For $1≤p≤\infty$, find the values of the parameter \lambda for which $\displaystyle\lim_{\epsilon \rightarrow 0^+} \left(\displaystyle \frac{1}{{\epsilon}^{\lambda}}{\int_{0}^{\epsilon}f(x)dx} \right)=0$, $∀ f∈{L}^{p}[0,1]$
I cannot see how to approach this problem. The title of the section from which this problem comes from is "the inequalities of Young, Holder, and Minkowski" from Royden's Real Analysis, but I do not see how it applies here. Any suggestions will be appreciated.
By Hölder's inequality $$\frac{1}{\epsilon^\lambda}\int_0^\epsilon f=\frac{1}{\epsilon^\lambda}\int_0^1 f\chi_{[0,\epsilon]}\leq\frac{||f||_{L^p}||\chi_{[0,\epsilon]}||_{L^q}}{\epsilon^\lambda}=\frac{||f||_{L^p}{\epsilon^\frac{1}{q}}}{\epsilon^\lambda}$$ so it is sufficient for $\lambda<\frac{1}{q}$