CHARACTERISATIONS OF HARDY GROWTH SPACES WITH DOUBLING WEIGHTS

2014 ◽  
Vol 90 (2) ◽  
pp. 275-282 ◽  
Author(s):  
EVGUENI DOUBTSOV
Keyword(s):  
Unit Disc ◽  
Holomorphic Functions ◽  
Lipschitz Functions ◽  
Doubling Condition ◽  
Integral Means ◽  
Doubling Weights ◽  
Growth Space ◽  
Solid Hull

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H(\mathbb{D})$ denote the space of holomorphic functions on the unit disc $\mathbb{D}$. Given $p>0$ and a weight $\omega $, the Hardy growth space $H(p, \omega )$ consists of those $f\in H(\mathbb{D})$ for which the integral means $M_p(f,r)$ are estimated by $C\omega (r)$, $0<r<1$. Assuming that $p>1$ and $\omega $ satisfies a doubling condition, we characterise $H(p, \omega )$ in terms of associated Fourier blocks. As an application, extending a result by Bennett et al. [‘Coefficients of Bloch and Lipschitz functions’, Illinois J. Math. 25 (1981), 520–531], we compute the solid hull of $H(p, \omega )$ for $p\ge 2$.

scholarly journals A note on a space Hp, a of holomorphic functions

1987 ◽  
Vol 35 (3) ◽  
pp. 471-479
Author(s):  
H. O. Kim ◽  
S. M. Kim ◽  
E. G. Kwon
Keyword(s):  
Hardy Space ◽  
Complex Plane ◽  
Unit Disc ◽  
Weak Type ◽  
Holomorphic Functions ◽  
Embedding Theorem ◽  
Type Operator ◽  
Taylor Coefficients ◽  
Bounded Holomorphic

For 0 < p < ∞ and 0 ≤a; ≤ 1, we define a space Hp, a of holomorphic functions on the unit disc of the complex plane, for which Hp, 0 = H∞, the space of all bounded holomorphic functions, and Hp, 1 = Hp, the usual Hardy space. We introduce a weak type operator whose boundedness extends the well-known Hardy-Littlewood embedding theorem to Hp, a, give some results on the Taylor coefficients of the functions of Hp, a and show by an example that the inner factor cannot be divisible in Hp, a.

scholarly journals A note on the phase retrieval of holomorphic functions

2020 ◽  
pp. 1-8
Author(s):  
Rolando Perez
Keyword(s):  
Unit Circle ◽  
Connected Domain ◽  
Unit Disc ◽  
Phase Retrieval ◽  
Holomorphic Functions ◽  
Nevanlinna Class

Abstract We prove that if f and g are holomorphic functions on an open connected domain, with the same moduli on two intersecting segments, then $f=g$ up to the multiplication of a unimodular constant, provided the segments make an angle that is an irrational multiple of $\pi $ . We also prove that if f and g are functions in the Nevanlinna class, and if $|f|=|g|$ on the unit circle and on a circle inside the unit disc, then $f=g$ up to the multiplication of a unimodular constant.

Boundary Behavior and Quasi-Normality of Finitely Valent Holomorphic Functions

1973 ◽  
Vol 25 (4) ◽  
pp. 812-819
Author(s):  
David C. Haddad
Keyword(s):  
Complex Number ◽  
Unit Disc ◽  
Boundary Behavior ◽  
Dense Subset ◽  
Holomorphic Functions ◽  
Asymptotic Values

A function denned in a domain D is n-valent in D if f(z) — w0 has at most n zeros in D for each complex number w0. Let denote the class of nonconstant, holomorphic functions f in the unit disc that are n-valent in each component of the set . MacLane's class is the class of nonconstant, holomorphic functions in the unit disc that have asymptotic values at a dense subset of |z| = 1.

scholarly journals Some remarks on angular ranges and sequences of ρ-points for holomorphic functions

1975 ◽  
Vol 58 ◽  
pp. 69-82
Author(s):  
H. Yoshida
Keyword(s):  
Unit Disc ◽  
Holomorphic Functions ◽  
Maximum Modulus ◽  
Spherical Derivative

In this paper, we will give examples of holomorphic functions in the unit disc having singular connections between the growth of maximum modulus and angular ranges (Theorem A) as well as singular connections between the growth of spherical derivative and sequences of ρ-points (Theorem B).

scholarly journals Zeros of Lipschitz functions analytic in the unit disc.

1971 ◽  
Vol 18 (2) ◽  
pp. 129-139 ◽  
Author(s):  
B. A. Taylor ◽  
D. L. Williams
Keyword(s):  
Unit Disc ◽  
Lipschitz Functions

Composition operators on weighted spaces of holomorphic functions on the upper half plane

2018 ◽  
Vol 122 (1) ◽  
pp. 141
Author(s):  
Wolfgang Lusky
Keyword(s):  
Half Plane ◽  
Composition Operator ◽  
Unit Disc ◽  
Composition Operators ◽  
Bounded Operator ◽  
Holomorphic Functions ◽  
Weighted Spaces ◽  
Weight Functions ◽  
Spaces Of Holomorphic Functions ◽  
Upper Half Plane

We consider moderately growing weight functions $v$ on the upper half plane $\mathbb G$ called normal weights which include the examples $(\mathrm{Im} w)^a$, $w \in \mathbb G$, for fixed $a > 0$. In contrast to the comparable, well-studied situation of normal weights on the unit disc here there are always unbounded composition operators $C_{\varphi }$ on the weighted spaces $Hv(\mathbb G)$. We characterize those holomorphic functions $\varphi \colon \mathbb G \rightarrow \mathbb G$ where the composition operator $C_{\varphi } $ is a bounded operator $Hv(\mathbb G) \rightarrow Hv(\mathbb G)$ by a simple property which depends only on $\varphi $ but not on $v$. Moreover we show that there are no compact composition operators $C_{\varphi }$ on $Hv(\mathbb G)$.

scholarly journals Majorization Results for Certain Subfamilies of Analytic Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Muhammad Arif ◽  
Miraj Ul-Haq ◽  
Omar Barukab ◽  
Sher Afzal Khan ◽  
Saleem Abullah
Keyword(s):  
Holomorphic Function ◽  
Analytic Functions ◽  
Unit Disc ◽  
Holomorphic Functions ◽  
Open Unit ◽  
Open Unit Disc

Let h 1 z and h 2 z be two nonvanishing holomorphic functions in the open unit disc with h 1 0 = h 2 0 = 1 . For some holomorphic function q z , we consider the class consisting of normalized holomorphic functions f whose ratios f z / z q z and q z are subordinate to h 1 z and h 2 z , respectively. The majorization results are obtained for this class when h 1 z is chosen either h 1 z = cos z or h 1 z = 1 + sin z or h 1 z = 1 + z and h 2 z = 1 + sin z .

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