Let $f,h,g\in L^{2}(\mathbb R^2).$ Prove the inequality $$\int_{\mathbb R^3}\vert f(x,y)g(y,z)h(z,x)\vert d(x,y,z)\leq\|f\|_2\|g\|_2\|h\|_2.$$
Using Fubini's Theorem and the Cauchy-Schwarz Inequality I obtained the following string of inequalities $$\int_{\mathbb R^3}\vert f(x,y)g(y,z)h(z,x)\vert\,d(x,y,z)\leq\|f\|_2\int_{\mathbb R}\left[\int_{\mathbb R}\int_{\mathbb R}\vert g(y,z)h(z,x)\vert^2\,dx\,dy\right]^{1/2}dz,$$ $$\int_{\mathbb R^3}\vert f(x,y)g(y,z)h(z,x)\vert\,d(x,y,z)\leq\|g\|_2\int_{\mathbb R}\left[\int_{\mathbb R}\int_{\mathbb R}\vert f(x,y)h(z,x)\vert^2\,dy\,dz\right]^{1/2}dx,$$ $$\int_{\mathbb R^3}\vert f(x,y)g(y,z)h(z,x)\vert\,d(x,y,z)\leq\|h\|_2\int_{\mathbb R}\left[\int_{\mathbb R}\int_{\mathbb R}\vert f(x,y)g(y,z)\vert^2\,dx\,dz\right]^{1/2}dy,$$ which I think, upon multiplication of all three, lead to nothing.
After that I tried using again Fubini's Theorem and this time Minkowski's Integral Inequality, but that leads to a similar string of inequalities, once again leading nowhere.
Can anyone please recommend me something to try to get started on this problem?
Thank you for your time and I apologize for my failure to solve the problem or come up with a more clever attempt.