Let $p\in(1,2)$, and $f, g:[0, \tau]\rightarrow\mathbb{R}$ are a real valued functions. I like to upper bound $\int_{t}^{u} e^{ ps}|f(s)|^{p-2}|g(s)|^{2}ds$ for $t, u \in[0,\tau]$ in the following way: \begin{equation*} \int_{t}^{u} e^{ ps}|f(s)|^{p-2}|g(s)|^{2}ds \leq c\int_{t}^{u}e^{p's}|f(s)|^{\boldsymbol{q}}\,ds+c'\int_{t}^{u} e^{ p''s}|g(s)|^{\boldsymbol{q'}}\,ds, \end{equation*} where $c,c'$ are constants and $p',p'', \boldsymbol{q}, \boldsymbol{q'}$ are exponents such that $\boldsymbol{q}\leq p$ and $\boldsymbol{q'}\leq p$. The problem that I cannot apply Young's inequality ($ab\leq\frac{a^{n}}{n}+\frac{a^{m}}{m}$), for $n=\frac{2}{p}$ and $m=\frac{p-2}{p}$ since $\frac{p-2}{p}<0$.
I would be very grateful if you could help me.