A convexity property of zeros of Bessel functions

Fork=1, 2,... letj _vk andc _vk be thek-th positive zeros of the Bessel function $$C_v \left( x \right) = C_v \left( {\alpha ;x} \right) = J_v \left( x \right)\cos \alpha - Y_v \left( x \right)\sin \alpha , 0 \leqslant \alpha< \pi$$ whereY _v (X) is the Bessel function of the second kind. Using the notationj _vκ =C _vk withκ=k?α/π introduced in 3] we show that the functionj _vκ +f(v) is convex with respect toυ≥0 forκ≥0.7070..., wheref(υ) is defined in the theorem of section 2. As an application we find the inequality 0 >j _0κ +j _1κ ? 2κπ > log 8/9, where κ≥0.7070....