Math 461 Bard College

# 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.)