On the Fekete–Szegö type functionals for functions which are convex in the direction of the imaginary axis

Zaprawa, Paweł

doi:10.5802/crmath.144

Analyse et géométrie complexes

On the Fekete–Szegö type functionals for functions which are convex in the direction of the imaginary axis

Zaprawa, Paweł ¹

¹ Lublin University of Technology, Department of Mathematics, Faculty of Mechanical Engineering, Nadbystrzycka 38D, 20-618, Lublin, Poland

Comptes Rendus. Mathématique, Tome 358 (2020) no. 11-12, pp. 1213-1226.

Résumé

In this paper we consider two functionals of the Fekete–Szegö type: $Φ_{f} (μ) = a_{2} a_{4} - μ a_{3}^{2}$ and $Θ_{f} (μ) = a_{4} - μ a_{2} a_{3}$ for analytic functions $f (z) = z + a_{2} z^{2} + a_{3} z^{3} + ...$ , $z \in Δ$ , ( $Δ = {z \in ℂ : | z | < 1}$ ) and for real numbers $μ$ . For $f$ which is univalent and convex in the direction of the imaginary axis, we find sharp bounds of the functionals $Φ_{f} (μ)$ and $Θ_{f} (μ)$ . It is possible to transfer the results onto the class $𝒦_{ℝ} (i)$ of functions convex in the direction of the imaginary axis with real coefficients as well as onto the class $𝒯$ of typically real functions. As corollaries, we obtain bounds of the second Hankel determinant in $𝒦_{ℝ} (i)$ and $𝒯$ .

Reçu le : 2019-08-14
Révisé le : 2020-06-24
Accepté le : 2020-11-03
Publié le : 2021-01-25

DOI : 10.5802/crmath.144

Classification : 30C50

Affiliations des auteurs :

Zaprawa, Paweł ¹

¹ Lublin University of Technology, Department of Mathematics, Faculty of Mechanical Engineering, Nadbystrzycka 38D, 20-618, Lublin, Poland

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     title = {On the {Fekete{\textendash}Szeg\"o} type functionals for functions which are convex in the direction of the imaginary axis},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1213--1226},
     publisher = {Acad\'emie des sciences, Paris},
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Zaprawa, Paweł. On the Fekete–Szegö type functionals for functions which are convex in the direction of the imaginary axis. Comptes Rendus. Mathématique, Tome 358 (2020) no. 11-12, pp. 1213-1226. doi : 10.5802/crmath.144. https://www.numdam.org/articles/10.5802/crmath.144/

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