scholarly journals A New Roper-Suffridge Extension Operator on a Reinhardt Domain

2011 ◽  
Vol 2011 ◽  
pp. 1-14 ◽  
Author(s):  
Jianfei Wang ◽  
Cailing Gao
Keyword(s):  
Positive Integer ◽  
Unit Disc ◽  
Extension Operator ◽  
Reinhardt Domain ◽  
Starlike Mappings

We introduce a new Roper-Suffridge extension operator on the following Reinhardt domaingiven bywherefis a normalized locally biholomorphic function on the unit discD,pjare positive integer,ajare complex constants, andj=2,…,n. Some conditions forajare found under which the operator preserves almost starlike mappings of orderαand starlike mappings of orderα, respectively. In particular, our results reduce to many well-known results when allαj=0.

scholarly journals LOWER BOUND TO A PROBLEM OF MOCANU ON DIFFERENTIAL SUBORDINATION

1997 ◽  
Vol 27 (3) ◽  
pp. 209-217
Author(s):  
S. PONNUSAMY
Keyword(s):  
Lower Bound ◽  
Analytic Functions ◽  
Unit Disc ◽  
Differential Subordination ◽  
The Family ◽  
Alpha Beta ◽  
Starlike Mappings

Let $s^*$ denote the family of starlike mappings in the unit disc $\Delta$. Let $\mathcal{R}(\alpha, \beta)$ denote the family of normalized analytic functions in $\Delta$ satisfying the condition Re$(f'(z)+\alpha f''(z))>\beta$, $z \in\Delta$ for some $\alpha > 0$. In this note, among other things, we give a lower bound to the problem of Mocanu aimed at determining $\inf\{\alpha : \mathcal{R}(\alpha,0) \subset S^*\}$.

Roper-Suffridge extension operator on a reinhardt domain

2014 ◽  
Vol 34 (6) ◽  
pp. 1761-1774
Author(s):  
Hongjun LI ◽  
Shuxia FENG
Keyword(s):  
Extension Operator ◽  
Reinhardt Domain

scholarly journals Structure of the kernels associated with invariant subspaces of the Bergman shift

2003 ◽  
Vol 67 (3) ◽  
pp. 445-457
Author(s):  
George Chailos
Keyword(s):  
Positive Integer ◽  
Unit Circle ◽  
Unit Disc ◽  
Invariant Subspaces ◽  
Meromorphic Continuation ◽  
Zero Set ◽  
Boundary Behaviour ◽  
Bergman Shift ◽  
Vector Valued ◽  
General Vector

In this article we consider index 1 invariant subspaces M of the operator of multiplication by ζ(z) = z, Mζ, on the Bergman space of the unit disc . It turns out that there is a positive sesquianalytic kernel lλ defined on  ×  which determines M uniquely. Here we study the boundary behaviour and some of the basic properties of the kernel lλ. Among other things, we show that if the lower zero set of M, Z̲(M), is nonempty, the kernel lλ for fixed λ ∈  has a meromorphic continuation across \Z̲(M), where  is the unit circle. Furthermore we consider some special type of kernels lλ and by studying their structure we obtain information for the invariant subspaces related to them. Lastly, and after introducing the general for the invariant subspaces related to them. Lastly, and after introducing the general vector valued setting, we discuss some analogous results for the case of , where m is a positive integer.

The Order of Monochromatic Subgraphs with a Given Minimum Degree

10.37236/1725 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Yair Caro ◽  
Raphael Yuster
Keyword(s):  
Positive Integer ◽  
Minimum Degree

Let $G$ be a graph. For a given positive integer $d$, let $f_G(d)$ denote the largest integer $t$ such that in every coloring of the edges of $G$ with two colors there is a monochromatic subgraph with minimum degree at least $d$ and order at least $t$. Let $f_G(d)=0$ in case there is a $2$-coloring of the edges of $G$ with no such monochromatic subgraph. Let $f(n,k,d)$ denote the minimum of $f_G(d)$ where $G$ ranges over all graphs with $n$ vertices and minimum degree at least $k$. In this paper we establish $f(n,k,d)$ whenever $k$ or $n-k$ are fixed, and $n$ is sufficiently large. We also consider the case where more than two colors are allowed.

Extensions of Rings Having McCoy Condition

2009 ◽  
Vol 52 (2) ◽  
pp. 267-272 ◽  
Author(s):  
Muhammet Tamer Koşan
Keyword(s):  
Positive Integer ◽  
Associative Ring ◽  
Nonzero Element ◽  
Ring Extensions

AbstractLet R be an associative ring with unity. Then R is said to be a right McCoy ring when the equation f (x)g(x) = 0 (over R[x]), where 0 ≠ f (x), g(x) ∈ R[x], implies that there exists a nonzero element c ∈ R such that f (x)c = 0. In this paper, we characterize some basic ring extensions of right McCoy rings and we prove that if R is a right McCoy ring, then R[x]/(xn) is a right McCoy ring for any positive integer n ≥ 2.

EVALUATION OF CONVOLUTION SUMS AND FOR k = a · b = 21, 33, AND 35

2021 ◽  
pp. 1-20
Author(s):  
K. PUSHPA ◽  
K. R. VASUKI
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Divisor Function ◽  
Convolution Sums

Abstract The article focuses on the evaluation of convolution sums $${W_k}(n): = \mathop \sum \nolimits_{_{m < {n \over k}}} \sigma (m)\sigma (n - km)$$ involving the sum of divisor function $$\sigma (n)$$ for k =21, 33, and 35. In this article, our aim is to obtain certain Eisenstein series of level 21 and use them to evaluate the convolution sums for level 21. We also make use of the existing Eisenstein series identities for level 33 and 35 in evaluating the convolution sums for level 33 and 35. Most of the convolution sums were evaluated using the theory of modular forms, whereas we have devised a technique which is free from the theory of modular forms. As an application, we determine a formula for the number of representations of a positive integer n by the octonary quadratic form $$(x_1^2 + {x_1}{x_2} + ax_2^2 + x_3^2 + {x_3}{x_4} + ax_4^2) + b(x_5^2 + {x_5}{x_6} + ax_6^2 + x_7^2 + {x_7}{x_8} + ax_8^2)$$ , for (a, b)=(1, 7), (1, 11), (2, 3), and (2, 5).

Sign in / Sign up

Export Citation Format

Share Document