In the complex field, special functions are closely related to geometric holomorphic functions. Koebe function is a notable contribution to the study of the geometric function theory (GFT), which is a univalent function. This sequel introduces a new class that includes a more general Koebe function which is holomorphic in a complex domain. The purpose of this work is to present a new operator correlated with GFT. A new generalized Koebe operator is proposed in terms of the convolution principle. This Koebe operator refers to the generality of a prominent differential operator, namely the Ruscheweyh operator. Theoretical investigations in this effort lead to a number of implementations in the subordination function theory. The ti

The main objectives of this pepper are to introduce new classes. We have attempted to obtain coefficient estimates, radius of convexity, Distortion and  Growth theorem and other related results for the classes

    In this paper, we derive some subordination and superordination results for certain subclasses of p− valent analytic functions that defined by generalized Fox-wright functions using the principle of differential subordination, ----------producing best dominant univalent solutions. We have also derived inclusion relations and solved majorization problem.

In this paper, the construction of Hermite wavelets functions and their operational matrix of integration is presented. The Hermite wavelets method is applied to solve nth order Volterra integro diferential equations (VIDE) by expanding the unknown functions, as series in terms of Hermite wavelets with unknown coefficients. Finally, two examples are given

The main objective of this paper is to introduce and study the generality differential operator involving the q-Mittag-Leffler function on certain subclasses of analytic functions.  Also, we  investigate the inclusion properties of these classes, by using the concept of subordination between analytic functions.

In this paper, making use of the q-R uscheweyh differential  operator , and  the  notion of t h e J anowski f unction, we study some subclasses of  holomorphic   f- unction s . Moreover , we obtain so me geometric characterization like co efficient es timat es , rad ii of starlikeness ,distortion theorem , close- t o- convexity , con vexity, ext reme point s, neighborhoods, and the i nte gral mean inequalities of func tions affiliation to these c lasses

          The authors introduced and addressed  several new subclasses  of the family of meromorphically multivalent -star-like functions in the punctured unit disk  in this study, which makes use of several higher order -derivatives. Many fascinating properties and characteristics are extracted systematically for each of these newly identified function classes. Distortion theorems and radius problems are among these characteristics and functions. A number of coefficient inequalities for functions belonging to the subclasses are studied, and discussed, as well as a suitable condition for them is set. The numerous results are presented in this study and the previous works on this

       In this paper we offer two new subclasses of an open unit disk of r-fold symmetric bi-univalent functions. The Taylor-Maclaurin coefficients  have their coefficient bounds calculated. Furthermore, for functions in , we have solved Fekete-  functional issues. For the applicable classes, there are also a few particular special motivator results.

The aim of this paper is prove a theorem on the Riesz mean of expansions with respect to Riesz bases, which extends the previous results of Loi and Tahir on the Schrodinger operator to the operator of 4-th order.