F., G-643-117.6/1999, and INTAS-99-00089. The second author thanks the DAAD for supporting a visit to Kiel in June–July 2002. Both authors thank Günter Frank for helpful discussions.  相似文献   

19.
 We study traveling waves of a discrete system
where f and g are Lipschitz continuous with g increasing and f monostable, i.e., f(0)=f(1)=0 and f>0 on (0,1). We show that there is a positive c min such that a traveling wave of speed c exists if and only if cc min. Also, we show that traveling waves are unique up to a translation if f′(0)>0>f′(1) and g′(0)>0. The tails of traveling waves are also investigated. Received: 28 February 2002 / Published online: 28 March 2003 This work was partially supported by the National Science Council of the Republic of China under the grants NSC 89-2735-M-001D-002 and 89-2115-M-003-014. Chen thanks the support from the National Science Foundation Grant DMS-9971043.  相似文献   

20.
Let E and F be Banach spaces, f: UEF be a map of C r (r ⩾ 1), x 0U, and ft (x 0) denote the FréLechet differential of f at x 0. Suppose that f′(x 0) is double split, Rank(f′(x 0)) = ∞, dimN(f′(x 0)) > 0 and codimR(f′(x 0)) s> 0. The rank theorem in advanced calculus asks to answer what properties of f ensure that f(x) is conjugate to f′(x 0) near x 0. We have proved that the conclusion of the theorem is equivalent to one kind of singularities for bounded linear operators, i.e., x 0 is a locally fine point for f′(x) or generalized regular point of f(x); so, a complete rank theorem in advanced calculus is established, i.e., a sufficient and necessary condition such that the conclusion of the theorem to be held is given.   相似文献   

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