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Volume 62 , issue 1 ( 2010 )back
Fractional double Newton step properties for polynomials with all real zeros1--9
A. Melman

Abstract

When doubling the Newton step for the computation of the largest zero of a real polynomial with all real zeros, a classical result shows that the iterates never overshoot the largest zero of the derivative of the polynomial. Here we show that when the Newton step is extended by a factor θ with 1<θ<2, the iterates cannot overshoot the zero of a different function. When θ=2, our result reduces to the one for the double-step case. An analogous property exists for the smallest zero.

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Keywords: Newton; overshoot; polynomial; double; fractional; step; zero; root.

MSC: 65H05

On bitopological full normality11--18
M. K. Bose and Ajoy Mukharjee

Abstract

The notion of bitopological full normality is introduced. Along with other results, we prove a bitopological version of A. H. Stone's theorem on paracompactness: A Hausdorff topological space is paracompact if and only if it is fully normal.

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Keywords: Pairwise paracompact spaces; pairwise fully normal spaces; shrinkable pairwise open cover.

MSC: 54E55

Extremal non-compactness of weighted composition operators on the disk algebra19--22
Harish Chandra and Bina Singh

Abstract

Let A(D) denote the disk algebra and Wψ,ϕ be weighted composition operator on A(D). In this paper we obtain a condition on ψ and ϕ for Wψ,ϕ to exhibit extremal non-compactness. As a consequence we show that the essential norm of a composition operator on A(D) is either 0 or 1.

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Keywords: Essential norm; composition operator; inner function; extremal non-compactness.

MSC: 47B33

On certain multivalent functions with negative coefficients defined by using a differential operator23--35
M. K. Aouf

Abstract

In this paper, we introduce the subclass Sj(n,p,q,α) of analytic and p-valent functions with negative coefficients defined by new operator Dnp. In this paper we give some properties of functions in the class Sj(n,p,q,α) and obtain numerous sharp results including (for example) coefficient estimates, distortion theorem, radii of close-to-convexity, starlikeness and convexity and modified Hadamard products of functions belonging to the class Sj(n,p,q,α). Finally, several applications involving an integral operator and certain fractional calculus operators are also considered.

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Keywords: Multivalent functions; differential operator; modified-Hadamard product; fractional calculus.

MSC: 30C45

The Schur-harmonic-convexity of dual form of the Hamy symmetric function37--46
Junxia Meng, Yuming Chu and Xiaomin Tang

Abstract

We prove that the dual form of the Hamy symmetric function $$ H_n(x, r)=H_n(x_1, x_2, \dots, x_n; r)=\prod_{1\leq i_1<\cdots

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Keywords: Dual form; Hamy symmetric function; Schur convex; Schur harmonic convex.

MSC: 26B25, 05E05, 26D20

On variation topology47--50
R. G. Vyas

Abstract

Let I be a real interval and X be a Banach space. It is observed that spaces ΛBV(p)([a,b],R), LBV(I,X) (locally bounded variation), BV0(I,X) and LBV0(I,X) share many properties of the space BV([a,b],R). Here we have proved that the space ΛBV(p)0(I,X) is a Banach space with respect to the variation norm and the variation topology makes LΛBV(p)0(I,X) a complete metrizable locally convex vector space (i.e\. a Fréchet space).

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Keywords: ΛBV(p); Banach space; complete metrizable locally convex vector space; Fréchet space.

MSC: 26A45, 46A04

Certain subclasses of analytic functions defined by a family of linear operators51--61
A. O. Mostafa

Abstract

In this paper, we obtain some applications of first order differential subordination and superordination results involving Dziok-Srivastava operator and other linear operators for certain normalized analytic functions in the open unit disc.

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Keywords: Analytic functions; differential subordination; superordination; sandwich theorems; Dziok-Srivastava operator.

MSC: 30C45

Cauchy operator on Bergman space of harmonic functions on unit disk63--67
Milutin R. Dostanić

Abstract

We find the exact asymptotic behaviour of singular values of the operator CPh, where C is the integral Cauchy's operator and Ph integral operator with the kernel K(z,ζ)=(1|z|2|ζ|2)2π|1z¯ζ|42π |z|2|ζ|2|1z¯ζ|2.

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Keywords: Bergman space; Cauchy operator; asymptotics of eigenvalues.

MSC: 47G10, 45P05

Quasi continuous selections of upper Baire continuous mappings69--76
Milan Matejdes

Abstract

The paper deals with the existence problem of selections for a closed valued and c-upper Baire continuous multifunction F. The main goal is to find a minimal usco multifunction intersecting F and its selection that is quasi continuous everywhere except at points of a nowhere dense set. The methods are based on properties of minimal multifunctions and a cluster multifunction generated by a cluster process with respect to the system of all sets of second category with the Baire property.

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Keywords: Quasi-continuity; Baire continuity; usco multifunction; minimal multifunction; selection; cluster point; cluster multifunction.

MSC: 54C60, 54C65, 26E25

Some remarks on almost Lindel{ö}f spaces and weakly Lindel{ö}f spaces77--83
Yan-Kui Song and Yun-Yun Zhang

Abstract

A space X is almost Lindel{ö}f (weakly Lindel{ö}f) if for every open cover \CalU of X, there exists a countable subset \CalV of \CalU such that {¯V:V\CalV}=X (respectively, ¯\CalV=X). In this paper, we investigate the relationships among almost Lindel{ö}f spaces, weakly Lindel{ö}f spaces and Lindel{ö}f spaces, and also study topological properties of almost Lindel{ö}f spaces and weakly Lindel{ö}f spaces.

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Keywords: Lindel{ö}f; almost Lindel{ö}f; weakly Lindel{ö}f.

MSC: 54D20, 54E18

On a class of sequences related to the lp space defined by a sequence of Orlicz functions85--93
Ayhan Esi

Abstract

In this article we introduce the space m(Ω,ϕ,q) on generalizing the sequence space m(ϕ) using the sequence of Orlicz functions. We study its different properties and obtain some inclusion results involving the space m(Ω,ϕ,q).

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Keywords: Seminorm; Orlicz function.

MSC: 40A05, 46A45, 46E30