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

Dziok, Jacek1 ; Raina, Ravinder Krishna2 ; Sokół, Janusz3
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, Tome 353 (2015) no. 5, pp. 433-438.

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∈(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∈U, on ait :

1+zf″(z)f′(z)≺H(z,t),
où H(z,t)=1/(1−2tz+z2) 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∈(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+zf″(z)f′(z)≺H(z,t)=11−2tz+z2(z∈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.

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DOI : 10.1016/j.crma.2015.02.001
Dziok, Jacek 1 ; Raina, Ravinder Krishna 2 ; Sokół, Janusz 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 
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Dziok, Jacek; Raina, Ravinder Krishna; Sokół, Janusz. Application of Chebyshev polynomials to classes of analytic functions. Comptes Rendus. Mathématique, Tome 353 (2015) no. 5, pp. 433-438. doi : 10.1016/j.crma.2015.02.001. https://www.numdam.org/articles/10.1016/j.crma.2015.02.001/

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