Volume 63 , issue 1 ( 2011 ) | back |

2-nsr lemma and quotient space in 2-normed space | 1$-$6 |

**Abstract**

In this paper we discuss properties of compactness and compact operator on $2$-normed space. Also we consider a result which is similar to Riesz Lemma and its applications in $2$-normed space. We introduce quotient space from the finite dimensional subspace of a $2$-normed space.

**Keywords:** $2$-normed space, locally bounded, compact operator, Riesz Lemma, $e$-bounded operator.

**MSC:** 41A65, 41A15

Fixed point theorems for some generalized contractive multi-valued mappings and fuzzy mappings | 7$-$26 |

**Abstract**

In this paper, first we give a theorem which generalizes the Banach contraction principle and fixed point theorems given by many authors, and then a fixed point theorem for a multi-valued $(\theta, L)$-weak contraction. We extend the notion of $(\theta,L)$-weak contraction to fuzzy mappings and obtain some fixed point theorems. A coincidence point theorem for a hybrid pair of mappings $f:X\to X$ and $T:X\to W(X)$ is established. Later on we prove a fixed point theorem for a different type of fuzzy mapping.

**Keywords:** Contraction mapping; fixed point; Banach contraction principle; Hausdorff metric; fuzzy set; fuzzy mapping.

**MSC:** 03E72, 47H10, 54H25

A note on rate of approximation for certain Bézier-Durrmeyer operators | 27$-$32 |

**Abstract**

The present paper deals with certain Bézier-Durrmeyer type sequence of linear positive operators $M_{n,\alpha}(f,x)$, having different basis functions in summation and integration. We estimate the rate of convergence of these operators $M_{n,\alpha}(f,x)$, for functions having derivatives of bounded variation.

**Keywords:** Linear positive operators; Baskakov and Szász basis functions; summation-integral type operators; Bézier variant; rate of convergence.

**MSC:** 41A25, 41A30

Weighted composition operators on weighted Bergman spaces of infinite order with the closed range property | 33$-$39 |

**Abstract**

We study under which conditions weighted composition operators acting on weighted Bergman spaces of infinite order are bounded from below or, equivalently, have closed range.

**Keywords:** Weighted composition operator; weighted Bergman space of infinite order.

**MSC:** 47B33, 47B38

A note on star compact spaces with a $G_{\delta}$-diagonal | 41$-$43 |

**Abstract**

In this note we give an example of a Hausdorff, star compact space with a $G_\delta$-diagonal which is not metrizable, which answers negatively a question of van Mill, Tkachuk and Wilson (Problem 4.8 in [J. van Mill, V.V. Tkachuk, R.G. Wilson, Classes defined by stars and neighbourhood assignments, Topology Appl. 154 (2007), 2127--2134]).

**Keywords:** Star compact space, $G_\delta$-diagonal.

**MSC:** 54D20

Certain classes of multiple generating functions for some sets of polynomials in several variables | 45$-$54 |

**Abstract**

In this paper some generating functions for some sets of polynomials in several variables are established. In these classes of generating functions an arbitrary sequence of multivariable functions is considered. The generating functions so derived are shown here to lead some known results of Raina, Raina and Bajpai and Zeitlin and are capable to provide as special cases, a large number of new summation formulas and generating functions for simpler sequences, extended polynomials and generalized Lauricella functions.

**Keywords:** Lagrange's expansions, generating functions, summation formulae, Lauricella function.

**MSC:** 33C70, 33C47, 33C65

Radius estimates of a subclass of univalent functions | 55$-$58 |

**Abstract**

For analytic functions $f$ normalized by $f(0)=f'(0)-1=0$ in the open unit disk $U$, a class $P_{\alpha}(\lambda)$ of $f$ defined by $|D^{\alpha}_{z}(\frac{z}{f(z)})|\leq \lambda$, where $D^{\alpha}_{z}$ denotes the fractional derivative of order $\a$, $m \leq \alpha < m+1$, $m \in N_{0} $, is introduced. In this article, we study the problem when $\frac{1}{r} f(rz) \in P_{\alpha}(\lambda)$, $3 \leq \alpha < 4$.

**Keywords:** Analytic functions, univalent functions, Cauchy-Schwarz inequality, fractional differential operator.

**MSC:** 30C45

Szász-Mirakjan type operators of two variables providing a better estimation on $[0,1]\times[0,1]$ | 59$-$66 |

**Abstract**

This paper deals with a modification of the classical Szász-Mirakjan type operators of two variables. It introduces a new sequence of non-polynomial linear operators which hold fixed the polynomials $x^{2}+\alpha x$ and $y^{2}+\beta y$ with $\alpha ,\beta \in [0,\infty)$ and we study the convergence properties of the new approximation process. Also, we compare it with Szász-Mirakjan type operators and show an improvement of the error of convergence in $[0,1] \times [0,1]$. Finally, we study statistical convergence of this modification.

**Keywords:** Szász-Mirakjan type operators, $A$-statistical convergence for
double sequences, Korovkin-type approximation theorem, modulus of contiunity.

**MSC:** 41A25, 41A36

Optimal fourth order family of iterative methods | 67$-$72 |

**Abstract**

In this work, we construct a family of optimal fourth order iterative methods requiring three evaluations. During each iterative step, methods need evaluation of two derivatives and one function. According to the Kung and Traub conjecture an optimal iterative method without memory based on $3$ evaluations could achieve an optimal convergence order of $4$. The proposed iterative family of methods are especially appropriate for finding zeros of functions whose derivative is easy to evaluate. For example, polynomial functions and functions defined via integrals.

**Keywords:** Iterative method; fourth order; Newton method; convergence; nonlinear; optimal; derivative.

**MSC:** 41H25, 65D99

Coefficient inequalities for certain classes of analytic functions of complex order | 73$-$78 |

**Abstract**

Let $\Cal{Q}_{b}(\Phi ,\Psi ;\alpha )$ be the class of normalized analytic functions defined in the open unit disk and satisfying $$ \RE\left\{ 1+\frac{1}{b}\left( \frac{f(z)\ast \Phi (z)}{f(z)\ast \Psi (z)}-1\right) \right\} >\alpha $$ for nonzero complex number $b$ and for $0\leq \alpha <1$. Sufficient condition, involving coefficient inequalities, for $f(z)$ to be in the class $\Cal{Q}_{b}(\Phi ,\Psi ;\alpha )$ is obtained. Our main result contains some interesting corollaries as special cases.

**Keywords:** Analytic functions; Starlike and convex functions of complex order; Hadamard product; Coefficient inequalities.

**MSC:** 30C45