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1.
The Grunsky and Teichmüller norms ϰ(f) and k(f) of a holomorphic univalent function f in a finitely connected domain D ∋ ∞ with quasiconformal extension to are related by ϰ(f) ≤ k(f). In 1985, Jürgen Moser conjectured that any univalent function in the disk Δ* = {z: |z| > 1} can be approximated locally uniformly by functions with ϰ(f) < k(f). This conjecture has been recently proved by R. Kühnau and the author. In this paper, we prove that approximation is possible in a stronger sense, namely, in the norm on the space of Schwarzian derivatives. Applications of this result to Fredholm eigenvalues are given. We also solve the old Kühnau problem on an exact lower bound in the inverse inequality estimating k(f) by ϰ(f), and in the related Ahlfors inequality. To Reiner Kühnau on his 70th birthday  相似文献   

2.
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   

3.
GivenF(z),f 1(z), ..,f n(z) defined on a finite point setE, and givenB — the set of generalised polynomials Σ k =1/n a kfk(z) — the definition of a juxtapolynomial is extended in the following manner: for a fixedλ(0<λ≦1),f(z) εB is called a generalizedλ-weak juxtapolynomial toF(z) onE if and only if there exists nog(z) εB for whichg(z)=F(z) wheneverf(z)=F(z) and |g(z)−F(z) |<λ|f(z)−F(z)| wheneverf(z)≠F(z). The properties of suchf(z) are investigated with particular attention given to the real case. This note is an extension of a part of the author’s M.Sc. Thesis under the supervision of Prof. B. Grünbaum to whom the author wishes to express his sincerest appreciation. The author also wishes to thank Dr. J. Lindenstrauss for his valuable remarks in the preparation of this paper.  相似文献   

4.
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   

5.
Summary The author considers a schlicht pseudo-conformal mapping of a domainB in the (z 1 , z 2 )-space onto the Reinhardt circular domainC in the (ξ 1 , ξ 2 )-space by a pair of functions (see (1)§ 1). The domainB is assumed to include the bicylinder ((2)§ 2) and to omit four planes zk=ak, zk=bk, k=1, 2. Upper bounds for a sequence of the coefficients μv (k) of the developments (1) are given, see p. 304. The upper bounds depend only on ak, bk, k=1, 2, and on the radii rk of the bicylinder ((2)§ 2). The bounds are obtained by using the method of the kernel function. The result can be considered as an analogue to the inequalities of Grunsky in the theory of functions of one complex variable. To Enrico Bompiani on his scientific Jubiles This paper was prepared under the sponsorship of the N. S. F.  相似文献   

6.
Classes of functionsU k, which generalize starlike functions in the same manner that the classV k of functions with boundary rotation bounded by generalizes convex functions, are defined. The radius of univalence and starlikeness is determined. The behavior off α(z) = ∫ 0 z [f'(t)]α dt is determined for various classes of functions. It is shown that the image of |z|<1 underV kfunctions contains the disc of radius 1/k centered at the origin, andV k functions are continuous in |z|≦1 with the exception of at most [k/2+1] points on |z|=1.  相似文献   

7.
Let function f(z) ≠ 0 be analytic in the unit disk and have sparse nonzero Taylor coefficients. Then the rate of decay of the function f as x → 1 − 0 depends on the rate of sparseness of its nonzero Taylor coefficients. In this paper, we consider the case f(z) = $ \sum\nolimits_{k = 0}^\infty {a_k z^{n_k } } $ \sum\nolimits_{k = 0}^\infty {a_k z^{n_k } } , where n k A 0(k + 2) p logb(k + 2).  相似文献   

8.
We study the approximation of functions f(z) that are analytic in a neighborhood of zero by finite sums of the form H n (z) = H n (h, f, {λ k }; z) = Σ k=1 n λ k h(λ k z), where h is a fixed function that is analytic in the unit disk |z| < 1 and the numbers λ k (which depend on h, f, and n) are calculated by a certain algorithm. An exact value of the radius of the convergence H n (z) → f(z), n, and an order-sharp estimate for the rate of this convergence are obtained; an application to numerical analysis is given.  相似文献   

