The main goal of this paper is to introduce the higher derivatives multivalent harmonic function class, which is defined by the general linear operator. As a result, geometric properties such as coefficient estimation, convex combination, extreme point, distortion theorem and convolution property are obtained. Finally, we show that this class is invariant under the Bernandi-Libera-Livingston integral for harmonic functions.

            In this paper, the Normality set  will be investigated. Then, the study highlights some concepts properties and important results. In addition, it will prove that every operator with normality set has non trivial invariant subspace of  .

Recently, numerous the generalizations of Hurwitz-Lerch zeta functions are investigated and introduced. In this paper, by using the extended generalized Hurwitz-Lerch zeta function, a new Salagean’s differential operator is studied. Based on this new operator, a new geometric class and yielded coefficient bounds, growth and distortion result, radii of convexity, star-likeness, close-to-convexity, as well as extreme points are discussed.

 الأثر V بالنسبة إلى   sinshT و خواصه قد تم دراسته في هذا البحث حيث تم دراسة علاقة الأثر المخلص والاثر المنتهى التولد والاثر المنفصل وربطها بالمؤثرات المتباينة حيث تم بهنة العلاقات التالية ان الاثر اذا وفقط اذا مقاس في حالة كون المؤثر هو عديم القوة وكذلك في حالة كون المؤثر شامل فان الاثر هو منتهي التولد اي ان الغضاء هو منتهي التولد وايضا تم برهن ان الاثر مخلص لكل مؤثر مقيد وك\لك قد تم التحقق من انه لاي مؤثر مقي

This study delves into the properties of the associated act V over the monoid S of sinshT. It examines the relationship between faithful, finitely generated, and separated acts, as well as their connections to one-to-one and onto operators. Additionally, the correlation between acts over a monoid and modules over a ring is explored. Specifically, it is established that  functions as an act over S if and only if  functions as module, where T represents a nilpotent operator. Furthermore, it is proved that when T is onto operator and  is finitely generated, is guaranteed to be finite-dimensional. Prove that for any bounded operator the following,  is acting over S if and only if  is a module where T is a nilpotent operator, is a

We study in this paper the composition operator that is induced by ?(z) = sz + t. We give a characterization of the adjoint of composiotion operators generated by self-maps of the unit ball of form ?(z) = sz + t for which |s|?1, |t|<1 and |s|+|t|?1. In fact we prove that the adjoint is a product of toeplitz operators and composition operator. Also, we have studied the compactness of C? and give some other partial results.