scholarly journals Inequalities for products of polynomials I

2009 ◽  
Vol 104 (1) ◽  
pp. 147 ◽  
Author(s):  
I.E. Pritsker ◽  
S. Ruscheweyh
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Compact Sets ◽  
Sharp Constants ◽  
Best Constant ◽  
Connected Sets

We study inequalities connecting the product of uniform norms of polynomials with the norm of their product. This circle of problems include the Gelfond-Mahler inequality for the unit disk and the Kneser-Borwein inequality for the segment $[-1,1]$. Furthermore, the asymptotically sharp constants are known for such inequalities over arbitrary compact sets in the complex plane. It is shown here that this best constant is smallest (namely: 2) for a disk. We also conjecture that it takes its largest value for a segment, among all compact connected sets in the plane.

On BMOA for Riemann Surfaces

1981 ◽  
Vol 33 (5) ◽  
pp. 1255-1260 ◽  
Author(s):  
Thomas A. Metzger
Keyword(s):  
Analytic Function ◽  
Unit Disk ◽  
Complex Plane ◽  
Riemann Surfaces ◽  
Hardy Class ◽  
Image Position

Let Δ denote the unit disk in the complex plane C. The space BMO has been extensively studied by many authors (see [3] for a good exposition of this topic). Recently, the subspace BMOA (Δ) has become a topic of interest. An analytic function f, in the Hardy class H2(A), belongs to BMOA (Δ) if(1)whereIt is known (see [3, p. 96]) that (1) is equivalent to(2)

scholarly journals Growth Theorems for a Subclass of Strongly Spirallike Functions

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yan-Yan Cui ◽  
Chao-Jun Wang ◽  
Si-Feng Zhu
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Starlike Functions ◽  
Geometric Properties ◽  
Strongly Starlike Functions ◽  
Strongly Starlike ◽  
Spirallike Functions ◽  
Analytic Mappings

In this paper we consider a subclass of strongly spirallike functions on the unit diskDin the complex planeC, namely, strongly almost spirallike functions of typeβand orderα. We obtain the growth results for strongly almost spirallike functions of typeβand orderαon the unit diskDinCby using subordination principles and the geometric properties of analytic mappings. Furthermore we get the growth theorems for strongly almost starlike functions of orderαand strongly starlike functions on the unit diskDofC. These growth results follow the deviation results of these functions.

scholarly journals Strong Differential Subordination Results for Multivalent Analytic Functions Associated with Dziok-Srivastava Operator

2019 ◽  
pp. 159-168
Author(s):  
Abbas Kareem Wanas ◽  
Hala Abbas Mehdi
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Analytic Functions ◽  
Differential Subordination ◽  
Open Unit ◽  
Open Unit Disk ◽  
Closed Unit Disk ◽  
Strong Differential Subordination

In this paper, by making use of the principle of strong subordination, we establish some interesting properties of multivalent analytic functions defined in the open unit disk and closed unit disk of the complex plane associated with Dziok-Srivastava operator.

scholarly journals Lipschitz Continuity for the Solutions of Triharmonic Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Zhou Yu ◽  
Xiao Bing
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Dirichlet Boundary ◽  
Lipschitz Continuity ◽  
Boundary Value ◽  
Jordan Domain ◽  
Lipschitz Continuous ◽  
Equivalent Conditions

Let D be the unit disk in the complex plane C and denote T=∂D. Write Hom+T,∂Ω for the class of all sense-preserving homeomorphism of T onto the boundary of a C2 convex Jordan domain Ω. In this paper, five equivalent conditions for the solutions of triharmonic equations ∂z∂z¯3ω=ff∈CD¯ with Dirichlet boundary value conditions ωzz¯zz¯T=γ2∈CT,ωzz¯T=γ1∈CT and ωT=γ0∈Hom+T,∂Ω to be Lipschitz continuous are presented.

scholarly journals On a Uniqueness Theorem

1969 ◽  
Vol 35 ◽  
pp. 151-157 ◽  
Author(s):  
V. I. Gavrilov
Keyword(s):  
Unit Circle ◽  
Uniqueness Theorem ◽  
Unit Disk ◽  
Complex Plane ◽  
Open Unit ◽  
Open Unit Disk ◽  
Extended Complex ◽  
Extended Complex Plane

1. Let D be the open unit disk and r be the unit circle in the complex plane, and denote by Q the extended complex plane or the Rie-mann sphere.