9.
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   

10.
We prove the following result: If the function Max (log|ω -f 1(z)|, ..., log|ω -f k(z)|) is plurisubharmonic in the open setD×ℂ (D open of ℂ n ), thenf 1,...,f k are analytic functions iff 1,...,f k are continuous functions onD(k≥2). We prove also some other results.  相似文献   

11.
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   

12.
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   

13.
LetW(D) denote the set of functionsf(z)=Σ n=0 A n Z n a nzn for which Σn=0 |a n |<+∞. Given any finite set lcub;f i (z)rcub; i=1 n inW(D) the following are equivalent: (i) The generalized shift sequence lcub;f 1(z)z kn ,f 2(z)z kn+1, …,f n (z)z (k+1)n−1rcub; k=0 is a basis forW(D) which is equivalent to the basis lcub;z m rcub; m=0 . (ii) The generalized shift sequence is complete inW(D), (iii) The function has no zero in |z|≦1, wherew=e 2πiti /n.  相似文献   

14.
For β > 0 and an integer r ≥ 2, denote by [(H)\tilde]¥,br\tilde H_{\infty ,\beta }^r those 2π-periodic, real-valued functions f on ℝ, which are analytic in S β := {z: |Im z| < β} and satisfy the restriction |f (r)(z)|≤1, zS β . The optimal quadrature formulae about information composed of the values of a function and its kth (k = 1, ..., r − 1) derivatives on free knots for the classes [(H)\tilde]¥,br\tilde H_{\infty ,\beta }^r are obtained, and the error estimates of the optimal quadrature formulae are exactly determined.  相似文献   

15.
In this paper the closed convex hulls of the compact familiesC β(p), of multivalently close to convex functions of order β andV 0 k (p), of multivalent functions of bounded boundary rotation, have been determined, respectively for β≥1 andk≥2(p+1)/p. Extreme points of these convex hulls are partially characterised. For a fixed pointz 0D={z:|z|<1}, a new familyC β(p, z0) is defined through Montel normalisation and its closed convex hull is also foud. Sharp coefficient estimates for functions subordinate to or majorised by some function inC β(p) orC' β(p) are discussed for β>0. It is shown that iff is subordinate to some function inC β(p) then each Taylor coefficient off is dominated by the corresponding coefficient of the function .  相似文献   

16.
In this paper, we find all the forms of meromorphic functions f(z) that share the value 0 CM, and share b(z)IM with g(z)=a1(z)f(z)+a2(z)f(z). And a1(z), a2(z) and b(z) (a2(z),b(z)?0) be small functions with respect to f(z). As an application, we show that some of nonlinear differential equations have no transcendental meromorphic solution.  相似文献   

17.
We consider the complex differential equations of the form
Ak(z)f(k)+Ak−1(z)f(k−1)+?+A1(z)f+A0(z)f=F(z),  相似文献   

18.
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   

19.
In this paper, using the matrix skills and operator theory techniques we characterize the commutant of analytic Toeplitz operators on Bergman space. For f(z) = z^ng(z) (n ≥1), g(z) = b0 + b1z^p1 +b2z^p2 +.. , bk ≠ 0 (k = 0, 1, 2,...), our main result is =A′(Mf) = A′(Mzn)∩A′(Mg) = A′(Mz^s), where s = g.c.d.(n,p1,p2,...). In the last section, we study the relation between strongly irreducible curve and the winding number W(f,f(α)), α ∈ D.  相似文献   

20.
We prove that, for every sequence (a k) of complex numbers satisfying the conditions Σ(1/|a k |) < ∞ and |a k+1| − |a k | ↗ ∞ (k → ∞), there exists a continuous functionl decreasing to 0 on [0, + ∞] and such thatf(z) = Π(1 −z/|a k |) is an entire function of finitel-index.  相似文献   

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