scholarly journals Forbidden Families of Minimal Quadratic and Cubic Configurations

10.37236/6902 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Attila Sali ◽  
Sam Spiro
Keyword(s):  
Extremal Function ◽  
Asymptotic Growth ◽  
Column Permutation ◽  
Simple Matrix ◽  
The Extremal Function

A matrix is simple if it is a (0,1)-matrix and there are no repeated columns. Given a (0,1)-matrix $F$, we say a matrix $A$ has $F$ as a configuration, denoted $F\prec A$, if there is a submatrix of $A$ which is a row and column permutation of $F$. Let $|A|$ denote the number of columns of $A$. Let $\mathcal{F}$ be a family of matrices. We define the extremal function $\text{forb}(m, \mathcal{F}) = \max\{|A|\colon A \text{ is an }m-\text{rowed simple matrix and has no configuration } F\in\mathcal{F}\}$. We consider pairs $\mathcal{F}=\{F_1,F_2\}$ such that $F_1$ and $F_2$ have no common extremal construction and derive that individually each $\text{forb}(m, F_i)$ has greater asymptotic growth than $\text{forb}(m, \mathcal{F})$, extending research started by Anstee and Koch.

scholarly journals Forbidden Berge Hypergraphs

10.37236/6482 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
R. P. Anstee ◽  
Santiago Salazar
Keyword(s):  
Extremal Function ◽  
Column Permutation ◽  
Vertex Graph ◽  
Simple Matrix ◽  
The Extremal Function

A simple matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix $F$, we say that a (0,1)-matrix $A$ has $F$ as a Berge hypergraph if there is a submatrix $B$ of $A$ and some row and column permutation of $F$, say $G$, with $G\le B$. Letting $\|A\|$ denote the number of columns in $A$, we define the extremal function $\mathrm{Bh}(m,{ F})=\max\{\|A\|\,:\, A \hbox{ }m\hbox{-rowed simple matrix and no Berge hypergraph }F\}$. We determine the asymptotics of $\mathrm{Bh}(m,F)$ for all $3$- and $4$-rowed $F$ and most $5$-rowed $F$. For certain $F$, this becomes the problem of determining the maximum number of copies of $K_r$ in a $m$-vertex graph that has no $K_{s,t}$ subgraph, a problem studied by Alon and Shikhelman.

scholarly journals A majorant problem

1992 ◽  
Vol 15 (3) ◽  
pp. 441-447
Author(s):  
Ronen Peretz
Keyword(s):  
Unit Disc ◽  
Extremal Function ◽  
Fourier Coefficients ◽  
Positive Real Part ◽  
Complex Vector ◽  
Positive Real ◽  
Zero Sets ◽  
The Extremal Function

Letf(z)=∑k=0∞akzk,a0≠0be analytic in the unit disc. Any infinite complex vectorθ=(θ0,θ1,θ2,…)such that|θk|=1,k=0,1,2,…, induces a functionfθ(z)=∑k=0∞akθkzkwhich is still analytic in the unit disc.In this paper we study the problem of maximizing thep-means:∫02π|fθ(reiϕ)|pdϕover all possible vectorsθand for values ofrclose to0and for allp<2.It is proved that a maximizing function isf1(z)=−|a0|+∑k=1∞|ak|zkand thatrcould be taken to be any positive number which is smaller than the radius of the largest disc centered at the origin which can be inscribed in the zero sets off1. This problem is originated by a well known majorant problem for Fourier coefficients that was studied by Hardy and Littlewood.One consequence of our paper is that forp<2the extremal function for the Hardy-Littlewood problem should be−|a0|+∑k=1∞|ak|zk.We also give some applications to derive some sharp inequalities for the classes of Schlicht functions and of functions of positive real part.

scholarly journals The extremal function for partial bipartite tilings

2012 ◽  
Vol 33 (5) ◽  
pp. 807-815 ◽  
Author(s):  
Codruţ Grosu ◽  
Jan Hladký
Keyword(s):  
Extremal Function ◽  
The Extremal Function

Unpredictability of the coefficients of m-fold symmetric bi-starlike functions

2014 ◽  
Vol 25 (07) ◽  
pp. 1450064 ◽  
Author(s):  
Samaneh G. Hamidi ◽  
Jay M. Jahangiri
Keyword(s):  
Extremal Function ◽  
Univalent Functions ◽  
Starlike Functions ◽  
Square Root ◽  
Koebe Function ◽  
The Extremal Function

