In this paper , we studied and introduced a new subclass of analytic univalent functions defined by differ – INTEGRAL OPERATOR . We obtain distortion bounds, extreme points , and some theorem of this subclass .

In this paper, new generalized variants of Ostrowski’s type identities involving the Atangana-Baleanu-Katugampola FRACTIONAL INTEGRAL OPERATOR for differentiable convex and twice differentiable convex functions are presented. Using these equalities or lemmas along with several known identities, new inequalities for convex functions and the Atangana-Baleanu-Katugampola, FRACTIONAL INTEGRAL OPERATORs are proved. By making appropriate choices of parameters, some connections between our results and various other findings are also recognized in the paper. Finally, some applications to unique means for positive real numbers are offered.

In this paper, several INTEGRAL inequalities for the functions whose derivatives are s-convex functions in the fourth sense are obtained by means of Caputo FRACTIONAL derivative and Caputo-Fabrizio INTEGRAL OPERATOR. Also the Hermite-Hadamard type inequalities for s-convex functions and their products are stated via Caputo-Fabrizio INTEGRAL OPERATOR.

In this paper, we establish sharp maximal function inequalities for the Toeplitz type OPERATOR related to some general FRACTIONAL INTEGRAL OPERATORs. As an application, we obtain the boundedness of the OPERATOR on Lebesgue, Morrey and Triebel-Lizorkin spaces. The OPERATORs include Littlewood-Paley OPERATOR, Marcinkiewicz OPERATOR and Bochner-Riesz OPERATOR.

In this investigation, using Opoola differential OPERATOR ($D^{m}(\mu,\beta,t)f(z)$), a new INTEGRAL OPERATOR: $I_{t,\beta,\mu}^{m,\sigma}(f_{1},...,f_{n})(z): A^{n}\rightarrow A$  is defined in the unit disk, $U=\left\lbrace z\in C:\left|z\right|<1\right\rbrace$; and we investigated the Univalence conditions of this generalized OPERATOR. Finally, a number of corollaries and remarks which show the extension of our results are presented.

In this paper, we consider a general INTEGRAL OPERATOR Gn(z). The main object of the present paper is to study some properties of this INTEGRAL OPERATOR on the classes S* (a), K (a), M (b), N (b) and KD (m, b).

We give a brief survey of recent work on INTEGRAL forms in vertex OPERATOR algebras (VOAs).

Let $\mathfrak{A}$ be an algebra. A \textit{derivation} on $\mathfrak{A}$ is a linear mapping $\delta:\mathfrak{A}\to\mathfrak{A}$ such that $\delta(ab)=\delta(a)b+a\delta(b)$ for every $a,b\in\mathfrak{A}$. As a dual to this notion, we consider a linear mapping $\Delta:\mathfrak{A}\to\mathfrak{A}$ with the property $\Delta(a)\Delta(b)=\Delta(\Delta(a)b+a\Delta(b))$ for every $a,b\in\mathfrak{A}$ and we call it an \textit{integration}. In this paper, we give some examples, counterexamples and facts concerning integrations on algebras. Furthermore, we state and prove a characterization for integrations on finite dimensional matrix algebras.

The proposed study is focused to introduce a novel INTEGRAL transform op-erator, called Generalized Bivariate (GB) transform. The proposed trans-form includes the features of the recently introduced Shehu transform, ARA transform, and Formable transform. It expands the repertoire of existing Laplace-type bivariate transforms. The primary focus of the present work is to elaborate fashionable properties and convolution theorems for the proposed transform OPERATOR. The existence, inversion, and duality of the proposed transform have been established with other existing transforms. Implementation of the proposed transform has been demonstrated by ap-plying it to different types of differential and INTEGRAL equations. It validates the potential and trustworthiness of the GB transform as a mathematical tool. Furthermore, weighted norm inequalities for INTEGRAL convolutions have been constructed for the proposed transform OPERATOR.