مختصر البحث:
The main concern of the present thesis is to study the properties of some classes of Bazilević and Non-Bazilević functions and extending the previous results for these classes that contain univalent and multivalent functions of complex order in the…
The main concern of the present thesis is to study the properties of some classes of Bazilević and Non-Bazilević functions and extending the previous results for these classes that contain univalent and multivalent functions of complex order in the open unit disk U={z:z∈C;|z|<1} and meromorphic functions in the punctured open unit disk〖 U〗^*={z:z∈C;0<|z|<1},
In the geometric function theory relying on differential inequalities, the concept of subordination is an important concept that is newly introduced in this subject which is dealing with function as the complex form instead of the real deal and part of the imagination alone.
Moreover, we have relied on the term of subordination in the definition of a new types of Non-Bazilević functions H_(α,β)^(q,λ) (γ,μ,σ,A,B,g(z)), M_(b,t)^(μ,m) (δ,φ) involving operators which depend on both concepts q-Mittag-Leffler, k- symmetric points respectively, also defined new classes of Bazilević functions K_(λ_1,λ_2,l)^(m,ξ) (δ,β,ψ) and S_(δ,w,l)^(m,σ) (λ,ξ,g) associative with new linear operators.
Finding several new sufficient conditions by using the method of differential subordination for analytic functions in the open unit disk.
As well as, coefficient bounds and Feket-Szego ̈ inequalities for functions which belongs to the class K_(λ_1,λ_2,l)^(m,ξ) (δ,β,ψ) are obtained and finding the distortion theorem for functions belongs to the class S_(δ,w,l)^(n,σ) ( λ,ξ).
The class F_(λ,β)^m (η,σ,γ) of harmonic Bazilević univalent functions involving with linear operator L_(λ,β)^m f 〖(z)〗^α introduced and obtained a sufficient condition for the harmonic functions to be in the class F_(λ,β)^m (η,σ,γ) with a necessary and sufficient condition for functions in the class NF_(λ,β)^m (η,σ,γ), the distortion theorem, combination, convolution properties and extreme points are obtained.
Further, two subclasses R_(λ,α)^(n,β) (ξ,μ,γ) of meromorphic Bazilević functions and 〖 Q〗_(q,ξ)^(m,λ) (α,δ) of p- valent meromorphic Bazilević functions are defined and investigated with some sufficient conditions for the functions which belong to these classes are given.
Finally, through the concept of Chebyshev polynomials expansions a family consisting of Non-Bazilević analytic functions N(α,λ,t) is described and obtained various geometric properties, such as coefficient bounds and a Fekete-Szego ̈ inequality for functions belong to this class.
Also by using Faber polynomials, a new class of Bi-Bazilević univalent functions involving differential operator is defined, as well as the determination of upper bounds for functions belong to this class.
