Volume 62 , issue 1 ( 2010 ) | back |

Fractional double Newton step properties for polynomials with all real zeros | 1$-$9 |

**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 $\theta$ with $1 < \theta < 2$, the iterates cannot overshoot the zero of a different function. When $\theta=2$, our result reduces to the one for the double-step case. An analogous property exists for the smallest zero.

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

**MSC:** 65H05

On bitopological full normality | 11$-$18 |

**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.

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

**MSC:** 54E55

Extremal non-compactness of weighted composition operators on the disk algebra | 19$-$22 |

**Abstract**

Let $A({D})$ denote the disk algebra and $W_{\psi,\phi}$ be weighted composition operator on $A({D})$. In this paper we obtain a condition on $\psi$ and $\phi$ for $W_{\psi,\phi}$ 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.

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

**MSC:** 47B33

On certain multivalent functions with negative coefficients defined by using a differential operator | 23$-$35 |

**Abstract**

In this paper, we introduce the subclass $S_{j}(n,p,q,\alpha)$ of analytic and $p$-valent functions with negative coefficients defined by new operator $D_{p}^{n}$. In this paper we give some properties of functions in the class $S_{j}(n,p,q,\alpha)$ 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 $S_{j}(n,p,q,\alpha)$. Finally, several applications involving an integral operator and certain fractional calculus operators are also considered.

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

**MSC:** 30C45

The Schur-harmonic-convexity of dual form of the Hamy symmetric function | 37$-$46 |

**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

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

**MSC:** 26B25, 05E05, 26D20

On variation topology | 47$-$50 |

**Abstract**

Let $I$ be a real interval and $X$ be a Banach space. It is observed that spaces $\Lambda BV^{(p)}([a, b],R)$, $LBV(I,X)$ (locally bounded variation), $BV_0(I,X)$ and $LBV_0(I,X)$ share many properties of the space $BV([a,b],R)$. Here we have proved that the space $\Lambda BV^{(p)}_0(I,X)$ is a Banach space with respect to the variation norm and the variation topology makes $L\Lambda BV^{(p)}_0(I,X)$ a complete metrizable locally convex vector space (i.e\. a Fréchet space).

**Keywords:** $\Lambda BV^{(p)}$; Banach space; complete metrizable locally convex vector space; Fréchet space.

**MSC:** 26A45, 46A04

Certain subclasses of analytic functions defined by a family of linear operators | 51$-$61 |

**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.

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

**MSC:** 30C45

Cauchy operator on Bergman space of harmonic functions on unit disk | 63$-$67 |

**Abstract**

We find the exact asymptotic behaviour of singular values of the operator $CP_h$, where $C$ is the integral Cauchy's operator and $P_h$ integral operator with the kernel $$ K\left( z,\zeta\right) =\frac{\left( 1-\vert z\vert^2\vert\zeta\vert^2\right)^2} {\pi\vert 1-z\overline{\zeta }\vert^4}-\frac{2}{\pi }\ \frac{\vert z\vert^2\vert\zeta\vert^2} {\vert 1-z\overline{\zeta }\vert^2}. $$

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

**MSC:** 47G10, 45P05

Quasi continuous selections of upper Baire continuous mappings | 69$-$76 |

**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.

**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 spaces | 77$-$83 |

**Abstract**

A space $X$ is almost Lindel{ö}f (weakly Lindel{ö}f) if for every open cover $\Cal U$ of $X$, there exists a countable subset $\Cal V$ of $\Cal U$ such that $\bigcup\{\overline{V}:V\in \Cal V\}=X$ (respectively, $\overline{\bigcup\Cal V}=X$). 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.

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

**MSC:** 54D20, 54E18

On a class of sequences related to the $l^{p}$ space defined by a sequence of Orlicz functions | 85$-$93 |

**Abstract**

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

**Keywords:** Seminorm; Orlicz function.

**MSC:** 40A05, 46A45, 46E30