In the present study, we introduced general a subclass of bi-univalent functions by using the Bell numbers and $q-$Srivastava Attiya operator. Also, we investigate coefficient estimates and famous Fekete-Szegö inequality for functions belonging to this interesting class.

This article addresses the non-linear optimization problem of oscillatory processes governed by partial integro-differential equations involving a Fredholm integral operator. A distinctive feature of the problem is that both the objective functional and the functions describing external and boundary influences are non-linear with respect to the vector controls. The integro-differential equation describing the state of the oscillatory process includes Fredholm integral operator, which has a significant impact on the structure and properties of the solutions. The algorithm for constructing the complete solution to this problem, as well as the effect of the Fredholm integral operator on the solution of the corresponding boundary value problem, has been published in previous studies. This article is dedicated to the investigation of the convergence of approximate solutions to the exact solution of the considered non-linear optimization problem. The influence of the Fredholm integral operator on the convergence behavior of the approximations is examined. It is demonstrated that the presence of the integral operator necessitates the construction of three distinct types of approximations of the optimal process: “Resolvent” approximations, based on the resolvent of the kernel of the integral operator; Approximations by optimal controls, constructed through the approximation of control functions; Finite-dimensional approximations.

In this paper, we introduce the subclass Sβ (α, λ) of analytic functions and obtain coefficient inequality for functions belong to this class. Furthermore, we give sufficient conditions for starlikeness of reciprocal order of analytic functions. In the last part, we obtain the subordination results of a new subclass of analytic functions of reciprocal order, which are defined here by means of a Hadamard product of analytic functions. The results presented in this work improve or generalize the recent works of other authors and also give rise to several new results.

Let a normalized analytic function be given on the open unit disk. In this paper, we define and consider some familiar subsets of analytic functions associated with sine functions in the region of unit disk on the complex plane. For these classes, we aim to find the upper bounds of the modules of the Hankel determinants obtained from the coefficients of the functions belonging to some classes defined by subordination.