Geometric Function Theory .................................

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1. Geometric Function Theory Submitted to:- Dr Bushra Kanwal Submitted by:- Naina

2. (1.1) C lass of Normalized functions Let denote by the class of all analytic and normalized functions of the form where D := {z ∈ C : |z| < 1} is the open unit disc, and let be the subclass of consisting in the univalent functions in D. (1.2) Class of Starlike Function The subclass of defined by is called the class of starlike (univalent) functions in D.  

3. (1.3) Class of Convex Functions The subclass of defined by is called the class of convex (univalent) functions in D.  

4. Let denote the class   (1.4) the well-known Carathéodory class The family of holomorphic functions ρ in that satisfies the condition and of the form Lemma (1.5) If has the form then Lemma (1.6) And for any complex number Ϛ we have  

5. Theorem Statement: If has the form f(z)= then Proof, Since then their exist a Schwartz function that is is analytic in D and satisfy the condition and for all z D, Such that  

6. Since has the form it’s follows that From the fact that and for all z D if we define the function p by We obtain that p and  

7. Lemma, If p has the form then, So using this lemma,  

8. Theorem Statement: If f has the form f(z)= then Proof, Since then their exist a Schwartz function that is is analytic in D and satisfy the condition and for all z D, Such that Since has the form it’s follows that  

9. From the fact that and for all z D if we define the function p by We obtain that p and According to above relation we get Now equating the corresponding coefficient of we get  

10. Lemma, If p has the form then, So using this lemma,