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.

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 Boundary Behaviour of Functions With Hadamard Gaps

1972 ◽  
Vol 48 ◽  
pp. 173-181 ◽  
Author(s):  
K. G. Binmore ◽  
R. Hornblower
Keyword(s):  
Unit Disc ◽  
Open Unit ◽  
Boundary Behaviour ◽  
Open Unit Disc ◽  
Hadamard Gaps ◽  
Boundary Properties

In this paper we discuss the boundary properties of a function f which is analytic in the open unit disc Δ and has Hadamard gaps—i.e. (1)where. (2)

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 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.

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.

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

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.

scholarly journals Radial growth and boundedness for Bloch functions

1990 ◽  
Vol 42 (1) ◽  
pp. 33-39 ◽  
Author(s):  
A. Bonilla ◽  
F. Perez Gonzalez
Keyword(s):  
Complex Plane ◽  
Wide Class ◽  
Unit Disc ◽  
Radial Growth ◽  
Bloch Space ◽  
Sufficient Conditions ◽  
Open Unit ◽  
Open Unit Disc ◽  
Boundedness Condition ◽  
Significant Extension

Let B be the Bloch space of all those functions f holomorphic in the open unit disc D of the complex plane satisfying . We establish sufficient conditions for the boundedness of functions f belonging to B satisfying a certain uniform radial boundedness condition, and, by introducing a wide class of subsets E of ∂D, which we call negligible sets for boundedness, we show that if f ∈ B and there is a constant K > 0 such that , then f is bounded in D. Hence a significant extension of a theorem of Goolsby is obtained.

An argument of a function in H1/2

2012 ◽  
Vol 55 (2) ◽  
pp. 507-511
Author(s):  
Takahiko Nakazi ◽  
Takanori Yamamoto
Keyword(s):  
Hardy Space ◽  
Simple Proof ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc

AbstractLet H1/2 be the Hardy space on the open unit disc. For two non-zero functions f and g in H1/2, we study the relation between f and g when f/g ≥ 0 a.e. on ∂D. Then we generalize a theorem of Neuwirth and Newman and Helson and Sarason with a simple proof.

scholarly journals Harmonic univalent functions defined by post quantum calculus operators

2019 ◽  
Vol 11 (1) ◽  
pp. 5-17 ◽  
Author(s):  
Om P. Ahuja ◽  
Asena Çetinkaya ◽  
V. Ravichandran
Keyword(s):  
Unit Disc ◽  
Univalent Functions ◽  
Extreme Points ◽  
Open Unit ◽  
Open Unit Disc ◽  
Quantum Calculus ◽  
Special Cases ◽  
The Family ◽  
Covering Theorems

Abstract We study a family of harmonic univalent functions in the open unit disc defined by using post quantum calculus operators. We first obtained a coefficient characterization of these functions. Using this, coefficients estimates, distortion and covering theorems were also obtained. The extreme points of the family and a radius result were also obtained. The results obtained include several known results as special cases.

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