A Class of Analytic Starlike Functions Associated with Petal Like Region on the Positive Half of Complex Plane

2020 ◽  
Vol 87 (3-4) ◽  
pp. 165
Author(s):  
Rajesh Kumar Maurya ◽  
Poonam Sharma
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Algebraic Structure ◽  
Starlike Functions ◽  
Open Mapping ◽  
Open Unit ◽  
Open Unit Disk ◽  
Hankel Determinant ◽  
Open Mapping Theorem ◽  
Positive Half

In the light of Riemann open mapping theorem, if we map open unit disk U conformally onto a region then depending on the geometry of boundary of we can always extract a subclass of H[a, n] by subordinating various functionals of the function f ∈ H[a, n]. Depending upon the geometry of the range set attempts have been made to find some algebraic structure in such classes, for that Hankel determinant of coefficients of functions pertaining to these classes have been studied, bounds of various coefficients have been determined and also based on the subordination principle we have determined radius |z| &lt; r ;z ∈ U for which f belongs to such a class. In this paper our focus would be on n−PS<sup>*</sup> defined as n − PS<sup>*</sup> = {f ∈ A : Re {zf<sup>'</sup>(z)/f(z)} &gt; 0,|(zf<sup>'</sup>(z)/f(z))<sup>n</sup> - 1|&lt;1}.

LIMITS OF FRACTIONAL DERIVATIVES AND COMPOSITIONS OF ANALYTIC FUNCTIONS

2016 ◽  
Vol 103 (1) ◽  
pp. 104-115 ◽  
Author(s):  
THOMAS H. MACGREGOR ◽  
MICHAEL P. STERNER
Keyword(s):  
Fractional Derivative ◽  
Unit Disk ◽  
Power Function ◽  
Complex Plane ◽  
Analytic Functions ◽  
Fractional Derivatives ◽  
Hadamard Product ◽  
Open Unit ◽  
Open Unit Disk

Suppose that the function $f$ is analytic in the open unit disk $\unicode[STIX]{x1D6E5}$ in the complex plane. For each $\unicode[STIX]{x1D6FC}>0$ a function $f^{[\unicode[STIX]{x1D6FC}]}$ is defined as the Hadamard product of $f$ with a certain power function. The function $f^{[\unicode[STIX]{x1D6FC}]}$ compares with the fractional derivative of $f$ of order $\unicode[STIX]{x1D6FC}$. Suppose that $f^{[\unicode[STIX]{x1D6FC}]}$ has a limit at some point $z_{0}$ on the boundary of $\unicode[STIX]{x1D6E5}$. Then $w_{0}=\lim _{z\rightarrow z_{0}}f(z)$ exists. Suppose that $\unicode[STIX]{x1D6F7}$ is analytic in $f(\unicode[STIX]{x1D6E5})$ and at $w_{0}$. We show that if $g=\unicode[STIX]{x1D6F7}(f)$ then $g^{[\unicode[STIX]{x1D6FC}]}$ has a limit at $z_{0}$.

scholarly journals The Bloch space and Besov spaces of analytic functions

1996 ◽  
Vol 54 (2) ◽  
pp. 211-219 ◽  
Author(s):  
Karel Stroethoff
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Besov Spaces ◽  
Analytic Functions ◽  
Elementary Proof ◽  
Bloch Space ◽  
Little Bloch Space ◽  
Spaces Of Analytic Functions

We shall give an elementary proof of a characterisation for the Bloch space due to Holland and Walsh, and obtain analogous characterisations for the little Bloch space and Besov spaces of analytic functions on the unit disk in the complex plane.

scholarly journals Products of composition and n-th differentiation operators from α-bloch space to Qp space

Filomat ◽  
2013 ◽  
Vol 27 (5) ◽  
pp. 761-766
Author(s):  
Haiying Li ◽  
Cui Wang ◽  
Tianyu Xue ◽  
Xiangbo Zhang
Keyword(s):  
Unit Disk ◽  
Complex Plane ◽  
Composition Operators ◽  
Bloch Space ◽  
Open Unit ◽  
Open Unit Disk ◽  
Th Differentiation ◽  
Qp Space

Let ? be an analytic self-map of the open unit disk D on the complex plane and ? > 0, p ? 0, n ? N. In this paper, the boundedness and compactness of the products of composition operators and nth differentiation operators C?Dn from a-Bloch space B? and B?0 to Qp space are investigated.

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