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    Fractional calculus operator and its applications to certain classes of analytic functions. A study on fractional derivative operator in analytic and multivalent functions.

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    somia PhD Thesis (viva).pdf (2.125Mb)
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    Publication date
    2014-05-07
    Author
    Amsheri, Somia M.A.
    Supervisor
    Zharkova, Valentina V.
    Keyword
    Univalent
    Multivalent
    Convex
    Starlike
    Coefficient bounds
    Differential subordination
    Differential superordination
    Strong differential subordination
    Strong differential superordination
    Fractional calculus operator
    Analytic functions
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    Rights
    Creative Commons License
    The University of Bradford theses are licenced under a Creative Commons Licence.
    Institution
    University of Bradford
    Department
    Department Of Mathematics, School Of Computing, Informatics And Media
    Awarded
    2013
    
    Metadata
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    Abstract
    The main object of this thesis is to obtain numerous applications of fractional derivative operator concerning analytic and -valent (or multivalent) functions in the open unit disk by introducing new classes and deriving new properties. Our finding will provide interesting new results and indicate extensions of a number of known results. In this thesis we investigate a wide class of problems. First, by making use of certain fractional derivative operator, we define various new classes of -valent functions with negative coefficients in the open unit disk such as classes of -valent starlike functions involving results of (Owa, 1985a), classes of -valent starlike and convex functions involving the Hadamard product (or convolution) and classes of -uniformly -valent starlike and convex functions, in obtaining, coefficient estimates, distortion properties, extreme points, closure theorems, modified Hadmard products and inclusion properties. Also, we obtain radii of convexity, starlikeness and close-to-convexity for functions belonging to those classes. Moreover, we derive several new sufficient conditions for starlikeness and convexity of the fractional derivative operator by using certain results of (Owa, 1985a), convolution, Jack¿s lemma and Nunokakawa¿ Lemma. In addition, we obtain coefficient bounds for the functional of functions belonging to certain classes of -valent functions of complex order which generalized the concepts of starlike, Bazilevi¿ and non-Bazilevi¿ functions. We use the method of differential subordination and superordination for analytic functions in the open unit disk in order to derive various new subordination, superordination and sandwich results involving the fractional derivative operator. Finally, we obtain some new strong differential subordination, superordination, sandwich results for -valent functions associated with the fractional derivative operator by investigating appropriate classes of admissible functions. First order linear strong differential subordination properties are studied. Further results including strong differential subordination and superordination based on the fact that the coefficients of the functions associated with the fractional derivative operator are not constants but complex-valued functions are also studied.
    URI
    http://hdl.handle.net/10454/6320
    Type
    Thesis
    Qualification name
    PhD
    Collections
    Theses

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