Mohammed Alarifi, Najla
(2017)
The U Radius And Hankel Determinant For Analytic Functions, And Product Of Logharmonic Mappings.
PhD thesis, Universiti Sains Malaysia.
Abstract
This thesis studies geometric and analytic properties of complexvalued analytic functions and logharmonic mappings in the open unit disk D. It investigates four
research problems. As a precursor to the first, let U be the class consisting of normalized analytic functions f satisfying (z= f (z))2 f ′(z)−1 < 1: All functions f ∈ U are univalent. In the first problem, the U radius is determined for several classes of analytic functions. These include the classes of functions f satisfying the inequality Re f (z)=g(z) > 0; or  f (z)=g(z)−1 < 1 in D; for g belonging to a certain class of
analytic functions. In most instances, the exact U radius are found. A recent conjecture by Obradovi´c and Ponnusamy concerning the radius of univalence for a product involving univalent functions is also shown to hold true. The second problem deals with the Hankel determinant of analytic functions. For a normalized analytic function f ; let z f ′(z)= f (z) or 1+z f ′′(z)= f ′(z) be subordinate to a given analytic function
φ in D. Further let F be its kthroot transform, that is, F(z) = z[f(zk)=zk]1k
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