HOW SHARP IS BERNSTEIN’S INEQUALITY FOR JACOBI POLYNOMIALS?∗

Abstract. Bernstein’s inequality for Jacobi polynomials P (α,β)n, established in 1987 by P. Baratella for the region R1/2 = {|α | ≤ 1/2, |β | ≤ 1/2}, and subsequently supplied with an improved constant by Y. Chow, L. Gatteschi, and R. Wong, is analyzed here analytically and, above all, computationally with regard to validity and sharpness, not only in the original regionR1/2, but also in larger regionsRs = {−1/2 ≤ α ≤ s,−1/2 ≤ β ≤ s}, s> 1/2. Computation suggests that the inequality holds with new, somewhat larger, constants in any region Rs. Best constants are provided for s = 1:.5: 4 and s = 5: 1: 10. Our work also sheds new light on the so-called Erdélyi–Magnus–Nevai conjecture for orthonormal Jacobi polynomials, adding further support for its validity and suggesting.66198126... as the best constant implied in the conjecture. Key words. Bernstein’s inequality, Jacobi polynomials, sharpness, Erdélyi–Magnus–Nevai conjectur