Second Hankel determinant for close-to-convex functions

Dorina Răducanu; Paweł Zaprawa

doi:10.1016/j.crma.2017.09.006

Complex analysis

Second Hankel determinant for close-to-convex functions
[Deuxième déterminant de Hankel pour les fonctions presque convexes]

Présenté par : the Editorial Board

Dorina Răducanu ¹ ; Paweł Zaprawa ²

¹ Faculty of Mathematics and Computer Science, Transilvania University of Braşov, Iuliu Maniu 50, 500091 Braşov, Romania
² Faculty of Mechanical Engineering, Department of Mathematics, Lublin University of Technology, Nadbystrzycka 38D, 20-618 Lublin, Poland

Comptes Rendus. Mathématique, Volume 355 (2017) no. 10, pp. 1063-1071.

Résumé
Abstract

Aucune estimation précise de l'expression $| a_{2} a_{4} - {a_{3}}^{2} |$ pour la classe $C$ des fonctions presque convexes n'était connue jusqu'à présent. Dans cette Note, nous présentons des estimations de cette expression, nommée deuxième déterminant de Hankel pour la classe $C_{0}$ , c'est-à-dire la sous-classe $C$ , composée des fonctions f qui vérifient, dans le disque unité, l'inégalité $Re (z f^{'} (z) / g (z)) > 0$ avec une fonction étoilée g.

De plus, nous formulons quelques remarques à propos du deuxième déterminant de Hankel pour la classe $S$ des fonctions univalentes. Nous démontrons que $\max {| a_{2} a_{4} - a_{3}^{2} | : f \in S}$ est plus grand que 1.

So far, the sharp bound of the expression $| a_{2} a_{4} - a_{3}^{2} |$ for the class $C$ of close-to-convex functions has remained unknown. In this paper, we obtain the estimation of this expression, called the second Hankel determinant, for $C_{0}$ , i.e. the subset of $C$ consisting of functions f that satisfy in the unit disk the inequality $Re (z f^{'} (z) / g (z)) > 0$ with a starlike function g.

Moreover, some remarks on the second Hankel determinant for the class $S$ of univalent functions are made. It is proven that $\max {| a_{2} a_{4} - a_{3}^{2} | : f \in S}$ is greater than 1.

Reçu le : 2017-03-31
Accepté le : 2017-09-06
Publié le : 2017-10-01

DOI : 10.1016/j.crma.2017.09.006

Affiliations des auteurs :

Dorina Răducanu ¹ ; Paweł Zaprawa ²

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     title = {Second {Hankel} determinant for close-to-convex functions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1063--1071},
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     year = {2017},
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     language = {en},
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Dorina Răducanu; Paweł Zaprawa. Second Hankel determinant for close-to-convex functions. Comptes Rendus. Mathématique, Volume 355 (2017) no. 10, pp. 1063-1071. doi : 10.1016/j.crma.2017.09.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.09.006/

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