Some results for the family of univalent functions related with Limaçon domain

Afis Saliu; Khalida Inayat Noor; Saqib Hussain; Maslina Darus; Afis Saliu; Khalida Inayat Noor; Saqib Hussain; Maslina Darus

doi:10.3934/math.2021204

AIMS Mathematics

2021, Volume 6, Issue 4: 3410-3431. doi: 10.3934/math.2021204

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Research article

Some results for the family of univalent functions related with Limaçon domain

1.
Department of Mathematics, COMSATS University Islamabad, Park Road, Tarlai Kalan, Islamabad 45550, Pakistan
2.
Department of Mathematics, COMSATS University Islamabad, Abbottabad Campus 22060, Pakistan
3.
Department of Mathematical Sciences, Universiti Kebangsaan Malaysia, Bangi 43600, Selangor, Malaysia

Received: 07 October 2020 Accepted: 30 November 2020 Published: 19 January 2021
MSC : 30C45, 30C50, 30C55

The investigation of univalent functions is one of the fundamental ideas of Geometric function theory (GFT). However, the class of these functions cannot be investigated as a whole for some particular kind of problems. As a result, the study of its subclasses has been receiving numerous attentions. In this direction, subfamilies of the class of univalent functions that map the open unit disc onto the domain bounded by limaçon of Pascal were recently introduced in the literature. Due to the several applications of this domain in Mathematics, Statistics (hypothesis testing problem) and Engineering (rotary fluid processing machines such as pumps, compressors, motors and engines.), continuous investigation of these classes are of interest in this article. To this end, the family of functions for which $ \frac{\varsigma f^{\prime}(\varsigma)}{f(\varsigma)} $ and $ \frac{(\varsigma f^{\prime}(\varsigma))^{\prime}}{f^{\prime}(\varsigma)} $ map open unit disc onto region bounded by limaçon are studied. Coefficients bounds, Fekete Szeg $ \ddot{ \rm{o}} $ inequalities and the bounds of the third Hankel determinants are derived. Furthermore, the sharp radius for which the classes are linked to each other and to the notable subclasses of univalent functions are found. Finally, the idea of subordination is utilized to obtain some results for functions belonging to these classes.
- univalent functions,
- Schwarz functions,
- Limaçon domain,
- subordination,
- Hankel determinan
Citation: Afis Saliu, Khalida Inayat Noor, Saqib Hussain, Maslina Darus. Some results for the family of univalent functions related with Limaçon domain[J]. AIMS Mathematics, 2021, 6(4): 3410-3431. doi: 10.3934/math.2021204

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Abstract

The investigation of univalent functions is one of the fundamental ideas of Geometric function theory (GFT). However, the class of these functions cannot be investigated as a whole for some particular kind of problems. As a result, the study of its subclasses has been receiving numerous attentions. In this direction, subfamilies of the class of univalent functions that map the open unit disc onto the domain bounded by limaçon of Pascal were recently introduced in the literature. Due to the several applications of this domain in Mathematics, Statistics (hypothesis testing problem) and Engineering (rotary fluid processing machines such as pumps, compressors, motors and engines.), continuous investigation of these classes are of interest in this article. To this end, the family of functions for which $ \frac{\varsigma f^{\prime}(\varsigma)}{f(\varsigma)} $ and $ \frac{(\varsigma f^{\prime}(\varsigma))^{\prime}}{f^{\prime}(\varsigma)} $ map open unit disc onto region bounded by limaçon are studied. Coefficients bounds, Fekete Szeg $ \ddot{ \rm{o}} $ inequalities and the bounds of the third Hankel determinants are derived. Furthermore, the sharp radius for which the classes are linked to each other and to the notable subclasses of univalent functions are found. Finally, the idea of subordination is utilized to obtain some results for functions belonging to these classes.

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