scholarly journals On the Boundary Dieudonné–Pick Lemma

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1108
Author(s):  
Olga Kudryavtseva ◽  
Aleksei Solodov
Keyword(s):  
Fixed Point ◽  
Interior Point ◽  
Sharp Estimate ◽  
General Boundary ◽  
Extremal Functions ◽  
Angular Derivative ◽  
Additional Information ◽  
Carathéodory Theorem ◽  
Class Of Functions ◽  
Boundary Fixed Point

The class of holomorphic self-maps of a disk with a boundary fixed point is studied. For this class of functions, the famous Julia–Carathéodory theorem gives a sharp estimate of the angular derivative at the boundary fixed point in terms of the image of the interior point. In the case when additional information about the value of the derivative at the interior point is known, a sharp estimate of the angular derivative at the boundary fixed point is obtained. As a consequence, the sharpness of the boundary Dieudonné–Pick lemma is established and the class of the extremal functions is identified. An unimprovable strengthening of the Osserman general boundary lemma is also obtained.

scholarly journals Contractors and Linear Matrix Inequalities

2015 ◽  
Vol 1 (3) ◽  
Author(s):  
Jeremy Nicola ◽  
Luc Jaulin
Keyword(s):  
Fixed Point ◽  
Large Class ◽  
Linear Matrix Inequalities ◽  
Interior Point ◽  
Interior Point Methods ◽  
Linear Constraints ◽  
Linear Matrix ◽  
Nonlinear Constraints ◽  
Matrix Inequalities ◽  
Convex Constraints

Linear matrix inequalities (LMIs) comprise a large class of convex constraints. Boxes, ellipsoids, and linear constraints can be represented by LMIs. The intersection of LMIs are also classified as LMIs. Interior-point methods are able to minimize or maximize any linear criterion of LMIs with complexity, which is polynomial regarding to the number of variables. As a consequence, as shown in this paper, it is possible to build optimal contractors for sets represented by LMIs. When solving a set of nonlinear constraints, one may extract from all constraints that are LMIs in order to build a single optimal LMI contractor. A combination of all contractors obtained for other non-LMI constraints can thus be performed up to the fixed point. The resulting propogation is shown to be more efficient than other conventional contractor-based approaches.

scholarly journals On a maximal outer area problem for a class of meromorphic univalent fuctions

1986 ◽  
Vol 34 (3) ◽  
pp. 433-445 ◽  
Author(s):  
Stephen M. Zemyan
Keyword(s):  
Unit Disk ◽  
Extremal Function ◽  
Additional Information ◽  
Area Problem ◽  
Class Of Functions ◽  
Outer Area

For 0 < p < 1, let Sp denote the class of functions f (z) which are meromorphic and univalent in the unit disk U, with the normalisations f (0) = 0, f′(0) = 1 and f (p) = ∞, and let Sp (a) denote subclass of Sp consisting of those functions in Sp whose residue at the pole in equal to a. In this paper, we determine, for values of the residue a in a certain disk Δp, the greatest possible outer area over all functions in the class Sp (a). We also determine additional information concerning extremal function if the reside a dose not lie in Δp.

Stability results for fixed point sets of α*- Ψ contractive multivalued mappings

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3665-3670
Author(s):  
Binayak Choudhury ◽  
Chaitali Bandyopadhyay ◽  
Rajendra Pant
Keyword(s):  
Fixed Point ◽  
Contraction Mapping ◽  
Stability Result ◽  
Multivalued Mappings ◽  
Point Sets ◽  
Generalized Contraction ◽  
Class Of Functions ◽  
Stability Results ◽  
Fixed Point Sets

In this paper, we established a stability result for fixed point sets associated with a sequence of multivalued mappings which belong to class of functions obtained by a multivalued extension of certain generalized contraction mapping. Certain other aspects of these mappings are already studied in the existing literatures. We also construct an illustrative example.

scholarly journals Solving Fixed-Point Problems with Inequality and Equality Constraints via a Non-Interior Point Homotopy Path-Following Method

2017 ◽  
Vol 2017 ◽  
pp. 1-9
Author(s):  
Menglong Su ◽  
Yufeng Shang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Interior Point ◽  
Fixed Point Theorems ◽  
Path Following ◽  
Fixed Point Problems ◽  
Constructive Proof ◽  
Nonconvex Sets ◽  
Smooth Path

In recent years, fixed-point theorems have attracted increasing attention and have been widely investigated by many authors. Moreover, determining a fixed point has become an interesting topic. In this paper, we provide a constructive proof of the general Brouwer fixed-point theorem and then obtain the existence of a smooth path which connects a given point to the fixed point. We also present a non-interior point homotopy algorithm for solving fixed-point problems on a class of nonconvex sets by numerically tricking this homotopy path.

scholarly journals Starlikeness associated with parabolic regions

2005 ◽  
Vol 2005 (4) ◽  
pp. 561-570 ◽  
Author(s):  
Rosihan M. Ali
Keyword(s):  
Unit Disk ◽  
Sharp Estimate ◽  
Starlike Functions ◽  
Starlike Function ◽  
Extremal Functions ◽  
Sharp Growth ◽  
Coefficient Problems ◽  
Distortion Theorems ◽  
Convolution Properties ◽  
The Right