In 1984, Libera and Zlotkiewicz proved that the inverse of the square-root transform of the Koebe function is the extremal function for the inverses of odd univalent functions. The purpose of this paper is to point out that this is not the case for the m-fold symmetric bi-starlike functions by demonstrating the unpredictability of the coefficients of such functions.

scholarly journals Convolutions of prestarlike functions

1983 ◽  
Vol 6 (1) ◽  
pp. 59-68 ◽  
Author(s):  
O. P. Ahuja ◽  
H. Silverman
Keyword(s):  
Unit Disk ◽  
Extremal Function ◽  
Extreme Points ◽  
Distortion Theorems ◽  
The Extremal Function ◽  
Class Of Functions

The convolution of two functionsf(z)=∑n=0∞anznandg(z)=∑n=0∞bnzndefined as(f∗g)(z)=∑n=0∞anbnzn. Forf(z)=z−∑n=2∞anznandg(z)=z/(1−z)2(1−γ), the extremal function for the class of functions starlike of orderγ, we investigate functionsh, whereh(z)=(f∗g)(z), which satisfy the inequality|(zh′/h)−1|/|(zh′/h)+(1-2α)|<β,0≤α<1,0<β≤1for allzin the unit disk. Such functionsfare said to beγ-prestarlike of orderαand typeβ. We characterize this family in terms of its coefficients, and then determine extreme points, distortion theorems, and radii of univalence, starlikeness, and convexity. All results are sharp.

scholarly journals Color critical hypergraphs and forbidden configurations

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Richard Anstee ◽  
Balin Fleming ◽  
Zoltán Füredi ◽  
Attila Sali
Keyword(s):  
Linear Algebra ◽  
Upper Bound ◽  
Sufficient Condition ◽  
Column Permutation ◽  
International Audience ◽  
Simple Matrix ◽  
Forbidden Configuration ◽  
Forbidden Configurations

International audience The present paper connects sharpenings of Sauer's bound on forbidden configurations with color critical hypergraphs. We define a matrix to be \emphsimple if it is a $(0,1)-matrix$ with no repeated columns. Let $F$ be $a k× l (0,1)-matrix$ (the forbidden configuration). Assume $A$ is an $m× n$ simple matrix which has no submatrix which is a row and column permutation of $F$. We define $forb(m,F)$ as the best possible upper bound on n, for such a matrix $A$, which depends on m and $F$. It is known that $forb(m,F)=O(m^k)$ for any $F$, and Sauer's bond states that $forb(m,F)=O(m^k-1)$ fore simple $F$. We give sufficient condition for non-simple $F$ to have the same bound using linear algebra methods to prove a generalization of a result of Lovász on color critical hypergraphs.

scholarly journals Note on Schiffer’s Variation in the Class of Univalent Functions in the Unit Disc

1968 ◽  
Vol 32 ◽  
pp. 273-276
Author(s):  
Kikuji Matsumoto
Keyword(s):  
Unit Disc ◽  
Extremal Function ◽  
Univalent Functions ◽  
The Real ◽  
Maximum Value ◽  
The Extremal Function

Let S denote the class of univalent functions f(z) in the unit disc D: | z | < 1 with the following expansion: (1) f(z) = z + a2z2 + a3z3 + · · · · anzn + · ··.We denote by fn(z) the extremal function in S which gives the maximum value of the real part of an and by Dn the image of D under w = fn(z).

New classes of certain analytic functions concerned with subordinations

2017 ◽  
Vol 33 (2) ◽  
pp. 153-160
Author(s):  
NICOLETA BREAZ ◽  
◽  
SHIGEYOSHI OWA ◽  
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Extremal Function ◽  
Open Unit ◽  
Open Unit Disk ◽  
The Extremal Function

Let A be the class of analytic functions f(z) in the open unit disk U which satisfy f(0) = 0 and f 0(0) = 1. Applying the extremal function for the subclass S∗(α) of A, new classes P∗(α) and Q∗(α) are considered using certain subordinations. The object of the present paper is to discuss some interesting properties for f(z) belonging to the classes P∗(α) and Q∗(α)

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