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.
Highlights
We consider the Oppenheim's problem: What are the best possible constants l1,l2, r1, r2. Such that l1 cos x r1 cos x r2 hold for all x ∈[−π,π ] \ {0}?
We present some new inequalities related to this problem for trigonometric functions
Hold for all x ∈ I; I = [−π,π ], [−b,b], 0 < b < π, [−π, −a] ∪[a,π ], 0 < a < π. The solution of this problem can be stated in the following theorem: Theorem 3.1
Summary
We consider the Oppenheim's problem: What are the best possible constants l1,l2 , r1, r2. We established in [5] some inequalities related to this type of problem for Dunkl kernels ψ α − i by answering to the following question: What are, for α ≥ − 1 , the best possible constants 2. Our aim is to solve the analogues of the Oppenheim's problem for the real and imaginary parts of Dunkl kernels ψ α 1. We present some new inequalities related to this problem for trigonometric functions. These inequalities and Sonine integral formula for Bessel functions allow us to get a new version of the solution of this type of problem for Bessel functions α. Sonine integral formula for Bessel functions, we solve the Oppenheim's problem for the imaginary parts of Dunkl kernels ψ1α. Where a1, a2 and a3 are as in Theorem 1.1 of [6]
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.