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Univalence and starlikeness of nonlinear integral transform of certain class of analytic functions

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Abstract

Let U(λ, µ) denote the class of all normalized analytic functions f in the unit disk |z| < 1 satisfying the condition

$$ \frac{{f(z)}} {z} \ne 0and\left| {f'(z)\left( {\frac{z} {{f(z)}}} \right)^{\mu + 1} - 1} \right| < \lambda ,\left| z \right| < 1. $$

For f ∈ U(λ, µ) with µ ≤ 1 and 0 ≠ µ1 ≤ 1, and for a positive real-valued integrable function φ defined on [0, 1] satisfying the normalized condition ∫ 10 φ(t)dt = 1, we consider the transform G φ f (z) defined by

$$ G_\phi f(z) = z\left[ {\int_0^1 {\phi (t)\left( {\frac{{zt}} {{f(tz)}}} \right)^\mu dt} } \right]^{ - 1/\mu _1 } ,z \in \Delta . $$

In this paper, we find conditions on the range of parameters λ and µ so that the transform G φ 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 U(λ, µ).

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Civil Engineering, Bulevar Kralja Aleksandra 73, 11000, Belgrade, Serbia

    M. Obradović

  2. Department of Mathematics, Indian Institute of Technology Madras, Chennai, 600 036, India

    S. Ponnusamy & P. Vasundhra

Authors
  1. M. Obradović
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  2. S. Ponnusamy
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  3. P. Vasundhra
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Correspondence to S. Ponnusamy.

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Obradović, M., Ponnusamy, S. & Vasundhra, P. Univalence and starlikeness of nonlinear integral transform of certain class of analytic functions. Proc Math Sci 119, 593–610 (2009). https://doi.org/10.1007/s12044-009-0057-5

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  • DOI: https://doi.org/10.1007/s12044-009-0057-5

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