Turán Type Inequalities for Jacobi-Dunkl Polynomials

Frej Chouchene^1,

¹Department of Mathematics, University of Sousse, Higher School of Sciences and Technology of Hammam Sousse, 4011 Hammam Sousse, Tunisia

Cite this paper:
Frej Chouchene. Turán Type Inequalities for Jacobi-Dunkl Polynomials. Turkish Journal of Analysis and Number Theory. 2018; 6(3):72-83. doi: 10.12691/tjant-6-3-2

Abstract

In this paper we establish Turán type inequalities for Jacobi-Dunkl polynomials. These complete an earlier result proven by G. Gasper in [1], for Jacobi polynomials.

Keywords:
Jacobi polynomials Jacobi-Dunkl polynomials Turán type inequalities

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]	Gasper, G., An inequality of Turán type for Jacobi polynomials, Proc. Amer. Math. Soc., 32(2). 435-439. 1972.
[2]	Turán, P., On the zeros of the polynomials of Legendre, Časopis. Pĕst. Mat. Fys., 75. 113-122. 1950.
[3]	Szegö, G., On an inequality of P. Turán concerning Legendre polynomials, Bull. Amer. Math. Soc., 54. 401-405. 1948.
[4]	Alzer, H., Gerhold, S., Kauers, M. and Lupas, A., On Turán’s inequality for Legendre polynomials, Expo. Math., 25. 181-186. 2007.
[5]	Baricz, Á., Turán type inequalities for hypergeometric functions, Proc. Amer. Math. Soc., 136(9). 3223-3229. 2008.
[6]	Bustoz, J. and Savage, N., Inequalities for ultraspherical and Laguerre polynomials, SIAM J. Math. Anal., 10(5). 902-912. 1979.
[7]	Bustoz, J. and Ismail, M.E.H., Turán inequalities for ultraspherical and continuous q-ultraspherical polynomials. SIAM J. Math. Anal., 14. 807-818. 1983.
[8]	Bustoz, J. and Ismail, M.E.H., Turán inequalities for symmetric orthogonal polynomials, Internat. J. Math. Math. Sci., 20(1). 1-8. 1997.
[9]	Bustoz, J. and Pyung, I.S., Determinant inequalities for sieved ultraspherical polynomials, Internat. J. Math. Math. Sci., 25(11). 745-751. 2001.
[10]	Gasper, G., On the extension of Turán inequality for Jacobi polynomials. Duke Math. J., 38. 415-428. 1971.
[11]	Simic, S., Turán’s inequality for Appell polynomials, J. Inequal. Appl., 1-7. 2006.
[12]	Szasz, O., Inequalities concerning ultraspherical polynomials and Bessel functions, Proc. Amer. Math. Soc., 1. 256-267. 1950.
[13]	Csordas G. and Williamson, J., On polynomials satisfying a Turán type inequality. Proc. Amer. Math. Soc., 43(2). 367-372. 1974.
[14]	Danese, A., Some identities and inequalities involving ultraspherical polynomials. Duke Math. J. 26. 349-359. 1959.
[15]	Dimitrov, D.K., Higher order Turán inequalities. Proc. Amer. Math. Soc., 126. 2033-2037. 1998.
[16]	Szegö, G., An inequality for Jacobi polynomials, Studies in Mathematical Analysis and Related Topics, Stanford Univ. Press., Stanford, California, 392-398. 1962.
[17]	Venkatachaliengar, K. and Lakshmana Rao, S.K., On Turán’s inequality for ultraspherical polynomials, Proc. Amer. Math. Soc., 8. 1075-1087. 1957.
[18]	Luke, Y.L., The special functions and their approximations, Vol. 2, Academic Press, 1969.
[19]	Szegö, G., Orthogonal polynomials, Amer. Math. Soc. Colloq. Pub., Vol. 23, Amer. Math. Soc., Providence, R. I., 1967.
[20]	Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, G., Higher transcendental functions, Vol. 1, McGraw-Hill Book Company, 1953.
[21]	Luke, Y.L., The special functions and their approximations, Vol. 1, Academic Press, 1969.
[22]	Chouchene, F., Harmonic analysis associated with the Jacobi-Dunkl operator on J. Comput. Appl. Math., 178. 75-89. 2005.