$\int_{I}|f(x)||g(x)|^{2}dx \leq \bigg(\int_{I}|f(x)|dx\bigg)\bigg(\int_{I}|g(x)|^{2}dx\bigg)^{\beta}$

Question

$f \in L^{1}(I)$ and $g \in L^{2}(I)$. Is it possible to obtain an inequality of type $$ \int_{I}|f(x)||g(x)|^{2}dx \leq \bigg(\int_{I}|f(x)|dx\bigg)\bigg(\int_{I}|g(x)|^{2}dx\bigg)^{\beta}? $$ with $|I| < \infty$ and for some $\beta > 0$.