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]
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