Paper's Title:

On Singular Numbers of Hankel Matrices of Markov Functions

Author(s):

Vasily A. Prokhorov

Department of Mathematics and Statistics,
University of South Alabama,
Mobile, Alabama 36688-0002,
USA.
E-mail: prokhoro@southalabama.edu
URL: http://www.southalabama.edu/mathstat/people/prokhorov.shtml

Abstract:

Let E ⊂ (01,1) be a compact set and let μ be a positive Borel measure with support supp μ=E. Let

In the case when E=[a,b]⊂ (-1,1) and μ satisfies the condition dμ/dx>0 a.e. on E, we investigate asymptotic behavior of singular numbers σ_kn,n of the Hankel matrix D_n, where k_n/n→θ∈[0,1] as n→∞. Moreover, we obtain asymptotics of the Kolmogorov, Gelfand and linear k-widths, k=k_n, of the unit ball A_n,2 of P_n∩L₂(Γ) in the space L₂(μ,E), where Γ={z:|z|=1} and P_n is the class of all polynomials of the degree at most n.

Paper's Title:

On Certain Classes of Harmonic Univalent Functions Based on Salagean Operator

Author(s):

G. Murugusundaramoorthy, Thomas Rosy, and B. A. Stephen

Department of Applied Mathematics and Informatics,
Department of Mathematics, Vellore Institute of Technology,
Deemed University, Vellore - 632014, India.
gmsmoorthy@yahoo.com 

Department of Applied Mathematics and Informatics,
Department of Mathematics, Madras Christian College,
Chennai - 600059, India.
drthomasrosy@rediffmail.com 

Abstract:

We define and investigate a class of complex-valued harmonic univalent functions of the form f = h + g using Salagean operator where h and g are analytic in the unit disc U = { z : |z| < 1 }. A necessary and sufficient coefficient conditions are given for functions in these classes. Furthermore, distortion theorems, inclusion relations, extreme points, convolution conditions and convex combinations for this family of harmonic functions are obtained.

Paper's Title:

Properties of Certain Multivalent Functions Involving Ruscheweyh Derivatives

Author(s):

N-Eng Xu and Ding-Gong Yang

Department of Mathematics,
Changshu Institute of Technology,
Changshu, Jiangsu 215500,
China

xun@cslg.edu.cn
 

Abstract:

Let A_p(p∈ N) be the class of functions which are analytic in the unit disk. By virtue of the Ruscheweyh derivatives we introduce the new subclasses C_p(n,α,β,λ,μ) of A_p. Subordination relations, inclusion relations, convolution properties and a sharp coefficient estimate are obtained. We also give a sufficient condition for a function to be in C_p(n,α,β,λ,μ)

Paper's Title:

Polyanalytic Functions on Subsets of Z[i]

Author(s):

Abtin Daghighi

Linköping University,
SE-581 83,
Sweden.

E-mail: abtindaghighi@gmail.com

Abstract:

For positive integers q we consider the kernel of the powers L^q where L is one of three kinds of discrete analogues of the Cauchy-Riemann operator. The first two kinds are well-studied, but the third kind less so. We give motivations for further study of the third kind especially since its symmetry makes it more appealing for the cases q≥ 2.

From an algebraic perspective it makes sense that the chosen multiplication on the kernels is compatible with the choice of pseudo-powers. We propose such multiplications together with associated pseudo-powers. We develop a proof-tool in terms of certain sets of uniqueness.

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