A Geometric Extension of Schwarz’s Lemma and Applications

2016 ◽  
Vol 59 (01) ◽  
pp. 30-35 ◽  
Author(s):  
Galatia Cleanthous
Keyword(s):  
Holomorphic Function ◽  
Unit Disc ◽  
Covering Theorem ◽  
Rectilinear Segment ◽  
Closed Unit Disc ◽  
Schwarz’S Lemma

Abstract Let f be a holomorphic function of the unit disc , preserving the origin. According to Schwarz’s Lemma, |f'(0)| ≤ 1, provided that . We prove that this bound still holds, assuming only that f() does not contain any closed rectilinear segment [0, eiϕ], ϕ ∊ [0, zπ], i.e., does not contain any entire radius of the closed unit disc. Furthermore, we apply this result to the hyperbolic density and give a covering theorem.

scholarly journals A Criterion for Normalcy

1968 ◽  
Vol 32 ◽  
pp. 277-282 ◽  
Author(s):  
Paul Gauthier
Keyword(s):  
Meromorphic Function ◽  
Holomorphic Function ◽  
Unit Disc ◽  
Meromorphic Functions

Gavrilov [2] has shown that a holomorphic function f(z) in the unit disc |z|<1 is normal, in the sense of Lehto and Virtanen [5, p. 86], if and only if f(z) does not possess a sequence of ρ-points in the sense of Lange [4]. Gavrilov has also obtained an analagous result for meromorphic functions by introducing the property that a meromorphic function in the unit disc have a sequence of P-points. He has shown that a meromorphic function in the unit disc is normal if and only if it does not possess a sequence of P-points.

p-Radial Exceptional Sets and Conformal Mappings

2007 ◽  
Vol 50 (4) ◽  
pp. 579-587
Author(s):  
Piotr Kot
Keyword(s):  
Holomorphic Function ◽  
Unit Disc ◽  
Conformal Mappings ◽  
Exceptional Sets ◽  
Image Position

AbstractFor p > 0 and for a given set E of type Gδ in the boundary of the unit disc ∂ we construct a holomorphic function f ∈ such thatIn particular if a set E has a measure equal to zero, then a function f is constructed as integrable with power p on the unit disc .

scholarly journals Representation with majorant of the Schwarz lemma at the boundary

2017 ◽  
Vol 101 (115) ◽  
pp. 191-196
Author(s):  
Bülent Örnek ◽  
Tuğba Akyel
Keyword(s):  
Holomorphic Function ◽  
Blaschke Product ◽  
Unit Disc ◽  
Sufficient Conditions ◽  
Schwarz Lemma ◽  
Local Behavior ◽  
Finite Blaschke Product ◽  
Boundary Points ◽  
Finite Set ◽  
The Schwarz Lemma

Let f be a holomorphic function in the unit disc and |f(z)?1| < 1 for |z| < 1. We generalize the uniqueness portion of Schwarz?s lemma and provide sufficient conditions on the local behavior of f near a finite set of boundary points that needed for f to be a finite Blaschke product.

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 .

Examples of adic spaces

2020 ◽  
pp. 27-34
Author(s):  
Peter Scholze ◽  
Jared Weinstein
Keyword(s):  
Unit Disc ◽  
Special Point ◽  
Open Unit ◽  
Open Unit Disc ◽  
Generic Point ◽  
Closed Unit Disc ◽  
Affine Line

This chapter discusses various examples of adic spaces. These examples include the adic closed unit disc; the adic affine line; the closure of the adic closed unit disc in the adic affine line; the open unit disc; the punctured open unit disc; and the constant adic space associated to a profinite set. The chapter focuses on one example: the adic open unit disc over Zp. The adic spectrum Spa Zp consists of two points, a special point and a generic point. The chapter then studies the structure of analytic points. It also clarifies the relations between analytic rings and Tate rings.

scholarly journals Composition operators

1978 ◽  
Vol 18 (3) ◽  
pp. 439-446 ◽  
Author(s):  
R.K. Singh ◽  
B.S. Komal
Keyword(s):  
Composition Operator ◽  
Unit Disc ◽  
Composition Operators ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Closed Unit Disc ◽  
Necessary And Sufficient ◽  
Made In

A study of centered composition operators on l2 is made in this paper. Also the spectrum of surjective composition operators is computed. A necessary and sufficient condition is obtained for the closed unit disc to be the spectrum of a surjective composition operator.

scholarly journals Radius Problems for Starlike Functions Associated with the Tan Hyperbolic Function

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Khalil Ullah ◽  
Saira Zainab ◽  
Muhammad Arif ◽  
Maslina Darus ◽  
Meshal Shutaywi
Keyword(s):  
Holomorphic Function ◽  
Convex Functions ◽  
Unit Disc ◽  
Starlike Functions ◽  
Open Unit ◽  
Hyperbolic Function ◽  
Open Unit Disc ◽  
Starlike And Convex Functions ◽  
Radius Problems

The aim of this particular article is at studying a holomorphic function f defined on the open-unit disc D = z ∈ ℂ : z < 1 for which the below subordination relation holds z f ′ z / f z ≺ q 0 z = 1 + tan h z . The class of such functions is denoted by S tan h ∗ . The radius constants of such functions are estimated to conform to the classes of starlike and convex functions of order β and Janowski starlike functions, as well as the classes of starlike functions associated with some familiar functions.

The Maximum Modulus of Normal Meromorphic Functions and Applications to Value Distribution

1970 ◽  
Vol 22 (4) ◽  
pp. 803-814 ◽  
Author(s):  
Paul Gauthier
Keyword(s):  
Holomorphic Function ◽  
Unit Disc ◽  
Meromorphic Functions ◽  
Normal Family ◽  
Holomorphic Functions ◽  
Maximum Modulus ◽  
Value Distribution ◽  
Minimum Modulus ◽  
Image Position

Let f(z) be a function meromorphic in the unit disc D = (|z| < 1). We consider the maximum modulusand the minimum modulusWhen no confusion is likely, we shall write M(r) and m(r) in place of M(r,f) and m(r,f).Since every normal holomorphic function belongs to an invariant normal family, a theorem of Hayman [6, Theorem 6.8] yields the following result.THEOREM 1. If f(z) is a normal holomorphic function in the unit disc D, then(1)This means that for normal holomorphic functions, M(r) cannot grow too rapidly. The main result of this paper (Theorem 5, also due to Hayman, but unpublished) is that a similar situation holds for normal meromorphic functions.

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