I want to prove a relation and I need an inequality Let $A$ a rectifiable curve in the complex plan and $p>1$ and $f(x)$ an even nonnegative trignometric polynomal has the form $f(x) =\sum_{k=-n}^{n} c_k e^{ikx} $ and $g(x,z)$ a complex function with complex variable $ z$ and real variable $x\in [-\pi ,\pi] $ and here the original relation is $$\bigg(\sup_{A} \int_{A} \bigg|\int_0^\pi f(x) g(x,z) dx\bigg|^p |dz|\bigg)^{1/p} \le \int_0^\pi f(x) \bigg(\sup_{A} \int_{A} |g(x.z)|^p |dz| \bigg)^{1/p} dx$$
Thank you for your help
By Minkowski's inequality, for any rectifiable curve $\gamma$ $$ \Bigl(\int_{\gamma} \Bigl|\int_0^\pi f(x)\,g(x,z)\,dx\,\Bigr|^p\,|dz|\Bigr)^{1/p} \le \int_0^\pi |f(x)|\Bigl(\int_{\gamma} |g(x,z)|^p\,|dz|\Bigr)^{1/p}\,dx. $$ Taking first the sup on the right hand side over all curves $A$ we get $$ \Bigl(\int_{\gamma} \Bigl|\int_0^\pi f(x)\,g(x,z)\,dx\,\Bigr|^p\,|dz|\Bigr)^{1/p} \le \int_0^\pi |f(x)|\Bigl(\sup_{A}\int_{A} |g(x,z)|^p\,|dz|\Bigr)^{1/p}\,dx. $$ The right hand side does not depend on $\gamma$. Taking now the sup on the left hand side proves the desired inequality.