Differential Subordination And Some Selected Topics In Geometric Function Theory

Geometric Function Theory is the branch of complex analysis which deals and studies the geometric properties of analytic functions. It means that Geometric Function Theory is an area of Mathematics characterized by an intriguing mar-riage between geometry and analysis. Its origins date from the 19th century but new applications arise continually. Interest in Geometric Function Theory has experienced resurgence in recent decades as the methods of algebraic geometry and function theory on compact Riemann surfaces have been found relevance in constructing finite-gap solutions to non-linear integrable system [11]. Early string theory models rely on elements of Geometric Function Theory for the com-putation of so called Veneziano amplitudes [36]. Even new developments in the constructive approach to linear and non-linear boundary value and initial value problems using spectral analysis [23] are likely to lead to a role for Geometric Function Theory in the solution of a wide range of partial differential equations. Geometric Function Theory is a classical subject. Yet it continue to find new applications in an ever-growing variety of areas such as modern mathematical physics, more traditional fields of physics such as fluid dynamics, non-linear in-tegrable systems theory of partial differential equations.It is well known [31] that a number of important classes of univalent or mul-tivalent functions (e.g., convex function, starlike function) are related through their derivatives. These functions play an important part in problems from signal theory, in moment problem and in constructing quadrature formulas, see Ronning [91] and the references cited therein for some recent applications. In this thesis we shall introduce and investigate some new properties of certain classes of uni-valent or multivalent functions by using the method of differential subordination. It consists of five chapters.In Chapter 1, we briefly introduce the background and some preliminary concepts about the related work.In Chapter 2, we consider some new classes of analytic functions in U. It contains five sections. Firstly, in Section 2.1, we define a class Tn(A,B,γ,α) as following: f'(z)+αzf"(z)(?)h(z) (z∈U), where We obtain sharp bounds on Ref'(z),Re(?),|f(z)|, and coefficient estimates for functions f(z) in Tn(A, B,γ,α). Conditions for univalency and starlikeness, con-volution properties and the radius of convexity are also presented.Secondly, in Section 2.2, we discuss further properties of the class Tn(A, B,1,α). The main object of this section is to investigate a new problem, that is to find (?) Re{f'(z)+αzf"{z)}, where f(z) varies in the class (?) From this result some interesting consequences are obtained.In Section 2.3, by using the well-known Dziok-Srivastava operator Hp,q,s(α1) and differential subordination. we introduce a classΩp,q,s(α1,λ; h) of multivalent analytic functions as following: (?) whereλis a complex number and h(z) is a convex univalent function in U with h(0)=1. A number of inclusion relationships for the classΩp,q,s(α1,λ;h) are given. We also focus on some other interesting properties such as convolution properties and integral operator of the classΩp,q,s(α1,λ;h).Section 2.4 is mainly concerned with multivalent analytic functions associ-ated with an extended fractional differintegral operatorΩz(λ.p). In 1978, Owa [75] introduced the definitions of fractional calculus (that is, fractional inte-grals Dz-λand fractional derivatives Dzλof an arbitrary order). Very recently, Patel and Mishra [82] defined the extended fractional differintegral operatorΩz{λ,p):A(p)→A(p)(-∞<λ-p and 0≤α< 1) of meromorphic starlike functions involving a linear operator Dn+P-1. It should be remarked in passing that the operator Dn+p-1 is analogous to that of Ruscheweyh derivative [93]. In particular, we have Lp(n+p, 1)f(z)=Dn+p-1f(z) for f(z)∈ ∑p. We obtain some inclusion relationships for the class Tn+p-1(α). Further, integral transforms of functions in Tn+p-1(α) and convolution conditions are also considered.Section 3.3 is mainly concerned with the classesΩp,q,s(α1; A, B) andΩp,q,s+(α1; A, B) of meromorphically multivalent functions associated with the gen-eralized hypergeometric function. The object of this section is to present several inclusion and other properties of functions in the classesΩp,q,s(α1; A, B) andΩp,q,s+(α1; A, B). The most interesting one is that we apply the familiar concept of neighborhoods of analytic functions (see [30,94]) to meromorphically p-valent functions in the class∑p.In Chapter 4, we consider several argument inequalities for certain analytic functions. The basic tool of our research is the technique of differential subordi-nation. All the results obtained in this chapter are new.In Section 4.1, we derive some sufficient conditions for p-valent strongly starlike functions of order a. All results obtained in this section are sharp. In particular, we have a result which was given earlier by Nunokawa [67] (see also Nunokawa and Thomas [71]) by using another method.LetΩdenote the class of functions k(z) which are analytic in U with k(0)=1 and k(z)≠0 (z∈U). In Section4.2, by making use of the method of differential subordination, we derive certain sharp conditions connecting k(z) and zk'(z) for k(z)∈Ωto satisfy (?) Especially, we observe that the result in Theorem 4.2.6 withα1≠α2,λ=p=1 and b=-1 is better than the unique result obtained by Takahashi and Nunokawa [117] by using a different technique of proof.In Section 4.3 we improve the main result obtained by Nunokawa et al. in a recent paper [69].The purpose of Chapter 5 is to investigate some properties of certain uni-valent functions. In Section 5.1, we introduce and study a new class S(a) (0<α≤2) of univalent functions. It should be remarked that the introduction of the class S(α) (0<α≤2) is motivated essentially by the work of Nunokawa et al. [68]. In particular we obtain a criterion for univalence which generalizes the only result of Nunokawa et al. [68]. In Section 5.2, we consider some new subclasses of strongly starlike functions and strongly convex functions defined by the Noor integral and study their properties.