Existence of two measurable functions s.t. $\int_{D}f^3dx = \int_{D}g^4dx = 1$ and $\int_{D}(f+g)^2dx = \pi^3 $

Question

Given $D = \lbrace (x,y) : x^2+y^2 \le 1 \rbrace $, say if we could find two measurable functions $f, g: D \to [0, + \infty]$ s.t. (then give an explicit example) : $$\int_{D}f^3dx = \int_{D}g^4dx = 1$$

and

$$\int_{D}(f+g)^2dx = \pi^3 $$

I tried to reason in terms of characteristic functions, but I am stuck on this exercise, some hints will be greatly appreciated. Thank you

Answer 1

Note that, if $f,g$ satisfy the first two equalities, then $$\int_Df^2\leq\left(\int_Df^3\right)^{2/3}|D|^{1/3}=\pi^{1/3},$$ and also $$\int_Dg^2\leq\left(\int_Dg^4\right)^{1/2}|D|^{1/2}=\pi^{1/2}.$$ So then $$\int_D(f+g)^2\leq 2\int_D(f^2+g^2)\leq 2\pi^{1/3}+2\pi^{1/2},$$ but this last term is less than $\pi^3$. Therefore, no such functions exist.