Remark on Integral Means of Derivatives of Blaschke Products

2018 ◽  
Vol 61 (3) ◽  
pp. 640-649 ◽  
Author(s):  
Atte Reijonen
Keyword(s):  
Blaschke Product ◽  
Unit Disc ◽  
Regularity Condition ◽  
Blaschke Products ◽  
Integral Means ◽  
Area Measure ◽  
Lebesgue Area ◽  
Derivatives Of

AbstractIf B is the Blachke product with zeros {zn}, then , whereMoreover, it is a well-known fact that, for 0 < p < ∞,is bounded if and only if Mp(r, ΨB) is bounded. We find a Blaschke product B0 such that Mp(r, ) and Mp(r, ) are not comparable for any < p < ∞. In addition, it is shown that, if 0 < p < ∞, B is a Carleson–Newman Blaschke product and a weight ω satisfies a certain regularity condition, thenwhere d A(z) is the Lebesgue area measure on the unit disc.

scholarly journals INTEGRAL MEANS OF THE DERIVATIVES OF BLASCHKE PRODUCTS

2008 ◽  
Vol 50 (2) ◽  
pp. 233-249 ◽  
Author(s):  
EMMANUEL FRICAIN ◽  
JAVAD MASHREGHI
Keyword(s):  
Blaschke Product ◽  
Blaschke Products ◽  
Integral Means ◽  
Rate Of Growth ◽  
Model Subspace ◽  
Derivatives Of

AbstractWe study the rate of growth of some integral means of the derivatives of a Blaschke product and we generalize several classical results. Moreover, we obtain the rate of growth of integral means of the derivative of functions in the model subspaceKBgenerated by the Blaschke productB.

Approximation by Unimodular Functions

1971 ◽  
Vol 23 (2) ◽  
pp. 257-269 ◽  
Author(s):  
Stephen Fisher
Keyword(s):  
Blaschke Product ◽  
Uniform Approximation ◽  
Unit Disc ◽  
Open Unit ◽  
Blaschke Products ◽  
Open Unit Disc ◽  
Pointwise Approximation ◽  
Finite Blaschke Product ◽  
Image Position ◽  
Restricted Zeros

The theorems in this paper are all concerned with either pointwise or uniform approximation by functions which have unit modulus or by convex combinations of such functions. The results are related to, and are outgrowths of, the theorems in [4; 5; 10].In § 1, we show that a function bounded by 1, which is analytic in the open unit disc Δ and continuous on may be approximated uniformly on the set where it has modulus 1 (subject to certain restrictions; see Theorem 1) by a finite Blaschke product; that is, by a function of the form*where |λ| = 1 and |αi| < 1, i = 1, …, N. In § 1 we also discuss pointwise approximation by Blaschke products with restricted zeros.

Integral Means and Zero Distributions of Blaschke Products

1972 ◽  
Vol 24 (5) ◽  
pp. 755-760 ◽  
Author(s):  
C. N. Linden
Keyword(s):  
Half Plane ◽  
Blaschke Product ◽  
Blaschke Products ◽  
Integral Means ◽  
Elementary Theorem ◽  
Image Position ◽  
Analogous Theorem

A sequence {zn} in D = {z: |z| < 1} is a Blaschke sequence if and only ifIf 0 appears m times in {zn} thenis the Blaschke product defined by {zn}. The set of all Blaschke products will be denoted by . If B ∊ it is well-known that B is regular in D, and |B(z, {zn})| < 1 when z ∊ D.For a given pair of values p in (0, ∞) and q in [0, ∞) we denote by ℐ(p, g) the class of all Blaschke products B(z, {zn}) such thatas r → 1 — 0. In the case q ≦ max(p — 1,0) the classes of functions B and ℐ(p, q) are identical: this is a particular case of an elementary theorem for functions subharmonic in a disc, the analogous theorem for functions subharmonic in a half-plane appearing in [1],

scholarly journals Derivatives of Blaschke Products and Model Space Functions

2019 ◽  
Vol 63 (4) ◽  
pp. 716-725
Author(s):  
David Protas
Keyword(s):  
Blaschke Product ◽  
Bergman Spaces ◽  
Model Space ◽  
Weighted Bergman Spaces ◽  
Blaschke Products ◽  
Distribution Of Zeros ◽  
The Relationship ◽  
Derivatives Of

AbstractThe relationship between the distribution of zeros of an infinite Blaschke product $B$ and the inclusion in weighted Bergman spaces $A_{\unicode[STIX]{x1D6FC}}^{p}$ of the derivative of $B$ or the derivative of functions in its model space $H^{2}\ominus \mathit{BH}^{2}$ is investigated.

scholarly journals INTEGRABILITY OF THE DERIVATIVE OF A BLASCHKE PRODUCT

2007 ◽  
Vol 50 (3) ◽  
pp. 673-687 ◽  
Author(s):  
Daniel Girela ◽  
José Ángel Peláez ◽  
Dragan Vukotić
Keyword(s):  
Blaschke Product ◽  
Bergman Spaces ◽  
Blaschke Products ◽  
Hardy And Bergman Spaces ◽  
Interpolating Blaschke Products ◽  
Derivatives Of

AbstractWe study the membership of derivatives of Blaschke products in Hardy and Bergman spaces, especially for the the interpolating Blaschke products and for those whose zeros lie in a Stolz domain. We obtain new and very simple proofs of some known results and prove new theorems that complement or extend the earlier works of Ahern, Clark, Cohn, Kim, Newman, Protas, Rudin, Vinogradov and others.

