scholarly journals Evolution families and the Loewner equation I: the unit disc

2012 ◽  
Vol 2012 (672) ◽  
pp. 1-37 ◽  
Author(s):  
Bracci Filippo ◽  
Manuel D. Contreras ◽  
Santiago Díaz-Madrigal
Keyword(s):  
Unit Disc ◽  
Loewner Equation ◽  
Evolution Families

scholarly journals Evolution families and the Loewner equation II: complex hyperbolic manifolds

2009 ◽  
Vol 344 (4) ◽  
pp. 947-962 ◽  
Author(s):  
Filippo Bracci ◽  
Manuel D. Contreras ◽  
Santiago Díaz-Madrigal
Keyword(s):  
Hyperbolic Manifolds ◽  
Loewner Equation ◽  
Evolution Families

The Growth and Borel Points of Random Algebroid Functions in the Unit Disc

2021 ◽  
Vol 41 (4) ◽  
pp. 1119-1129
Author(s):  
Daochun Sun ◽  
Yingying Huo ◽  
Fujie Chai
Keyword(s):  
Unit Disc ◽  
Algebroid Functions

Hankel determinants of second and third order for the class 𝓢 of univalent functions

2021 ◽  
Vol 71 (3) ◽  
pp. 649-654
Author(s):  
Milutin Obradović ◽  
Nikola Tuneski
Keyword(s):  
Unit Disc ◽  
Univalent Functions ◽  
Upper Bounds ◽  
Third Order ◽  
Hankel Determinants

Abstract In this paper we give the upper bounds of the Hankel determinants of the second and third order for the class 𝓢 of univalent functions in the unit disc.

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 SHARP LOGARITHMIC DERIVATIVE ESTIMATES WITH APPLICATIONS TO ORDINARY DIFFERENTIAL EQUATIONS IN THE UNIT DISC

2010 ◽  
Vol 88 (2) ◽  
pp. 145-167 ◽  
Author(s):  
I. CHYZHYKOV ◽  
J. HEITTOKANGAS ◽  
J. RÄTTYÄ
Keyword(s):  
Differential Equations ◽  
Unit Disc ◽  
Logarithmic Derivative ◽  
Maximum Modulus ◽  
Exceptional Set ◽  
Proximate Order ◽  
Upper Density ◽  
Linear Differential ◽  
Logarithmic Derivatives ◽  
Growth Of Solutions

AbstractNew estimates are obtained for the maximum modulus of the generalized logarithmic derivatives f(k)/f(j), where f is analytic and of finite order of growth in the unit disc, and k and j are integers satisfying k>j≥0. These estimates are stated in terms of a fixed (Lindelöf) proximate order of f and are valid outside a possible exceptional set of arbitrarily small upper density. The results obtained are then used to study the growth of solutions of linear differential equations in the unit disc. Examples are given to show that all of the results are sharp.

scholarly journals A note on n-harmonic majorants

1986 ◽  
Vol 34 (3) ◽  
pp. 461-472
Author(s):  
Hong Oh Kim ◽  
Chang Ock Lee
Keyword(s):  
Harmonic Function ◽  
Complex Plane ◽  
Unit Disc ◽  
Dirichlet Integral ◽  
Harmonic Majorant ◽  
Harmonic Majorants

Suppose D (υ) is the Dirichlet integral of a function υ defined on the unit disc U in the complex plane. It is well known that if υ is a harmonic function in U with D (υ) < ∞, then for each p, 0 < p < ∞, |υ|p has a harmonic majorant in U.We define the “iterated” Dirichlet integral Dn (υ) for a function υ on the polydisc Un of Cn and prove the polydisc version of the well known fact above:If υ is an n-harmonic function in Un with Dn (υ) < ∞, then for each p, 0 < p < ∞, |υ|p has an n-harmonic majorant in Un.

scholarly journals A coefficient inequality for the class of analytic functions in the unit disc

2003 ◽  
Vol 2003 (59) ◽  
pp. 3753-3759
Author(s):  
Yaşar Polatoğlu ◽  
Metın Bolcal
Keyword(s):  
Analytic Functions ◽  
Unit Disc ◽  
Coefficient Inequality

The aim of this paper is to give a coefficient inequality for the class of analytic functions in the unit discD={z||z| <1}.

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