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Starlike functions associated with a cardioid

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Abstract

Let \(\mathcal {S}^*_C\) be the class of normalized analytic functions f in the unit disc with \(zf'(z)/f(z)\) lying in the region bounded by the cardioid given by the equation \((9 x^2+9 y^2-18x+5)^2- 16 (9 x^2+9 y^2-6x+1)=0\). We determine the structural formula, coefficient estimates, growth results and various radii constants such as the radius of starlikeness, radius of lemniscate of Bernoulli starlikeness, radius of M-starlikeness and radius of \(\mathcal {M}(\beta )\)-starlikeness for functions in the class \(\mathcal {S}^*_C\). In addition, the \(\mathcal {S}^*_C\)-radii for functions belonging to several interesting classes are determined. All the results obtained are sharp.

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Authors and Affiliations

  1. Department of Mathematics, University of Delhi, Delhi, 110007, India

    Kanika Sharma, Naveen Kumar Jain & V. Ravichandran

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  1. Kanika Sharma
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  2. Naveen Kumar Jain
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  3. V. Ravichandran
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Correspondence to V. Ravichandran.

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The work presented here was supported by a grant from University of Delhi.

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Sharma, K., Jain, N.K. & Ravichandran, V. Starlike functions associated with a cardioid. Afr. Mat. 27, 923–939 (2016). https://doi.org/10.1007/s13370-015-0387-7

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  • DOI: https://doi.org/10.1007/s13370-015-0387-7

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