The main object of this paper is to introduce and study systematically the univalence criteria of a new family of integral operators by using a substantially general form of the widely-investigated Srivastava-Attiya operator. In particular, we derive several new sufficient conditions of univalence for this generalized Srivastava-Attiya operator. Relevant connections with other related earlier works are also pointed out.

In this paper, the generalized differential operator is applied to derive some subclasses of function class σ of bi-univalent functions defined in unit disk U. We estimate the bounds of the coefficients a ₂ and a ₃ for all functions which belong to the derived subclasses of σ.

In this paper, through the instrument of the well-known Chebyshev polynomials and subordination, we defined a family of functions, consisting of Bazilevie functions of type alpha, involving the Ruscheweyh derivative operator. Also, we investigate coefficient bounds and Fekete-Szego inequalities for this class.

The aim object of this paper is to study some classes of the differential subordination concept for the subclasses of univalent functions by using the convolution operator. Some interesting geometric properties like coefficient bound, growth and distortion theorems, closure theorem, radii of convexity, and starlikeness of functions are obtained. Extreme points and integral operators are also investigated.

In this paper, we introduce a new certain subclass N(α,λ,t) of Non-Bazilević analytic functions of type (1-α) by using the Chebyshev polynomials expansions. We investigated some basic useful characteristics for this class, also we obtain coefficient bounds and Fekete-Szegö inequalities for functions belong to this class. This class is considered a general case for some of the previously studied classes. Further we discuss its consequences.