LOGARITHMIC DERIVATIVE OF THE BLASCHKE PRODUCT WITH SLOWLY INCREASING COUNTING FUNCTION OF ZEROS

2021 ◽  
Vol 9 (1) ◽  
pp. 164-170
Author(s):  
Y. Gal ◽  
M. Zabolotskyi ◽  
M. Mostova
Keyword(s):  
Blaschke Product ◽  
Meromorphic Functions ◽  
Logarithmic Derivative ◽  
Counting Function ◽  
Finite System ◽  
Blaschke Products ◽  
Nevanlinna Characteristic ◽  
Number Of Zeros ◽  
Accumulation Points ◽  
Theory Of Value

The Blaschke products form an important subclass of analytic functions on the unit disc with bounded Nevanlinna characteristic and also are meromorphic functions on $\mathbb{C}$ except for the accumulation points of zeros $B(z)$. Asymptotics and estimates of the logarithmic derivative of meromorphic functions play an important role in various fields of mathematics. In particular, such problems in Nevanlinna's theory of value distribution were studied by Goldberg A.A., Korenkov N.E., Hayman W.K., Miles J. and in the analytic theory of differential equations -- by Chyzhykov I.E., Strelitz Sh.I. Let $z_0=1$ be the only boundary point of zeros $(a_n)$ %=1-r_ne^{i\psi_n},$ $-\pi/2+\eta<\psi_n<\pi/2-\eta,$ $r_n\to0+$ as $n\to+\infty,$ of the Blaschke product $B(z);$ $\Gamma_m=\bigcup\limits_{j=1}^{m}\{z:|z|<1,\mathop{\text{arg}}(1-z)=-\theta_j\}=\bigcup\limits_{j=1}^{m}l_{\theta_j},$ $-\pi/2+\eta<\theta_1<\theta_2<\ldots<\theta_m<\pi/2-\eta,$ be a finite system of rays, $0<\eta<1$; $\upsilon(t)$ be continuous on $[0,1)$, $\upsilon(0)=0$, slowly increasing at the point 1 function, that is $\upsilon(t)\sim\upsilon\left({(1+t)}/2\right),$ $t\to1-;$ $n(t,\theta_j;B)$ be a number of zeros $a_n=1-r_ne^{i\theta_j}$ of the product $B(z)$ on the ray $l_{\theta_j}$ such that $1-r_n\leq t,$ $0<t<1.$ We found asymptotics of the logarithmic derivative of $B(z)$ as $z=1-re^{-i\varphi}\to1,$ $-\pi/2<\varphi<\pi/2,$ $\varphi\neq\theta_j,$ under the condition that zeros of $B(z)$ lay on $\Gamma_m$ and $n(t,\theta_j;B)\sim \Delta_j\upsilon(t),$ $t\to1-,$ for all $j=\overline{1,m},$ $0\leq\Delta_j<+\infty.$ We also considered the inverse problem for such $B(z).$

Angular and Tangential Limits of Blaschke Products and their Successive Derivatives

1962 ◽  
Vol 14 ◽  
pp. 334-348 ◽  
Author(s):  
G. T. Cargo
Keyword(s):  
Blaschke Product ◽  
Common Knowledge ◽  
Holomorphic Functions ◽  
Blaschke Products ◽  
Radial Limit ◽  
Radial Limits ◽  
Almost Everywhere ◽  
Image Position ◽  
Bounded Holomorphic ◽  
Tangential Limits

In this paper, we shall be concerned with bounded, holomorphic functions of the formwhere(1)(2)and(3)B(z{an}) is called a Blaschke product, and any sequence {an} which satisfies (2) and (3) is called a Blaschke sequence. For a general discussion of the properties of Blaschke products, see (18, pp. 271-285) or (14, pp. 49-52).According to a theorem due to Riesz (15), a Blaschke product has radial limits of modulus one almost everywhere on C = {z: |z| = 1}. Moreover, it is common knowledge that, if a Blaschke product has a radial limit at a point, then it also has an angular limit at the point (see 14, p. 19 and 6, p. 457).

On Logarithmic Derivatives of Functions in a Class of Starlike Mappings

1993 ◽  
Vol 36 (1) ◽  
pp. 38-44
Author(s):  
Alan D. Gluchoff
Keyword(s):  
Integral Means ◽  
Koebe Function ◽  
Starlike Mappings ◽  
Derivatives Of ◽  
Logarithmic Derivatives

AbstractThe purpose of this paper is to prove some facts about integral means of (d2/dz2)(log[f(z)/z])—or equivalently f″/f, for f in a class of starlike mappings of a "singular" nature. In particular it is noted that the Koebe function is not extremal for the Hardy means Mp(r,f″/f) for functions in this class.

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