In the page 122 of Durrett's book, with definitions $\mu_k=\int x^kdF(x)$ and $v_k=\int |x|^kdF(x)$, If $\text{limsup}\frac{\mu_{2k}^{1/2k}}{2k}=r<\infty$, then by using Cauchy-Schwarz $v_{2k+1}^2\leq \mu_{2k} \mu_{2k+2}$. Why can we conclude that $\text{limsup}\frac{v_{k}^{1/k}}{k}=r<\infty$?
It's obvious that $v_{2k}=\mu_{2k}$, but I cannot show that $(\frac{v_{2k+1}^{1/(2k+1)}}{2k+1})^2\leq \frac{\mu_{2k}^{1/2k}}{2k}\frac{\mu_{2k+2}^{1/(2k+2)}}{2k+2}$.