Volume 65 , issue 1 ( 2013 ) | back |

On certain linear operator defined by basic hypergeometric functions | 1$-$7 |

**Abstract**

By employing the basic hypergeometric series, we introduce here a linear operator for analytic functions. By means of this linear operator, we define and investigate a class of analytic functions. Also, as an application of Jack's lemma, sufficient conditions for univalence, starlikeness and strong starlikeness of certain analytic functions are obtained.

**Keywords:** Basic hypergeometric function; subordination; starlike; close to convex; univalent function;
Sălăgean differential operator; Jack's lemma.

**MSC:** 30C45

A new characterization of spaces with locally countable $sn$-networks | 8$-$13 |

**Abstract**

In this paper we prove that a space $X$ is with a locally countable $sn$-network (resp., weak base) if and only if it is a compact-covering (resp., compact-covering quotient) compact and $ss$-image of a metric space, if and only if it is a sequentially-quotient (resp., quotient) $\pi$- and $ss$-image of a metric space, which gives a new characterization of spaces with locally countable $sn$-networks (or weak bases).

**Keywords:** Weak base; $sn$-network; $\sigma$-strong network; locally countable; compact-covering; compact map; $\pi$-map.

**MSC:** 54C10, 54D65, 54E40, 54E99

Certain subclasses of uniformly starlike and convex functions defined by convolution with negative coefficients | 14$-$28 |

**Abstract**

The aim of this paper is to obtain coefficient estimates, distortion theorems, convex linear combinations and radii of close-to-convexity, starlikeness and convexity for functions belonging to the class $TS(g,\lambda;\alpha,\beta)$. Furthermore partial sums $f_{n}(z)$ of functions $f(z)$ in the class $TS(g,\lambda;\alpha,\beta)$ are considered and sharp lower bounds for the ratios of real part of $f(z)$ to $f_{n}(z)$ and $f'(z)$ to $f_{n}'(z)$ are determined.

**Keywords:** Analytic function; Hadamard product; distortion theorems; partial sums.

**MSC:** 30C45

Common fixed points for generalized nonlinear contractive mappings in metric spaces | 29$-$34 |

**Abstract**

The purpose of this paper is to present a common fixed point theorem for generalized nonlinear contractive mappings in complete metric spaces by generalizing and extending some known results. As a consequence, a common fixed point theorem for a Banach operator pair is obtained.

**Keywords:** Common fixed point; $f$-weakly contractive maps; generalized $f$-weakly contractive maps; weakly compatible maps; Banach operator pair.

**MSC:** 47H10, 54H25

Functions from $L_p$-spaces and Taylor means | 35$-$45 |

**Abstract**

In this paper, we take up Taylor means to study the degree of approximation of $f\in L_p$ ($p\ge1$) under the $L_p$-norm and obtain a general theorem which is used to obtain four more theorems that improve some earlier results obtained by Mohapatra, Holland and Sahney [J. Approx. Theory 45 (1985), 363--374]. One of our theorems provides the Jackson order as the degree of approximation for a subspace of $Lip(\a,p)$ ($0<\a<1$, $p\ge1$) and generalizes a result due to Chui and Holland [J. Approx. Theory 39 (1983), 24--38].

**Keywords:** Taylor means; degree of approximation.

**MSC:** 41A25, 42A10, 40G10

Strong convergence of implicit iterates with errors for non-Lipschitzian asymptotically quasi-nonexpansive type mappings in Banach spaces | 46$-$57 |

**Abstract**

In this paper we prove that an implicit iterative process with errors converges strongly to a common fixed point for a finite family of asymptotically quasi-nonexpansive type mappings on unbounded sets in a uniformly convex Banach space. Our results unify, improve and generalize the corresponding results of Ud-din and Khan, Sun, Wittman, Xu and Ori and many others.

**Keywords:** Asymptotically quasi-nonexpansive type mapping; implicit iteration process with errors; common fixed point;
strong convergence; uniformly convex Banach space.

**MSC:** 47H09, 47H10

On the space of $p$-summable sequences | 58$-$63 |

**Abstract**

On the space $\ell^p$ of $p$-summable sequences (of real numbers), one can derive a norm from the 2-norm as indicated by Gunawan [H. Gunawan, {The space of $p$-summable sequences and its natural $n$-norms}, Bull. Austral. Math. Soc. {64} (2001), 137--147]. The purpose of this note is to establish the equivalence between such a norm and the usual norm on $\ell^p$. We show that our result is useful in understanding the topology of $\ell^p$ as a 2-normed space.

**Keywords:** $p$-summable sequence space; norm equivalence.

**MSC:** 46B05, 46B20, 46A45, 46B45

On the fine spectrum of generalized upper double-band matrices $\Delta^{uv}$ over the sequence space $l_1$ | 64$-$73 |

**Abstract**

The main purpose of this paper is to determine the fine spectrum of generalized upper triangle double-band matrices $\Delta^{uv}$ over the sequence space $\ell_{1}$.

