On the monotonicity of a function related to the local time of a symmetric Lévy process

Let ψ be the characteristic exponent of a symmetric Lévy process X. The functionh(x)=2π∫0∞1-cos(λx)ψ(λ)dλappears in various studies on the local time of X. We study monotonicity properties of the function h. In case when X is a subordinate Brownian motion, we show that x↦h(x) is a Bernstein function.