Sufficient conditions of starlikeness and convexity for functions of complex order

Let A be the class of normalised and analytic functions defined in the unit disc {:||1}.Uzz In this paper we study the expression 2()1,,()()zfzzzUbfzbfz for some (0) and (\{0}bb as a criteria for starlikeness and convexity at analytic functions of complex order

Differential Subordination And Coefficients Problems Of Certain Analytic Functions

Let A be the class of normalized analytic functions f on the unit disk D, in the form f(z) = z+ P1n =2 anzn: A function f in A is univalent if it is a one-to-one mapping. This thesis discussed ¯ve research problems. Lambangkan A sebagai kelas fungsi analisis ternormal pada cakera unit D berbentuk f(z) = z + P1n =2 anzn: Fungsi f dalam A adalah univalen jika fungsi tersebut ialah pemetaan satu ke satu. Tesis ini mengkaji lima masalah penye- lidikan

Subordination And Convolution Of Multivalent Functions And Starlikeness Of Integral Transforms

This thesis deals with analytic functions as well as multivalent functions de- �ned on the unit disk U. In most cases, these functions are assumed to be normalized, either of the form f(z) = z + 1X k=2 akzk; or f(z) = zp + 1X k=1 ak+pzk+p; p a �xed positive integer. Let A be the class of functions f with the �rst normalization, while Ap consists of functions f with the latter normalization. Five research problems are discussed in this work. First, let f(q) denote the q-th derivative of a function f 2 Ap. Using the theory of di�erential subordination, su�cient conditions are obtained for the following di�erential chain to hold: f(q)(z) �(p; q)z

Analytic and Harmonic Univalent Functions

Studies on analytic univalent functions became the focus of intense researchwith theBieberbachconjectureposed in 1916 concerning the size of the moduli of the Taylor coefficients of these functions. In efforts towards its resolution, the conjecture inspired the development of several ingeniously different mathematical techniques with powerful influence. These techniques include Lowner’s parametric representation method, the area method,Grunsky inequalities, and methods of variations.Despite the fact that the conjecture was affirmatively settled by de Branges in 1985, complex function theory continued to remain a highly active relevant area of research

Differential Subordination And Superordination For Analytic And Meromorphic Functions Defined By Linear Operators [QA331. N219 2007 f rb].

Suatu fungsi f yang tertakrif pada cakera unit terbuka U dalam satah kompleks C disebut univalen jika fungsi tersebut memetakan titik berlainan dalam U ke titik berlainan dalam C. A function f defined on the open unit disk U of the complex plane C is univalent if it maps different points of U to different points of C

Inclusion Properties Of Linear Operators And Analytic Functions

This thesis studies the class A of normalized analytic functions in the open unit disk U of the complex plane. The class of meromorphic functions in the punctured unit disk which does not include the origin is also studied. This thesis investigates six research problems. First, the classical subclasses of starlike, convex, close-toconvex and quasi-convex functions are extended by introducing new subclasses of analytic and meromorphic functions. The closure properties of these newly de ned classes are investigated and it is shown that these classes are closed under convolution with prestarlike functions and the Bernardi-Libera-Livingston integral operator. The univalence of functions f(z) = z + P1n=2 anzn 2 A is investigated by requiring the Schwarzian derivative S(f; z) and the second coe cient a2 of f to satisfy certain inequalities. New criterion for analytic functions to be strongly - Bazilevi c of nonnegative order is established in terms of the Schwarzian derivatives and the second coe cients. Also, similar conditions on the second coe cient of f and its Schwarzian derivative S(f; z) are obtained that would ensure the function f belongs to particular subclasses of S. For an analytic function f(z) = z+ P1n =2 anzn 2 A satisfying the inequality P1n =2 n(