scholarly journals On the Asymptotic Behavior of Functions Harmonic in a Disc

1966 ◽  
Vol 28 ◽  
pp. 187-191
Author(s):  
J. E. Mcmillan
Keyword(s):  
Asymptotic Behavior ◽  
Unit Circle ◽  
Complex Plane ◽  
Unit Disc ◽  
Open Unit ◽  
Continuous Curve ◽  
Open Unit Disc ◽  
Boundary Path

Let D be the open unit disc, and let C be the unit circle in the complex plane. Let f be a (real-valued) function that is harmonic in D. A simple continuous curve β: z(t) (0≦t<1) contained in D such that |z(t)|→1 as t→1 is a boundary path with end (the bar denotes closure).

scholarly journals On a conjecture of Z. Jianzhong

1992 ◽  
Vol 45 (1) ◽  
pp. 163-170
Author(s):  
Yasuo Matsugu
Keyword(s):  
Convex Function ◽  
Unit Ball ◽  
Unit Circle ◽  
Orlicz Space ◽  
Complex Plane ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Almost All

Let ϕ be a nonnegative, nondecreasing and nonconstant function defined on [0, ∞) such that Φ(t) = ϕ(et) is a convex function on (-∞, ∞). The Hardy-Orlicz space H (ϕ) is defined to be the class of all those functions f holomorphic in the open unit disc of the complex plane C satisfying The subclass H(ϕ)+ of H(ϕ) is defined to be the class of all those functions f ∈ H(ϕ) satisfying for almost all points eit of the unit circle. In 1990, Z. Jianzhong conjectured that H(ϕ)+ = H(ψ)+ if and only if H(ϕ) = H(ψ). In the present paper we prove that it is true not only on the unit disc of C but also on the unit ball of Cn.

Logarithmic growth filtrations for -modules over the bounded Robba ring

2021 ◽  
Vol 157 (6) ◽  
pp. 1265-1301
Author(s):  
Shun Ohkubo
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Unit Disc ◽  
Open Unit ◽  
Asymptotic Behavior Of Solutions ◽  
Newton Polygons ◽  
Open Unit Disc ◽  
Hypergeometric Differential Equation ◽  
Logarithmic Growth

In the 1970s, Dwork defined the logarithmic growth (log-growth for short) filtrations for $p$ -adic differential equations $Dx=0$ on the $p$ -adic open unit disc $|t|<1$ , which measure the asymptotic behavior of solutions $x$ as $|t|\to 1^{-}$ . Then, Dwork calculated the log-growth filtration for $p$ -adic Gaussian hypergeometric differential equation. In the late 2000s, Chiarellotto and Tsuzuki proposed a fundamental conjecture on the log-growth filtrations for $(\varphi ,\nabla )$ -modules over $K[\![t]\!]_0$ , which can be regarded as a generalization of Dwork's calculation. In this paper, we prove a generalization of the conjecture to $(\varphi ,\nabla )$ -modules over the bounded Robba ring. As an application, we prove a generalization of Dwork's conjecture proposed by Chiarellotto and Tsuzuki on the specialization property for log-growth Newton polygons.

scholarly journals The Closed Ideals in an Algebra of Analytic Functions

1957 ◽  
Vol 9 ◽  
pp. 426-434 ◽  
Author(s):  
Walter Rudin
Keyword(s):  
Banach Algebra ◽  
Complex Plane ◽  
Analytic Functions ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Complex Functions ◽  
Continuous Complex ◽  
Image Position

Let K and C be the closure and boundary, respectively, of the open unit disc U in the complex plane. Let be the Banach algebra whose elements are those continuous complex functions on K which are analytic in U, with norm (f ∊ ).

scholarly journals Coefficient inequality for transforms of starlike and convex functions

2015 ◽  
Vol 24 (1) ◽  
pp. 69-75
Author(s):  
D. VAMSHEE KRISHNA ◽  
◽  
B. VENKATESWARLU ◽  
T. RAMREDDY ◽  
◽  
...  
Keyword(s):  
Analytic Function ◽  
Complex Plane ◽  
Upper Bound ◽  
Convex Functions ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Toeplitz Determinants ◽  
Starlike And Convex Functions ◽  
Coefficient Inequality

The objective of this paper is to obtain an upper bound for the second Hankel functional associated with the k th root transform ... normalized analytic function f(z) belonging to starlike and convex functions, defined on the open unit disc in the complex plane, using Toeplitz determinants.

