Application of Chebyshev polynomials to classes of analytic functions

Jacek Dziok; Ravinder Krishna Raina; Janusz Sokół

doi:10.1016/j.crma.2015.02.001

Complex analysis

Application of Chebyshev polynomials to classes of analytic functions
[Application des polynômes de Chebyshev à des classes de fonctions analytiques]

Présenté par : the Editorial Board

Jacek Dziok ¹ ; Ravinder Krishna Raina ² ; Janusz Sokół ³

¹ Faculty of Mathematics and Natural Sciences, University of Rzeszów, Poland
² M.P. University of Agriculture and Technology, Udaipur, India
³ Department of Mathematics, Rzeszów University of Technology, Al. Powstańców Warszawy 12, 35-959 Rzeszów, Poland

Comptes Rendus. Mathématique, Volume 353 (2015) no. 5, pp. 433-438.

Résumé
Abstract

Notre propos dans cette Note est d'étudier quelques propriétés de base des polynômes de Chebyshev habituels en théorie des fonctions analytiques. Nous considérons plusieurs caractéristiques fondamentales pour les classes $H (t)$ , $t \in (1 / 2, 1]$ de fonctions f satisfaisant $f (0) = 0$ , $f^{'} (0) = 1$ , analytiques dans le disque unité ouvert $U = {z : | z | < 1}$ et telles que pour $z \in U$ , on ait :

1 + \frac{z f^{″} (z)}{f^{'} (z)} ≺ H (z, t),

où

H (z, t) = 1 / (1 - 2 t z + z^{2})

désigne la fonction génératrice des polynômes de Chebyshev de seconde espèce. Nous résolvons également le problème de Fekete–Szegö pour les classes considérées.

Our objective in this paper is to consider some basic properties of the familiar Chebyshev polynomials in the theory of analytic functions. We investigate some basic useful characteristics for a class $H (t)$ , $t \in (1 / 2, 1]$ , of functions f, with $f (0) = 0$ , $f^{'} (0) = 1$ , analytic in the open unit disc $U = {z : | z | < 1}$ satisfying the condition that

1 + \frac{z f^{″} (z)}{f^{'} (z)} ≺ H (z, t) = \frac{1}{1 - 2 t z + z^{2}} (z \in U),

where

H (z, t)

is the generating function of the second kind of Chebyshev polynomials. The Fekete–Szegö problem in the class is also solved.

Reçu le : 2014-11-28
Accepté le : 2015-02-09
Publié le : 2015-03-07

DOI : 10.1016/j.crma.2015.02.001

Affiliations des auteurs :

Jacek Dziok ¹ ; Ravinder Krishna Raina ² ; Janusz Sokół ³

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     author = {Jacek Dziok and Ravinder Krishna Raina and Janusz Sok\'o{\l}},
     title = {Application of {Chebyshev} polynomials to classes of analytic functions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {433--438},
     publisher = {Elsevier},
     volume = {353},
     number = {5},
     year = {2015},
     doi = {10.1016/j.crma.2015.02.001},
     language = {en},
}

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Jacek Dziok; Ravinder Krishna Raina; Janusz Sokół. Application of Chebyshev polynomials to classes of analytic functions. Comptes Rendus. Mathématique, Volume 353 (2015) no. 5, pp. 433-438. doi : 10.1016/j.crma.2015.02.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.02.001/

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