A parabolic starlike functionfof orderρin the unit disk is characterized by the fact that the quantityzf′(z)/f(z)lies in a given parabolic region in the right half-plane. Denote the class of such functions byPS∗(ρ). This class is contained in the larger class of starlike functions of orderρ. Subordination results forPS∗(ρ)are established, which yield sharp growth, covering, and distortion theorems. Sharp bounds for the first four coefficients are also obtained. There exist different extremal functions for these coefficient problems. Additionally, we obtain a sharp estimate for the Fekete-Szegö coefficient functional and investigate convolution properties forPS∗(ρ).

scholarly journals A Stochastic Maximin Fixed-Point Equation Related to Game Tree Evaluation

2007 ◽  
Vol 44 (3) ◽  
pp. 586-606
Author(s):  
Gerold Alsmeyer ◽  
Matthias Meiners
Keyword(s):  
Fixed Point ◽  
Distribution Function ◽  
Unique Fixed Point ◽  
Game Tree ◽  
Fixed Point Equation ◽  
Point Equation ◽  
Particular Solution ◽  
Minimax Game ◽  
Class Of Functions ◽  
Tree Evaluation

After suitable normalization the asymptotic root value W of a minimax game tree of order b ≥ 2 with independent and identically distributed input values having a continuous, strictly increasing distribution function on a subinterval of R appears to be a particular solution of the stochastic maximin fixed-point equation W ξ max1≤i≤bmin1≤j≤bWi,j, where Wi,j are independent copies of W and denotes equality in law. Moreover, ξ= g'(α) > 1, where g(x) := (1 − (1 − x)b)b and α denotes the unique fixed point of g in (0, 1). This equation, which takes the form F(t) = g(F(t/ξ)) in terms of the distribution function F of W, is studied in the present paper for a reasonably extended class of functions g so as to encompass more general stochastic maximin equations as well. A complete description of the set of solutions F is provided followed by a discussion of additional properties such as continuity, differentiability, or existence of moments. Based on these results, it is further shown that the particular solution mentioned above stands out among all other ones in that its distribution function is the restriction of an entire function to the real line. This extends recent work of Ali Khan, Devroye and Neininger (2005). A connection with another class of stochastic fixed-point equations for weighted minima and maxima is also discussed.

scholarly journals On analytic functions with reference to an integral operator

1983 ◽  
Vol 28 (2) ◽  
pp. 207-215 ◽  
Author(s):  
R. Parvatham ◽  
T.N. Shanmugam
Keyword(s):  
Integral Operator ◽  
Analytic Functions ◽  
Sharp Estimate ◽  
Radius Of Starlikeness ◽  
Image Position ◽  
Class Of Functions

Let E = {z: |z| < 1} and let H = {w : regular in E, w(0) = 0, |w(z)| < l, z ∈ E}.Let P(A, B) denote the class of functions in E which can be put in the form (1 + Aw(z))/(1 + Bw(z)), −1 ≤ A < B ≤ 1, w(z) ∈ H. Let S*(A, B) denote the class of functions f(z) of the form such that zf′(z)/f(z) ∈ P(A, B). If f(z) ∈ S*(A, B) and g(z) ∈ S*(C, D) then, in this paper the radius of starlikeness of order β (β ∈ [0, 1]) of the following integral operatoris determined. Conversely, a sharp estimate is obtained for the radius of starlikeness of the class of functionswhere g(z) and F(z) belong to the class S*(A, B).

scholarly journals A note on successive coefficients of spirallike functions

Filomat ◽  
2018 ◽  
Vol 32 (4) ◽  
pp. 1199-1207 ◽  
Author(s):  
Ming Li
Keyword(s):  
Unit Disk ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Upper Bounds ◽  
Extremal Functions ◽  
Lower And Upper Bounds ◽  
Bieberbach Conjecture ◽  
Precise Form ◽  
Spirallike Functions ◽  
Class Of Functions

Even there were several facts to show that ||an+1(f)|-|an(f)|| ? 1 is not true for the whole class of normalised univalent functions in the unit disk with with the form f(z) = z + ??,k=2 akzk. In 1978, Leung[7] proved ||an+1(f)|-|an(f)|| is actually bounded by 1 for starlike functions and by this result it is easy to get the conclusion |an| ? n for starlike functions. Since ||an+1(f)|-|an(f)|| ? 1 implies the Bieberbach conjecture (now the de Brange theorem), so it is still interesting to investigate the bound of ||an+1(f)|-|an(f)|| for the class of spirallike functions as this class of functions is closely related to starlike functions. In this article we prove that this functional is bounded by 1 and equality occurs only for the starlike case. We are also able to give a precise form of extremal functions. Furthermore we also try to find the sharp bound of ||an+1(f)|-|an(f)|| for non-starlike spirallike functions. By using the Carath?odory-Toeplitz theorem, we obtain the sharp lower and upper bounds of |an+1(f)|-|an(f)| for n = 1 and n = 2. These results disprove the expected inequality ||an+1(f)|-|an(f)||? cos ? for ?-spirallike functions.

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