$$ex a_ngeq 0 a vsm{n}a_nleq sqrt{pi}sex{vsm{n}a_n^2}^{1/4} sex{vsm{n}n^2a_n^2}^{1/4}, eex$$ $$ex int_0^infty |f(x)| d x leqsqrt{pi}sex{ int_0^infty f^2(x) d x }^{1/4}sex{ int_0^infty x^2f^2(x) d x }^{1/4}. eex$$
证明: 设 $$ex al=vsm{n}n^2a_n^2,quad eta=vsm{n}a_n^2, eex$$ 则 $$eex ea sex{vsm{n}a_n}^2&=sex{vsm{n}a_nsqrt{al+eta n^2}frac{1}{sqrt{al+eta n^2}}}^2 leq vsm{n}a_n^2(al+eta n^2)vsm{n}frac{1}{al+eta n^2}\ &leq 2al eta int_0^infty frac{1}{al+eta x^2} d x =pi aleta. eea eeex$$