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 Note on the uniqueness holomorphic function on the unit disk

Filomat ◽  
2018 ◽  
Vol 32 (6) ◽  
pp. 2321-2325 ◽  
Author(s):  
Tuğba Akyel ◽  
Tahir Azeroğlu
Keyword(s):  
Holomorphic Function ◽  
Unit Disk ◽  
Blaschke Product ◽  
Local Behavior ◽  
Finite Blaschke Product ◽  
Boundary Points ◽  
Finite Set

Let f be an holomorphic function the unit disk to itself. We provide conditions on the local behavior of f along boundary near a finite set of the boundary points that requires f to be a finite Blaschke product.

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.

scholarly journals New estimates for meromorphic functions

2021 ◽  
Vol 109 (123) ◽  
pp. 153-162
Author(s):  
Bülent Örnek
Keyword(s):  
Boundary Point ◽  
Unit Disc ◽  
Meromorphic Functions ◽  
Schwarz Lemma ◽  
Angular Derivative ◽  
Boundary Version ◽  
The Schwarz Lemma

A boundary version of the Schwarz lemma for meromorphic functions is investigated. For the function Inf(z) = 1/z +?? k=2 knck?2zk?2, belonging to the class of W, we estimate from below the modulus of the angular derivative of the function on the boundary point of the unit disc.

scholarly journals Uniqueness part of Schwarz lemma for driving point impedance functions

Filomat ◽  
2020 ◽  
Vol 34 (9) ◽  
pp. 2953-2959
Author(s):  
Nafi Örnek ◽  
Timur Düzenli
Keyword(s):  
Uniqueness Theorem ◽  
Schwarz Lemma ◽  
Boundary Points ◽  
Impedance Functions ◽  
Boundary Version ◽  
The Schwarz Lemma

In this paper, a boundary version of the uniqueness part of the Schwarz lemma for driving point impedance functions has been investigated. Also, more general results have been obtained for a different version of the Burns-Krantz uniqueness theorem. In these results, as different from the Burns-Krantz theorem, only the boundary points have been used as the conditions on the function. Also, more general majorants will be taken instead of power majorants in (1.1).

scholarly journals Behavior of meromorphic functions at the boundary of the unit disc

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3443-3452
Author(s):  
Bülent Örnek
Keyword(s):  
Meromorphic Function ◽  
Boundary Point ◽  
Unit Disc ◽  
Meromorphic Functions ◽  
Schwarz Lemma ◽  
Angular Derivative ◽  
Boundary Version ◽  
The Schwarz Lemma

In this paper, a boundary version of the Schwarz lemma for meromorphic functions is investigated. The modulus of the angular derivative of the meromorphic function Inf(z)=1/z+2nc0+3nc1z+4nc2z2+... that belongs to the class of M on the boundary point of the unit disc has been estimated from below.

scholarly journals On the Schwarz lemma at the upper half plane

Filomat ◽  
2019 ◽  
Vol 33 (10) ◽  
pp. 2995-3011
Author(s):  
Bülent Örnek
Keyword(s):  
Holomorphic Function ◽  
Real Axis ◽  
Half Plane ◽  
Simple Proof ◽  
Boundary Point ◽  
Schwarz Lemma ◽  
The Real ◽  
Upper Half Plane ◽  
Boundary Schwarz Lemma ◽  
The Schwarz Lemma

In this paper, we give a simple proof for the boundary Schwarz lemma at the upper half plane. Considering that f(z) is a holomorphic function defined on the upper half plane, we derive inequalities for the modulus of derivative of f (z), |f'(0)| by assuming that the f(z) function is also holomorphic at the boundary point z = 0 on the real axis with f(0)=Rf(i).

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.

scholarly journals Transversals in permutation groups and factorisations of complete graphs

2002 ◽  
Vol 65 (2) ◽  
pp. 277-288 ◽  
Author(s):  
Gil Kaplan ◽  
Arieh Lev
Keyword(s):  
Permutation Group ◽  
Directed Graph ◽  
Sufficient Conditions ◽  
Combinatorial Problems ◽  
Permutation Groups ◽  
Transitive Permutation Group ◽  
Complete Graphs ◽  
Frobenius Theorem ◽  
Disjoint Cycles ◽  
Finite Set

Let G be a transitive permutation group acting on a finite set of order n. We discuss certain types of transversals for a point stabiliser A in G: free transversals and global transversals. We give sufficient conditions for the existence of such transversals, and show the connection between these transversals and combinatorial problems of decomposing the complete directed graph into edge disjoint cycles. In particular, we classify all the inner-transitive Oberwolfach factorisations of the complete directed graph. We mention also a connection to Frobenius theorem.

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