Suppose $f\in L^p$, $f=f \chi_E+f\chi_{\tilde E}$ where $E=\{x:|f|> 1\}$ and $E$ has finite measure. Find an inequality for $\|f\|_p$ in terms of $\|f \chi_E\|_r$ and $\|f \chi_{\tilde E}\|_s$ where $r\le p$ and $s\ge p$. Assume the Lebesgue measure on $\mathbb R$ and ${\tilde E}=\mathbb R \setminus E$.
Since $E$ has finite measure, by Jensen's inequality, $\|f\chi_E\|_r \le k \|f\|_p$ for all $r\le p$ and $k=m(E)^{{1 \over r} - {1 \over p}}$. Since $|f\chi_{\tilde E}|\le 1$ and $s \ge p$, $\int |f\chi_{\tilde E}|^s \le \int |f|^p$ for all $s \ge p$. From this I don't see how to relate $\|f \chi_{\tilde E}\|_s$ and $\|f\|_p$.