Homework 8

Due Date: Friday, April 15

1. If \(a,b,c\in(0,\infty)\), prove that \[ \frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \,\geq\, \frac{3}{2}. \]
2. Let \((X,\mu)\) be a measure space, let \(f\), \(g\), and \(h\) be non-negative measurable functions on \(X\), and let \(p,q,r\in (1,\infty)\) so that \(1/p+1/q+1/r=1\). Prove that \[ \int_X fgh\,d\mu \,\leq\, \biggl(\int_X f^p\,d\mu\biggr)^{\!1/p}\biggl(\int_X g^q\,d\mu\biggr)^{\!1/q}\biggl(\int_X h^r\,d\mu\biggr)^{\!1/r} \]
3. Let \((X,\mu)\) be a measure space, and let \(1 \leq r < s < t < \infty\). Prove that there exist constants \(\alpha,\beta>0\) so that \[ \|f\|_s \,\leq\, \|f\|_r^\alpha\,\|f\|_t^\beta \] for every measurable function \(f\) on \(X\). (Thus any function that is both \(L^r\) and \(L^t\) must be \(L^s\) as well.)