Pólya-Radoux type results for some arithmetical functions

Let h:[1,∞)→[0,∞) be a function with the properties that there exists x0≥1 such that h∈C1([x0,∞)), h(x0)>0, h′(x)>0 for all x≥x0 and limx→∞⁡h(x)=∞. We prove that if a:N→[0,∞) and k≥1 then, ∑n≤xa(n)∽[h(x)]k if and only if for every function f:[0,1]→R such that xk−1f(x) is Riemann integrable the following equality holdslimx→∞⁡1[h(x)]k∑x0