• Univalence and Starlikeness of Nonlinear Integral Transform of Certain Class of Analytic Functions

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      https://www.ias.ac.in/article/fulltext/pmsc/119/05/0593-0610

    • Keywords

       

      Univalent; Bazilevič; star-like and spiral-like functions; integral transforms.

    • Abstract

       

      Let $\mathcal{U}(\lambda, \mu)$ denote the class of all normalized analytic functions 𝑓 in the unit disk $|z| < 1$ satisfying the condition

      \begin{equation*}\frac{f(z)}{z}\neq 0\quad\text{and}\quad\left|f'(z)\left(\frac{z}{f(z)}\right)^{\mu +1}-1\right| < \lambda,\quad |z| < 1.\end{equation*}

      For $f\in\mathcal{U}(\lambda, \mu)$ with $\mu\leq 1$ and $0\neq\mu_1\leq 1$, and for a positive real-valued integrable function 𝜑 defined on [0,1] satisfying the normalized condition $\int^1_0\varphi(t)dt=1$, we consider the transform $G_\varphi f(z)$ defined by

      \begin{equation*}G_\varphi f(z)=z\left[\int^1_0\varphi(t)\left(\frac{zt}{f(tz)}\right)^\mu dt\right]^{-1/\mu 1},\quad z\in\Delta.\end{equation*}

      In this paper, we find conditions on the range of parameters 𝜆 and 𝜇 so that the transform $G_\varphi f$ is univalent or star-like. In addition, for a given univalent function of certain form, we provide a method of obtaining functions in the class $\mathcal{U}(\lambda, \mu)$.

    • Author Affiliations

       

      M Obradović1 S Ponnusamy2 P Vasundhra2

      1. Department of Mathematics, Faculty of Civil Engineering, Bulevar Kralja Aleksandra 73, 11000 Belgrade, Serbia
      2. Department of Mathematics, Indian Institute of Technology Madras, Chennai 600 036, India
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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