Necessary and sufficient conditions for the operator equation I AXAX n  * , to have a real positive definite solution X are given. Based on these conditions, some properties of the operator A as well as relation between the solutions X andAare given.

          We obtain the coefficient estimates, extreme points, distortion and growth boundaries, radii of starlikeness, convexity, and close-to-convexity, according to the main purpose of this paper.

In this work, we give an identity that leads to establishing the operator . Also, we introduce the polynomials . In addition, we provide Operator proof for the generating function with its extension and the Rogers formula for . The generating function with its extension and the Rogers formula for the bivariate Rogers-Szegö polynomials  are deduced. The Rogers formula for  allows to obtain the inverse linearization formula for , which allows to deduce the inverse linearization formula for . A solution to a q-difference equation is introduced and the solution is expressed in terms of the operators . The q-difference method is used to recover an identity of the operator  and the generating function for the polynomials

Some relations of inclusion and their properties are investigated for functions of type " -valent that involves the generalized operator of Srivastava-Attiya by using the principle of strong differential subordination.

In this paper, we introduce and study the essential and closed fuzzy submodules of a fuzzy module X as a generalization of the notions of essential and closed submodules. We prove many basic properties of both concepts.

In this paper we have studied a generalization of  a class of ( w-valent ) functions with two fixed points involving hypergeometric function with generalization  integral operator . We obtain some results like, coefficient estimates and some theorems of this class.

The main purpose of this work is to introduce some types of fuzzy convergence sequences of operators defined on a standard fuzzy normed space (SFN-spaces) and investigate some properties and relationships between these concepts. Firstly, the definition of weak fuzzy convergence sequence in terms of fuzzy bounded linear functional is given. Then the notions of weakly and strongly fuzzy convergence sequences of operators  are introduced and essential theorems related to these concepts are proved. In particular, if ( ) is a strongly fuzzy convergent sequence with a limit  where linear operator from complete standard fuzzy normed space  into a standard fuzzy normed space  then  belongs to the set of all fuzzy bounded linear operators

In this paper, a differential operator is used to generate a subclass of analytic and univalent functions with positive coefficients. The studied class of the functions includes:  

which is defined in the open unit disk  satisfying the following condition

This leads to the study of properties such as coefficient bounds, Hadamard product, radius of close –to- convexity, inclusive properties, and (n, τ) –neighborhoods for functions belonging to our class.

The main goal of this paper is to study and discuss a new class of meromorphici "functions[ which are multivalent defined by [fractional  calculus operators. Coefficients iestimates , radiisi of satarlikeness , convexityi and closed-to-iconvexity are studied. Also distortion iand closure theorems for the classi" ,  are considered.