**Keywords:** Spectrum of an operator; matrix mapping; sequence space.

**MSC:** 47A10, 47B37

Fuzzy ideals in Laskerian rings | 74$-$81 |

**Abstract**

We introduce strongly primary fuzzy ideals and strongly irreducible fuzzy ideals in a unitary commutative ring and fixed their role in a Laskerian ring. We established that: A finite intersection of prime fuzzy ideals (resp. primary fuzzy ideals, irreducible fuzzy ideals and strongly irreducible fuzzy ideals) is a prime fuzzy ideal (resp. primary fuzzy ideal, irreducible fuzzy ideal and strongly irreducible fuzzy ideal). We also find that, a fuzzy ideal of a ring is prime if and only if it is semiprime and strongly irreducible. Furthermore we characterize that: (1) Every nonzero fuzzy ideal of a one dimensional Laskerian domain can be uniquely expressed as a product of primary fuzzy ideals with distinct radicals, (2) A unitary commutative ring is (strongly) Laskerian if and only if its localization is (strongly) Laskerian with respect to every fuzzy ideal.

**Keywords:** Fuzzy ideal; prime fuzzy ideal; primary fuzzy ideal; (strongly) irreducible fuzzy ideal; (strongly) Laskerian ring.

**MSC:** 13A15, 03E72, 13C12

Generalized distance and fixed point theorems in partially ordered probabilistic metric spaces | 82$-$93 |

**Abstract**

Recently, Ćirić, Mihe\c{t} and Saadati [Topoplogy Appl. 156 (2009), 2838-2844] proved a common fixed point theorem in partially ordered probabilistic metric spaces. In this paper, we consider the generalized distance in probabilistic metric spaces introduced by Saadati, et. al., [Bull. Iranian Math. Soc. 35:2 (2009), 97--117] and prove some fixed point theorems in partially ordered probabilistic metric spaces.

**Keywords:** Non-decreasing mapping; coincidence; fixed point; common fixed point; complete metric space; generalized distance.

**MSC:** 54H25, 47H10

Property (gz) for bounded linear operators | 94$-$103 |

**Abstract**

A bounded linear operator $T$ acting on a Banach space possesses property (gaw) if $\sigma(T)\setminus E_a(T)=\sigma_{BW}(T)$, where $\sigma_{BW}(T)$ is the B-Weyl spectrum of $T$, $\sigma(T)$ is the usual spectrum of $T$ and $E_a(T)$ is the set of all eigenvalues of $T$ which are isolated in the approximate point spectrum of $T$. In this paper we introduce and study the new spectral properties (z), (gz), (az) and (gaz) as a continuation of [M. Berkani, H. Zariouh, {\it New extended Weyl type theorems}, Mat. Vesnik {\bf 62} (2010), 145--154], which are related to Weyl type theorems. Among other results, we prove that $T$ possesses property (gz) if and only if $T$ possesses property (gaw) and $\sigma_{BW}(T)=\sigma_{SBF_+^-}(T)$; where $\sigma_{SBF_+^-}(T)$ is the essential semi-B-Fredholm spectrum of $T$.

**Keywords:** Property (z); property (gz); property (az); essential semi-B-Fredholm spectrum.

**MSC:** 47A53, 47A10, 47A11

Two-sided bounds for the complete Butzer-Flocke-Hauss Omega function | 104$-$121 |

**Abstract**

The main aim of this short note is to obtain two sided bounding inequalities for the real argument Butzer-Flocke-Hauss complete Omega function improving and developing a recent result by Pogány and Srivastava [Some two-sided bounding inequalities for the Butzer-Flocke-Hauss Omega function, Math. Inequal. Appl. 10 (2007), 587--595]. The main tools are the ODE whose particular solution is the Omega function and the related Čaplygin type differential inequality.

**Keywords:** Butzer-Flocke-Hauss complete Omega function, Čaplygin type differential
inequality,Čaplygin type comparison theorem, integral representation of the Omega function.

**MSC:** 34A40, 26D15, 33E30

Remarks on coupled fixed point theorem in cone metric spaces | 122$-$136 |

**Abstract**

In this paper, we first show that some coupled fixed point theorems in cone metric spaces are proper consequences of relevant fixed point theorems. Then we give and prove some corresponding coupled fixed point theorems in partially ordered cone metric spaces. Some examples are also given to illustrate our work.

**Keywords:** Coupled fixed point; mixed monotone mapping; partially ordered set; cone metric space; compatible mappings.

**MSC:** 47H10, 54H25

Growth of polynomials with prescribed zeros | 137$-$142 |

**Abstract**

In this paper, we study the growth of polynomials of degree $n$ having all their zeros on $|z|=k$, $k\leq 1$. Using the notation $M(p,t)=\max_{|z|=t}|p(z)|$, we measure the growth of $p$ by estimating $\big\{\frac{M(p,t)}{M(p,1)}\big\}^s$ from above for any $t\geq 1$, $s$ being an arbitrary positive integer. Also in this paper we improve the results recently proved by K. K. Dewan and Arty Ahuja [Growth of polynomials with prescribed zeros, J. Math. Ineq. 5 (2011), 355--361].

**Keywords:** Polar derivative; polynomial; Zygmund inequality; zeros.

**MSC:** 30A10, 30C10, 30D15, 41A17