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 article, a new class  of analytic functions which is defined by terms of a quasi-subordination is introduced. The coefficient estimates, including the classical  inequality of functions belonging to this class, are then derived. Also, several special improving results for the associated classes involving the subordination are presented.

          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 the present paper, by making use of the new generalized operator, some results of third order differential subordination and differential superordination consequence for analytic functions are obtained. Also, some sandwich-type theorems are presented.

In the present paper, we introduce two subclasses, S^*C(,,g,s,d) and  TS^*C(, ,g, s,d), of analytic functions . Coefficients bounds for these subclasses are calculated.

The main purpose of this article is to originate characteristic properties of the functions in the above subclasses.

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.

The concern of this article is the calculation of an upper bound of second Hankel determinant for the subclasses of functions defined by Al-Oboudi differential operator in the unit disc. To study special cases of the results of this article, we give particular values to the parameters A, B and λ

   In this work, a novel technique to obtain an accurate solutions to nonlinear form by multi-step combination with Laplace-variational approach (MSLVIM) is introduced. Compared with the  traditional approach for variational it overcome all difficulties and enable to provide us more an accurate solutions with extended of the convergence region as well as covering to larger intervals which providing us a continuous representation of approximate analytic solution and it give more better information of the solution over the whole time interval. This technique is more easier for obtaining the general Lagrange multiplier with reduces the time and calculations. It converges rapidly to exact formula with simply computable terms wit

     The nuclear size radii, density distributions and elastic electron scattering charge form factors for Fluorine isotopes (^{17,19,20,24,26}F) were studied using the radial wave functions (WF) of harmonic-oscillator (HO) potential and free mean field described by spherical Hankel functions (SHF) for the core and the valence parts, respectively for all aforementioned isotopes. The parameters for HO potential (size parameter ) and SHF were chosen to regenerate the available experimental size radii. It was found that using spherical Hankel functions in our work improved the calculated results quantities in comparison with empirical data.   

     In this paper, subclasses of  the function class  ∑  of analytic and bi-univalent functions associated with operator  L_q^(k,  λ)  are introduced and defined in the open unit disk △ by applying quasi-subordination. We obtain some results about the corresponding bound estimations of the coefficients  a_(2 )  and a_(3 ).