scholarly journals Boundary Schwarz lemma for holomorphic functions

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5553-5565
Author(s):  
Nafi Örnek ◽  
Burcu Gök
Keyword(s):  
Boundary Point ◽  
Unit Disc ◽  
Holomorphic Functions ◽  
Schwarz Lemma ◽  
Angular Derivative ◽  
Taylor Coefficient ◽  
Boundary Schwarz Lemma ◽  
Boundary Version

In this paper, a boundary version of Schwarz lemma is investigated. We take into consideration a function f (z) holomorphic in the unit disc and f (0) = 0 such that ?Rf? < 1 for ?z? < 1, we estimate a modulus of angular derivative of f (z) function at the boundary point b with f (b) = 1, by taking into account their first nonzero two Maclaurin coefficients. Also, we shall give an estimate below ?f'(b)? according to the first nonzero Taylor coefficient of about two zeros, namely z=0 and z0 ? 0. Moreover, two examples for our results are considered.

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 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 Applications of the Jack's lemma for the meromorphic functions at the boundary

2019 ◽  
Vol 38 (7) ◽  
pp. 219-226
Author(s):  
Tugba Akyel ◽  
Bulent Nafi Ornek
Keyword(s):  
Boundary Point ◽  
Meromorphic Functions ◽  
Schwarz Lemma ◽  
Jack’S Lemma ◽  
Boundary Version ◽  
Jack's Lemma ◽  
The Schwarz Lemma

In this paper, a boundary version of the Schwarz lemma for the class $\mathcal{% N(\alpha )}$ is investigated. For the function $f(z)=\frac{1}{z}% +a_{0}+a_{1}z+a_{2}z^{2}+...$ defined in the punctured disc $E$ such that $% f(z)\in \mathcal{N(\alpha )}$, we estimate a modulus of the angular derivative of the function $\frac{zf^{\prime }(z)}{f(z)}$ at the boundary point $c$ with $\frac{cf^{\prime }(c)}{f(c)}=\frac{1-2\beta }{\beta }$. Moreover, Schwarz lemma for class $\mathcal{N(\alpha )}$ is given.

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 On a class of analytic functions related to Robertson’s formula and subordination

2021 ◽  
Vol 27 (1) ◽  
Author(s):  
Adam Lecko ◽  
Gangadharan Murugusundaramoorthy ◽  
Srikandan Sivasubramanian
Keyword(s):  
Integral Representation ◽  
Analytic Functions ◽  
Boundary Point ◽  
Unit Disc ◽  
Starlike Functions ◽  
Analytic Formula ◽  
Growth Theorem

AbstractIn this paper, we define and study a class of analytic functions in the unit disc by modification of the well-known Robertson’s analytic formula for starlike functions with respect to a boundary point combined with subordination. An integral representation and growth theorem are proved. Early coefficients and the Fekete–Szegö functional are also estimated.

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.

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