A CONVEXITY PROPERTY OF ZEROS OF BESSEL-FUNCTIONS

For k=1, 2,... let jvkand cvkbe the k-th positive zeros of the Bessel function {Mathematical expression} where Yv(X) is the Bessel function of the second kind. Using the notation jvκ =Cvk with κ=k-α/π introduced in [3] we show that the function jvκ +f(v) is convex with respect to υ≥0 for κ≥0.7070..., where f(υ) is defined in the theorem of section 2. As an application we find the inequality 0 >j0κ +j1κ - 2κπ > log 8/9, where κ≥0.7070.... © 1990 Birkhäuser Verlag.