Journal of the Ramanujan Mathematical Society

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Uniform Close-to-Convexity Radius of Sections of Functions in the Close-to-Convex Family


Affiliations
1 Indian Statistical Institute (ISI), Chennai Centre, SETS (Society for Electronic Transactions and security), MGR Knowledge City, CIT Campus, Taramani, Chennai 600 113, India
     

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The authors consider the class F of normalized functions f analytic in the unit disk D and satisfying the condition

                    Re (1+zf '' (z)/f '(z))>-1/2, ∈ D.

Recently, Ponnusamy et al. [12] have shown that 1/6 is the uniform sharp bound for the radius of convexity of every section of each function in the class F. They conjectured that 1/3 is the uniform univalence radius of every section of f ∈ F. In this paper, we solve this conjecture affirmatively.


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  • Uniform Close-to-Convexity Radius of Sections of Functions in the Close-to-Convex Family

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Authors

S. Vaidhyanathan Bharanedhar
Indian Statistical Institute (ISI), Chennai Centre, SETS (Society for Electronic Transactions and security), MGR Knowledge City, CIT Campus, Taramani, Chennai 600 113, India
Saminathan Ponnusamy
Indian Statistical Institute (ISI), Chennai Centre, SETS (Society for Electronic Transactions and security), MGR Knowledge City, CIT Campus, Taramani, Chennai 600 113, India

Abstract


The authors consider the class F of normalized functions f analytic in the unit disk D and satisfying the condition

                    Re (1+zf '' (z)/f '(z))>-1/2, ∈ D.

Recently, Ponnusamy et al. [12] have shown that 1/6 is the uniform sharp bound for the radius of convexity of every section of each function in the class F. They conjectured that 1/3 is the uniform univalence radius of every section of f ∈ F. In this paper, we solve this conjecture affirmatively.