Stolz Angle Research Articles

Oscillation of Solutions of LDE\u2019s in Domains Conformally Equivalent to Unit Disc

Oscillation of solutions of f^{(k)} + a_{k-2} f^{(k-2)} + cdots + a_1 f' +a_0 f = 0 is studied in domains conformally equivalent to the unit disc. The results are applied, for example, to Stolz angles, horodiscs, sectors, and strips. The method relies on a new conformal transformation of higher order linear differential equations. Information on the existence of zero-free solution bases is also obtained.

A strong form of Plessner's theorem

Let f be a holomorphic, or even meromorphic, function on the unit disc. Plessner's theorem then says that, for almost every boundary point ζ, either (i) f has a finite nontangential limit at ζ, or (ii) the image f(S) of any Stolz angle S at ζ is dense in the complex plane. This paper shows that statement (ii) can be replaced by a much stronger assertion. This new theorem and its analogue for harmonic functions on halfspaces also strengthen classical results of Spencer, Stein and Carleson.

A Note on Recursive Interpolation for the Lipschitz Class

This note is framed in the field of complex analysis and deals with some types of interpolating sequences for Lipschitz functions in the unit disk. We introduce recursion between each point of a sequence and the next. We also add interpolation by the derivative, linking its values to those that the function takes. On the supposition that the sequences are quite contractive and lie in a Stolz angle, we relate the interpolating ones for each type to the uniformly separated sequences.

Multiple Interpolation and Principal Parts of a Laurent Series for Meromorphic Functions in the Unit Disk with Power Growth of the Nevanlinna Characteristic

In this paper we solve the multiple interpolation problem in the class of analytic functions in the unit disk with power growth of the Nevanlinna characteristic under the condition that interpolation nodes are contained in a finite union of Stolz angles and describe the principal parts of a Laurent series of meromorphic functions with the same restrictions on the Nevanlinna characteristic.

On Interpolation in the Class of Analytic Functions in the Unit Disk with the Nevanlinna Characteristic from Lp-spaces

Received 02.11.2015, received in revised form 08.12.2015, accepted 12.01.2016 In this paper we solve the interpolation problem for the class of analytic functions in the unit disk with the Nevanlinna characteristic from Lp-spaces under the condition that interpolation nodes are contained in a finite union of Stolz angles.

Stolz angle limit of a certain class of self-mappings of the unit disk

Let f be a mapping of the open unit disk U onto itself having a non-singular differentiable extension to the boundary point 1 which is a fixed point of f. For a∈U let p and q be Möbius transformations of the unit disk onto itself such that p(0)=a and q(f(a))=0. It is proved that the Stolz angle limit of p∘f∘q when a→1 is a diffeomorphic self-mapping g of the unit disk, which is a conjugate of an affine transformation. The convergence is almost uniform in U.

Inner Functions in Lipschitz, Besov, and Sobolev Spaces

We study the membership of inner functions in Besov, Lipschitz, and Hardy‐Sobolev spaces, finding conditions that enable an inner function to be in one of these spaces. Several results in this direction are given that complement or extend previous works on the subject from different authors. In particular, we prove that the only inner functions in either any of the Hardy‐Sobolev spaces with 1/p ≤ α < ∞ or any of the Besov spaces with 0 < p, q ≤ ∞ and α ≥ 1/p, except when p = ∞, α = 0, and 2 < q ≤ ∞ or when 0 < p < ∞, q = ∞, and α = 1/p are finite Blaschke products. Our assertion for the spaces , 0 < q ≤ 2, follows from the fact that they are included in the space VMOA. We prove also that for 2 < q < ∞, VMOA is not contained in and that this space contains infinite Blaschke products. Furthermore, we obtain distinct results for other values of α relating the membership of an inner function I in the spaces under consideration with the distribution of the sequences of preimages {I−1(a)}, |a | < 1. In addition, we include a section devoted to Blaschke products with zeros in a Stolz angle.

Interpolation of bounded sequences

This paper deals with an interpolation problem in the open unit disc ⅅ of the complex plane. We characterize the sequences in a Stolz angle of ⅅ, verifying that the bounded sequences are interpolated on them by a certain class of not bounded holomorphic functions on ⅅ, but very close to the bounded ones. We prove that these interpolating sequences are also uniformly separated, as in the case of the interpolation by bounded holomorphic functions.

On Blaschke products, Bloch functions and normal functions

We prove that if G is an analytic function in the unit disc such that G(z)→∞, as z→1, and B is an infinite Blaschke product whose sequence of zeros is contained in a Stolz angle with vertex at 1 then the function f=B⋅G is not a normal function.

Blaschke products in Lipschitz spaces

AbstractWe study the membership of Blaschke products in Lipschitz spaces, especially for interpolating Blaschke products and for those whose zeros lie in a Stolz angle. We prove several theorems that complement or extend the earlier works of Ahern and the author.

Interpolating Blaschke Products: Stolz and Tangential Approach Regions

A result of D.J. Newman asserts that a uniformly separated sequence contained in a Stolz angle is a finite union of exponential sequences. We extend this by obtaining several equivalent characterizations of such sequences. If the zeros of a Blaschke product B lie in a Stolz angle, then $B^\prime\in A^p$ for all $p<\frac 32$ and it has recently been shown that this result cannot be improved. Also, Newman's result can be used to prove that if B is an interpolating Blaschke product whose zeros lie in a Stolz angle, then $B^\prime\in\bigcap_{0<p<1}H^p\subset \bigcap_{0<p<2}A^p$ . In this paper we prove that if the zeros of an interpolating Blaschke product lie in a disk internally tangent to the unit circle, then $B^\prime\in \bigcap_{0<p<3/2}A^p$ and we show that this cannot be improved. We also obtain sharp results concerning the membership in Bergman spaces of the derivative of an interpolating Blaschke product whose sequence of zeros lies in other regions internally tangent to the unit circle.

