Mediterranean Journal of Mathematics

, Volume 13, Issue 2, pp 607–623

Maximal Area Integral Problem for Certain Class of Univalent Analytic Functions

  • Saminathan Ponnusamy
  • Swadesh Kumar Sahoo
  • Navneet Lal Sharma
Article

Abstract

One of the classical problems concerns the class of analytic functions f on the open unit disk |z| < 1 which have finite Dirichlet integral Δ(1, f), where
$$\Delta(r ,f) = \iint_{|z| < r} |f' (z)| ^ 2 \, {\rm d} x {\rm d}y \quad (0 < r \leq 1)$$
The class \({\mathcal{S} ^*(A,B)}\) of normalized functions f analytic in |z| < 1 and satisfies the subordination condition \({zf'(z)/f(z)\prec (1+Az)/(1+Bz)}\) in |z| < 1 and for some \({-1\leq B\leq 0}\) , \({A \in \mathbb{C}}\) with \({A\neq B}\) , has been studied extensively. In this paper, we solve the extremal problem of determining the value of
$$\max_{f\in \mathcal{S}^*(A,B)}\Delta(r,z/f)$$
as a function of r. This settles the question raised by Ponnusamy and Wirths (Ann Acad Sci Fenn Ser AI Math 39:721–731, 2014). One of the particular cases includes solution to a conjecture of Yamashita which was settled recently by Obradović et al. (Comput Methods Funct Theory 13:479–492, 2013).

Keywords

Analytic univalent convex star-like functions and spiral-like functions Dirichlet finite area integral Gaussian hypergeometric functions 

Mathematics Subject Classification

Primary 30C45 30C55 Secondary 33C05 

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Copyright information

© Springer Basel 2015

Authors and Affiliations

  • Saminathan Ponnusamy
    • 1
  • Swadesh Kumar Sahoo
    • 2
  • Navneet Lal Sharma
    • 2
  1. 1.Indian Statistical Institute (ISI), Chennai CentreSETS (Society for Electronic Transactions and Security), MGR Knowledge CityChennaiIndia
  2. 2.Discipline of MathematicsIndian Institute of Technology IndoreIndoreIndia

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