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.

The purpose of the present paper is to introduce a class $\boldsymbol{D}% _{\lambda, \delta }^{k, \alpha }C_{0}(\beta )$ of bi-concave functions defined by a differential operator. We find estimates on the Taylor-Maclaurin coefficients $\left\vert a_{2}\right\vert $ and $\left\vert a_{3}\right\vert $ for functions in this class. Several consequences of these results are also pointed out in the form of corollaries.

‎In the present paper, we investigate  the eigenvalues of an elliptic differential operator on compact Riemannian manifolds with boundary and derive a general inequality for these eigenvalues. Applying this inequality, we give universal estimates  for eigenvalues on compact domains of  complete submanifolds in an Euclidean space, and of complete manifolds admitting special functions. Finally, we find universal bounds on  the $(k+1)$-th eigenvalue on such objects in terms of the first $k$ eigenvalues independent of  the domains.

In this paper, we intend to introduce the Sturm-Liouville fractional problem and solve it using the collocation method based on Chebyshev cardinal polynomials. To this end, we first provide an introduction to the Sturm-Liouville fractional equation. Then the Chebyshev cardinal functions are introduced along with some of their properties and the operational matrices of the derivative, fractional integral, and Caputo fractional derivative are obtained for it. Here, for the first time, we solve the equation using the operational matrix of the fractional derivative without converting it to the corresponding integral equation. In addition to efficiency and accuracy, the proposed method is simple and applicable. The convergence of the method is investigated, and an example is presented to show its accuracy and efficiency.

We study the action and properties of a differential operator in the polydisk, extending some classical results from the unit disk. Using so called dyadic decomposition of the polydisk we find precise connections between quazinorms of holomorphic function in the polydisk with quazinorms on the subframe and the unit disk. All our results were previously well-known in the unit disk.

By using a certain operator Sn, we introduce a class of holomorphic functions Sn(b), and obtain some subordination results. We also show that the set Sn(b) is convex and obtain some new differential subordinations related to certain integral operators.

This paper deals with the numerical treatment of singularly perturbed delay differential equations having a delay on the first derivative term. The solution of the considered problem exhibits boundary layer behavior on the left or right side of the domain depending on the sign of the convective term. The term with the delay is approximated using Taylor series approximation, resulting in an asymptotically equivalent singularly perturbed boundary value problem. The uniformly convergent numerical scheme is developed using exponentially fitted finite difference method. The stability of the scheme is investigated using solution bound. The uniform convergence of the scheme is discussed and proved. Numerical examples are considered to validate the theoretical analysis.