Oppenheim's Problem and Some Inequalities Involving Bessel Functions and Dunkl Kernels

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. Oppenheim's Problem and Some Inequalities Involving Bessel Functions and Dunkl Kernels. Turkish Journal of Analysis and Number Theory. 2018; 6(3):61-71. doi: 10.12691/tjant-6-3-1

Abstract

In this paper, we establish some inequalities related to Oppenheim's problem for the real and imaginary parts of Dunkl kernels In order to prove our main results, we present some new inequalities involving Bessel functions of the first kind. Refinements of inequalities for Bessel functions are also given.

Keywords:
Oppenheim's problem Bessel functions modified Bessel functions Dunkl kernels

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]	Carver, W.B. and Oppenheim, A., Elementary problems and solutions: solutions: E 1277, Amer. Math. Monthly, 65(3). 206-209. 1958.
[2]	Mitrinović, D.S., Analytic inequalities, Springer-Verlag, Berlin, 1970.
[3]	Zhu, L., A solution of a problem of Oppenheim. Math. Inequal. Appl., 10(1). 57-61. 2007.
[4]	Baricz, Á., Functional inequalities involving Bessel and modified Bessel functions of the first kind, Expo. Math., 26(3). 279-293. 2008.
[5]	Chouchene, F., Oppenheim's problem and related inequalities for Dunkl kernels, Math. Inequal. Appl., 17(1). 1-40. 2014.
[6]	Baricz, Á. and Zhu, L., Extension of Oppenheim's problem to Bessel functions. J. Inequal. Appl., Art. ID 82038. 7 pp. 2007.
[7]	Baricz, Á., Generalized Bessel functions of the first kind, Lecture Notes in Mathematics, 1994. Springer-Verlag, Berlin, 2010. xiv+206 pp.
[8]	Chettaoui, C. and Trimèche, K., New type Paley-Wiener theorems for the Dunkl transform on . Integral Transforms Spec. Funct., 14(2). 97-115. 2003.
[9]	Dunkl, C.F., Differential-difference operators associated to reflection groups. Trans. Amer. Math. Soc., 311(1). 167-183. 1989.
[10]	Dunkl, C.F., Integral kernels with reflection group invariance, Canad. J. Math., 43. 1213-1227. 1991.
[11]	Dunkl, C.F., Hankel transforms associated to finite reflection groups, Contemp. Math., 138. 123-138. 1992.
[12]	Dunkl, C.F., Intertwining operators and polynomials associated with the symmetric group, Monatsh. Math., 126. 181-209. 1998.
[13]	Dunkl, C.F., Orthogonal polynomials of types A and B and related Calegero models, Commun. Math. Phys., 197. 451-487. 1998.
[14]	Mourou, M.A., Transmutation operators associated with a Dunkl type differential-difference operator on the real line and certain of their applications, Integral Transforms Spec. Funct., 12(1). 77-88. 2001.
[15]	Mourou, M.A. and Trimèche, K., Opérateurs de transmutation et théorème de Paley-Wiener associés à un opérateur aux dérivées et différences sur , C. R. Acad. Sci. Paris, Série I Math., 332. 397-400. 2001.
[16]	Mourou, M.A. and Trimèche, K., Transmutation operators and Paley-Wiener theorem associated with a differential-difference operator on the real line, Anal. Appl., 1. 43-70. 2003.
[17]	Rösler, M., Bessel-type signed hypergroups on . In: H. Heyer, A. Mukherjea (eds.). Probability measures on groups and related structures XI. Proceedings. Oberwolfach 1994, Singapore: World Scientific, 292-304. 1995.
[18]	Watson, G.N., A treatise on the theory of Bessel functions, Cambridge University Press, Cambridge, UK, 1962.