Fekete–Szegö inequality for certain spiral-like functions

Allu Vasudevarao

doi:10.1016/j.crma.2016.09.008

Complex analysis

Fekete–Szegö inequality for certain spiral-like functions
[Inégalité de Fekete–Szegö pour certaines fonctions spiralées]

Présenté par : the Editorial Board

Allu Vasudevarao ¹

¹ Department of Mathematics, Indian Institute of Technology Khargpur, Kharagpur-721 302, West Bengal, India

Comptes Rendus. Mathématique, Volume 354 (2016) no. 11, pp. 1065-1070.

Résumé
Abstract

Pour $| α | < π / 2$ , soit $S_{α}$ la classe des fonctions analytiques normalisées $f (z) = z + \sum_{n = 2}^{\infty} a_{n} z^{n}$ , non nulles dans le disque unité $D : = {z \in C; | z | < 1}$ et satisfaisant $Re P_{f} (z) > 0$ dans $D$ , où

P_{f} (z) = e^{i α} (1 + \frac{z f^{″} (z)}{f^{'} (z)}) .

Pour

f (z) \in S_{α}

, la fonction

z f^{'} (z)

est spiralée, notion introduite et étudiée de façon approfondie par M.S. Robertson [24]. Dans la présente Note, nous obtenons une borne supérieure précise de la fonctionnelle de Fekete–Szegö

| a_{3} - λ a_{2}^{2} |

, où λ est un paramètre complexe et

f \in S_{α}

.

For $| α | < π / 2$ , let $S_{α}$ denote the class of non-vanishing normalized analytic functions $f (z) = z + \sum_{n = 2}^{\infty} a_{n} z^{n}$ in the unit disk $D : = {z \in C : | z | < 1}$ satisfying $Re P_{f} (z) > 0$ in $D$ where

P_{f} (z) = e^{i α} (1 + \frac{z f^{″} (z)}{f^{'} (z)}) .

The class

S_{α}

consists of functions

f (z)

for which

z f^{'} (z)

is spiral-like, which has been introduced and extensively studied by M.S. Robertson [24]. In the present paper, we obtain the sharp upper bound for the Fekete–Szegö functional

| a_{3} - λ a_{2}^{2} |

for the complex parameter λ when

f \in S_{α}

.

Reçu le : 2016-08-12
Accepté le : 2016-09-20
Publié le : 2016-10-06

DOI : 10.1016/j.crma.2016.09.008

Affiliations des auteurs :

Allu Vasudevarao ¹

¹ Department of Mathematics, Indian Institute of Technology Khargpur, Kharagpur-721 302, West Bengal, India

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Allu Vasudevarao. Fekete–Szegö inequality for certain spiral-like functions. Comptes Rendus. Mathématique, Volume 354 (2016) no. 11, pp. 1065-1070. doi : 10.1016/j.crma.2016.09.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.09.008/

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