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

Presented by: 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.

Abstract
Résumé

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.

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.

Received: 2014-11-28
Accepted: 2015-02-09
Published online: 2015-03-07

DOI: 10.1016/j.crma.2015.02.001

Author's affiliations:

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

@article{CRMATH_2015__353_5_433_0,
     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},
}

TY  - JOUR
AU  - Jacek Dziok
AU  - Ravinder Krishna Raina
AU  - Janusz Sokół
TI  - Application of Chebyshev polynomials to classes of analytic functions
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 433
EP  - 438
VL  - 353
IS  - 5
PB  - Elsevier
DO  - 10.1016/j.crma.2015.02.001
LA  - en
ID  - CRMATH_2015__353_5_433_0
ER  -

%0 Journal Article
%A Jacek Dziok
%A Ravinder Krishna Raina
%A Janusz Sokół
%T Application of Chebyshev polynomials to classes of analytic functions
%J Comptes Rendus. Mathématique
%D 2015
%P 433-438
%V 353
%N 5
%I Elsevier
%R 10.1016/j.crma.2015.02.001
%G en
%F CRMATH_2015__353_5_433_0

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/

References
Cited by

[1] U. Bednarz; J. Sokół On the integral convolution of certain classes of analytic functions, Taiwanese J. Math., Volume 13 (2009) no. 5, pp. 1387-1396

[2] J. Dziok A general solution of the Fekete–Szegö problem, Bound. Value Probl. (2013) (Aricle ID 2013:98)

[3] J. Dziok; R.K. Raina; J. Sokół Certain results for a class of convex functions related to a shell-like curve connected with Fibonacci numbers, Comput. Math. Appl., Volume 61 (2011) no. 9, pp. 2605-2613

[4] J. Dziok; R.K. Raina; J. Sokół On alpha-convex functions related to shell-like functions connected with Fibonacci numbers, Appl. Math. Comput., Volume 218 (2011), pp. 966-1002

[5] A.W. Goodman On uniformly convex functions, Ann. Polon. Math., Volume 56 (1991), pp. 87-92

[6] R. Jurasinska; J. Sokół Some problems for certain family of starlike functions, Math. Comp. Modelling, Volume 55 (2012), pp. 2134-2140

[7] S. Kanas; H.M. Srivastava Linear operators associated with k-uniformly convex functions, Integral Transforms Spec. Funct., Volume 9 (2000) no. 2, pp. 121-132

[8] S. Kanas; A. Wiśniowska Conic regions and k-uniform convexity II, Folia Sci. Univ. Tech. Resov., Volume 22 (1998), pp. 65-78

[9] S. Kanas; A. Wiśniowska Conic regions and k-uniform convexity, J. Comput. Appl. Math., Volume 105 (1999), pp. 327-336

[10] S. Kanas; A. Wiśniowska Conic regions and k-starlike functions, Rev. Roumaine Math. Pures Appl., Volume 45 (2000), pp. 647-657

[11] A. Lecko; A. Wiśniowska Geometric properties of subclasses of starlike functions, J. Comput. Appl. Math., Volume 155 (2003), pp. 383-387

[12] W. Ma; D. Minda Uniformly convex functions, Ann. Polon. Math., Volume 57 (1992) no. 2, pp. 165-175

[13] F. Rønning Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., Volume 118 (1993), pp. 189-196

[14] J. Sokół On some subclass of strongly starlike functions, Demonstratio Math., Volume 31 (1998) no. 1, pp. 81-86

[15] J. Sokół On starlike functions connected with Fibonacci numbers, Folia Scient. Univ. Tech. Resoviensis, Volume 175 (1999), pp. 111-116

[16] E.T. Whittaker; G.N. Watson A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes of Analytic Functions with an Account of the Principal Transcendental Function, Cambridge University Press, 1963

Cited by Sources:

Comments - Policy