Averages of holomorphic mappings

1986 ◽  
Vol 100 (2) ◽  
pp. 371-381 ◽  
Author(s):  
V. Nestoridis
Keyword(s):  
Unit Circle ◽  
Lebesgue Measure ◽  
Unit Disc ◽  
Holomorphic Mappings ◽  
Open Unit ◽  
Open Unit Disc ◽  
Disc Algebra ◽  
Several Variables

In this paper we present two versions in several variables of the following result:Theorem 1([2, 3]). Let f be a function in the disc algebra (more generally in H1). Then for every point z0 in the open unit disc, there is an interval I on the unit circle T such that f(z0) = 1/|I| ∫Ifdσ, where 0 < |I| ≤ 2π denotes the length of I and σ the Lebesgue measure on T.

scholarly journals Tangential Boundary Properties of Arbitrary Functions in the Unit Disc

1972 ◽  
Vol 46 ◽  
pp. 111-120 ◽  
Author(s):  
Hidenobu Yoshida
Keyword(s):  
Complex Plane ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Boundary Properties ◽  
Angular Boundary ◽  
Boundary Behaviors

1. By the method of Dolzhenko’s paper, we studied relations between non-tangetial (angular) boundary behaviors and horocyclic boundary behaviors of arbitrary functions defined in the open unit disc of the complex plane in [8]. Vessey [5], [6] investigated the behavior of arbitrary functions on paths which are “more tangential” than horocycles. The purpose of the present paper is to prove the fact that is sharper than the results in Vessey [5], [6], and generalize the results in [8] to obtain the connection between behaviors on two “more tangential” angles.

Evaluation operators on the Bergman space

1995 ◽  
Vol 117 (3) ◽  
pp. 513-523 ◽  
Author(s):  
Kehe Zhu
Keyword(s):  
Bergman Space ◽  
Complex Plane ◽  
Analytic Functions ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Space Of Analytic Functions ◽  
Area Measure ◽  
Image Position

Let D be the open unit disc in the complex plane C and let dA be the normalized area measure on D. The Bergman space is the space of analytic functions f in D such that

On the Zeros of Functions with Derivatives in H1 and H∞

1970 ◽  
Vol 22 (2) ◽  
pp. 342-347 ◽  
Author(s):  
James Wells
Keyword(s):  
Unit Circle ◽  
Unit Disc ◽  
Open Unit ◽  
Hardy Class ◽  
Open Unit Disc ◽  
Zeros Of Functions ◽  
Limit Points ◽  
Image Position ◽  
Derivatives Of

Let {zk},0 < |zk| < 1, be a given sequence of points in the open unit disc D = {z: |z| < 1} and let E be its set of limit points on the unit circle T. In this note we consider the problem of finding conditions on the sequence {zk} which will ensure the existence of a function f analytic in D satisfying(A)and whose derivative f′ belongs to the Hardy class H1 or, alternatively, whose derivatives of all orders are bounded in D. We shall prove the following two theorems.THEOREM 1. If(1)(2)and(3)then there is a function f analytic in D which satisfies (A) and its derivative f′ belongs to H1.

The Essential Spectrum of the Essentially Isometric Operator

2014 ◽  
Vol 57 (1) ◽  
pp. 145-158
Author(s):  
H. S. Mustafayev
Keyword(s):  
Hilbert Space ◽  
Unit Circle ◽  
Essential Spectrum ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Isometric Operator ◽  
Disc Algebra ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space

AbstractLet T be a contraction on a complex, separable, infinite dimensional Hilbert space and let σ(T) (resp. σe(T)) be its spectrum (resp. essential spectrum). We assume that T is an essentially isometric operator; that is, IH -T*T is compact. We show that if D\σ(T) ≠ Ø, then for every f from the disc-algebraσe(f(T)) = f(σe(T))where D is the open unit disc. In addition, if T lies in the class C0.∪ C.0, thenσe(f(T)) = f(σ(T) ∩ Γ),where Γ is the unit circle. Some related problems are also discussed.

scholarly journals Coefficient inequality for transforms of certain subclass of analytic functions

2017 ◽  
Vol 21 (2) ◽  
pp. 185-193
Author(s):  
T. RamReddy ◽  
D. Shalini ◽  
D. Vamshee Krishna ◽  
B. Venkateswarlu
Keyword(s):  
Analytic Function ◽  
Complex Plane ◽  
Upper Bound ◽  
Analytic Functions ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Toeplitz Determinants ◽  
Coefficient Inequality

The objective of this paper is to obtain the best possible sharp upper bound for the second Hankel functional associated with the kth root transform [f(zk)]1/k of normalized analytic function f(z) when it belongs to certain subclass of analytic functions, defined on the open unit disc in the complex plane using Toeplitz determinants.

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