Equilibrium points of logarithmic potentials on convex domains

Let D D be a convex domain in C \mathbb {C} . Let a k &gt; 0 a_k &gt; 0 be summable constants and let z k ∈ D z_k \in D . If the z k z_k converge sufficiently rapidly to ζ ∈ ∂ D \zeta \in \partial D from within an appropriate Stolz angle, then the function ∑ k = 1 ∞ a k / ( z − z k ) \sum _{k=1}^\infty a_k /( z - z_k ) has infinitely many zeros in D D . An example shows that the hypotheses on the z k z_k are not redundant and that two recently advanced conjectures are false.

Uniformly discrete sequences in regions with tangential approach to the unit circle¶

A known result of Newman and Tse asserts that every uniformly discrete sequence contained in a Stolz angle is uniformly separated (see Newman, D.J., 1959, Interpolation in . Transactions of the American Mathematical Society, 92(3), 501–507; Tse, K.-F., 1971, Nontangential interpolating sequences and interpolation by normal functions. Proceedings of the American Mathematical Society, 29, 351–354). We prove that this statement no longer holds if the sequence is located in a tangential region of certain kind. It is well known that a uniformly discrete sequence need not be a Blaschke sequence. We show, however, that every uniformly discrete sequence inside a disc tangential to the unit circle must be a Blaschke sequence.

On the Membership in Bergman Spaces of the Derivative of a Blaschke Product With Zeros in a Stolz Domain

AbstractIt is known that the derivative of a Blaschke product whose zero sequence lies in a Stolz angle belongs to all the Bergman spaces Ap with 0 &lt; p &lt; 3/2. The question of whether this result is best possible remained open. In this paper, for a large class of Blaschke products B with zeros in a Stolz angle, we obtain a number of conditions which are equivalent to the membership of B′ in the space Ap (p &lt; 1). As a consequence, we prove that there exists a Blaschke product B with zeros on a radius such that B′ ∉ A3/2.

On Conversion of the L'Hopital Rule for Analytic Functions

We consider the question about the possibility of conversion of the L'Hopital rule for the limits of ratios of analytic functions at a boundary point of the analyticity domain in a Stolz angle.

On the zeros of functions in Bergman spaces and in some other related classes of functions

A well-known theorem of H.S. Shapiro and A.L. Shields implies that if f ≢ 0 is a function which belongs to the Bergman space A p ( 0 < p < ∞ ) and { z k } is a sequence of zeros of f which is contained in a Stolz angle, then { z k } satisfies the Blaschke condition. In this paper we improve this result. We consider a large class of regions contained in the unit disc D which touch ∂ D at a point ξ tangentially and we prove that the mentioned result remains true if we substitute a Stolz angle by any of these regions of tangential approach.

Hadamard products with power functions and multipliers of Hardy spaces

We consider Hadamard products of power functions P( z)=(1− z) − b with functions analytic in the open unit disk in the complex plane, and an integral representation is obtained when 0<Re b<2. Let μ n=∫ Δ ̄ ζ n dμ(ζ) where μ is a complex-valued measure on the closed unit disk Δ ̄ . Such sequences are shown to be multipliers of H p for 1⩽ p⩽∞. Moreover, if the support of μ is contained in a finite set of Stolz angles with vertices on the unit circle, we prove that { μ n } is a multiplier of H p for every p>0. When the support of μ is [0,1] we get the multiplier sequence ∫ 0 1t n dμ(t), which provides more concrete applications. We show that if the sequences { μ n } and { ν n } are related by an asymptotic expansion ν n μ n ≈ ∑ k=0 ∞ A k n k (n→∞) and μ n is a multiplier of H p into H q , then so is ν n . We ask whether {( n+1) iβ } is a multiplier of H p when β is a nonzero real number. It is clear that the question has an affirmative answer when p=2. The answer is shown to be negative when p=∞.

Boundary Behavior in a Stolz Angle of Analytic Functions on the Disk

In this paper, we study the behavior of analytic functions (on the disk) in a Stolz angle. In particular, we prove assertions of the type of the Hardy--Littlewood theorem and obtain estimates for the growth of functions in a Stolz angle.

Radial growth and exceptional sets for Cauchy–Stieltjes integrals

This paper considers the radial and nontangential growth of a function f given bywhere α&gt;0 and μ is a complex-valued Borel measure on the unit circle. The main theorem shows how certain local conditions on μ near eiθ affect the growth of f(z) as z→eiθ in Stolz angles. This result leads to estimates on the nontangential growth of f where exceptional sets occur having zero β-capacity.

Analytic functions and quasiconformal mappings in stolz angles and cones

Let f be analytic and bounded in the unit disk B 2 or quasiconformal in the unit ball Bn of , and let z 0 be a boundary point of B 2 or Bn . We assume that for a suitable majornat μ and for all z on ∂ B 2 when f is analytic, or for all z on an from 0 to z 0 when f is quasiconforml. We then obtain estimates for when z 1 and z 2 lie in a Stolz angle or cone with vertex at z 0.