Volume 64 , issue 3 ( 2012 ) | back |

Properties of some families of meromorphic multivalent functions associated with generalized hypergeometric functions | 173$-$189 |

**Abstract**

We introduce and study two subclasses $\Omega_{\lbrack\alpha_{1}]}(A,B,\lambda)$ and $\Omega_{\lbrack\alpha_{1}]}^{+}(A,B,\lambda)$ of meromorphic $p$-valent functions defined by certain linear operator involving the generalized hypergeometric function. The main object is to investigate the various important properties and characteristics of these subclasses of meromorphically multivalent functions. We extend the familiar concept of neighborhoods of analytic functions to these subclasses. We also derive many interesting results for the Hadamard products of functions belonging to the class $\Omega_{\lbrack\alpha_{1}]}^{+}(\alpha,\beta,\gamma,\lambda)$.

**Keywords:** Generalized hypergeometric function; Hadamard product; meromorphic functions; neighborhoods.

**MSC:** 30C45

On quasi-antiorder in semigroups | 190$-$199 |

**Abstract**

Partially ordered semigroups with apartness under an antiorder are investigated from the point of view of Bishop's constructive mathematics. We analyze quasi-antiorder relations on ordered semigroups under an antiorder. The connection between two quasi-antiorders on a semigroup is presented.

**Keywords:** Constructive algebra; semigroup with apartness; ordered semigroup; antiorder; quasi-antiorder.

**MSC:** 03F65, 20M99

Position vectors of curves in the galilean space G$_{3}$ | 200$-$210 |

**Abstract**

In this paper, we study the position vector of an arbitrary curve in Galilean 3-space ${G}_3$. We first determine the position vector of an arbitrary curve with respect to the Frenet frame. Also, we deduce in terms of the curvature and torsion, the natural representation of the position vector of an arbitrary curve. Moreover, we define a plane curve, helix, general helix, Salkowski curves and anti-Salkowski curves in Galilean space ${G}_3$. Finally, the position vectors of some special curves are obtained and sketching.

**Keywords:** Position vectors; Frenet equations; Galilean 3-space.

**MSC:** 53A35, 53B30, 53C50

More on evaluating determinants | 211$-$222 |

**Abstract**

This article provides a general technique for finding closed formulas for the determinants of families of matrices whose entries satisfy a three-term recurrence relation. The major purpose of this article is to generalize several published results about evaluating determinants. We also present new proofs for some known results due to Ch. Krattenthaler.

**Keywords:** Determinant; matrix factorization; recursive relation; generalized Pascal triangle.

**MSC:** 15B36, 15A15, 11C20

Solution of nonlinear integral equations via fixed point of generalized contractive condition | 223$-$231 |

**Abstract**

The main aim of our paper is to prove the existence of a solution of a system of simultaneous Voltera-Hammerstein nonlinear integral equations by the help of a common fixed point theorem satisfying a generalized contractive condition. For, we have used a common fixed point result of generalized contractive condition in a complete metric space for two pairs of weakly compatible mappings.

**Keywords:** Common fixed point; generalized contractive condition; nonlinear integral equation; weakly compatible mappings.

**MSC:** 47H10, 54H25

Doubly connected domination subdivision numbers of graphs | 232$-$239 |

**Abstract**

A set $S$ of vertices of a connected graph $G$ is a doubly connected dominating set (DCDS) if every vertex not in $S$ is adjacent to some vertex in $S$ and the subgraphs induced by $S$ and $V-S$ are connected. The doubly connected domination number $gc(G)$ is the minimum size of such a set. The doubly connected domination subdivision number sd$_{gc}(G)$ is the minimum number of edges that must be subdivided (each edge in $G$ can be subdivided at most once) in order to increase the doubly connected domination number. In this paper first we establish upper bounds on the doubly connected domination subdivision number in terms of the order $n$ of $G$ or of its edge connectivity number $\kappa'(G)$. We also prove that $gc(G)+sd_{gc}(G)\leq n$ with equality if and only if either $G=K_2$ or for each pair of adjacent non-cut vertices $u,vın V(G)$, $G[V(G)-\{u,v\}]$ is disconnected.

**Keywords:** Doubly connected domination number; doubly connected domination subdivision number.

**MSC:** 05C69

Integral and computational representation of summation which extends a Ramanujan's sum | 240$-$245 |

**Abstract**

A generalized sum, which contains Ramanujan's summation formula recorded in Hardy's article [G.H. Hardy, A chapter from Ramanujan's notebook, Proc. Camb. Phil. Soc. 21 (1923), 492--503] as a special case, has been represented in the form of Mellin-Barnes type contour integral. A computational representation formula is derived for this summation in terms of the unified Hurwitz-Lerch Zeta function.

**Keywords:** Ramanujan's summation formula; generalized hypergeometric functions ${}_pF_q$; Mellin-Barnes type path
integral; Hurwitz-Lerch Zeta function; computational representation.

**MSC:** 33C20, 40H05

Some properties of generalized Sz\`asz type operators of two variables | 246$-$257 |

**Abstract**

A generalization of Sz\`{a}sz type operators for two variables is constructed and the theorems on convergence and the degree of convergence are established. In addition, we consider the simultaneous approximation of these operators.

**Keywords:** Sz\`{a}sz type operator; modulus of smoothness; $K$-functional; rate of convergence; divided difference.

**MSC:** 41A25, 41A35, 41A36

A generalization of fixed point theorems in $S$-metric spaces | 258$-$266 |

**Abstract**

In this paper, we introduce $S$-metric spaces and give some of their properties. Also we prove a fixed point theorem for a self-mapping on a complete $S$-metric space.

**Keywords:** $S$-metric space; fixed point.

**MSC:** 54H25, 47H10