Univalence, strong starlikeness and integral transforms

Tom 86 / 2005

M. Obradović, S. Ponnusamy, P. Vasundhra Annales Polonici Mathematici 86 (2005), 1-13 MSC: 30C45, 30C55. DOI: 10.4064/ap86-1-1

Streszczenie

Let $\mathcal A$ represent the class of all normalized analytic functions $f$ in the unit disc ${\mit\Delta}$. In the present work, we first obtain a necessary condition for convex functions in ${\mit\Delta}$. Conditions are established for a certain combination of functions to be starlike or convex in ${\mit\Delta}$. Also, using the Hadamard product as a tool, we obtain sufficient conditions for functions to be in the class of functions whose real part is positive. Moreover, we derive conditions on $f$ and $\mu$ so that the non-linear integral transform $\int_0^z ({\zeta}/{f(\zeta)})^{\mu}\,d\zeta$ is univalent in ${\mit\Delta}$. Finally, we give sufficient conditions for functions to be strongly starlike of order $\alpha$.

Autorzy

  • M. ObradovićDepartment of Mathematics
    Faculty of Technology and Metallurgy
    4 Karnegijeva St.
    11000 Belgrad
    Serbia and Montenegro
    e-mail
  • S. PonnusamyDepartment of Mathematics
    Indian Institute of Technology
    Madras
    Chennai-600 036, India
    e-mail
  • P. VasundhraDepartment of Mathematics
    Indian Institute of Technology
    Madras
    Chennai-600 036, India
    e-mail

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