$\ell_{p^\prime}(\eta)$ and $\ell_p(\eta)$ are weighted Lebesgue sequence spaces and $\sum_{i\in\mathbb{N}}\eta_i=1$.
My professor gave the hint that we should use Jensen inequality. I tried to use $\phi(x)=|x|^{\frac{p^\prime}{p}}$, then suppose $\{a_i\}_{i\in\mathbb{N}}\in\ell_p(\eta)$ I got $$\sum_{i\in\mathbb{N}}\eta_i|a_i|^{p^\prime}\leq\left|\sum_{i\in\mathbb{N}}\eta_i|a_i|^p\right|^{\frac{p^\prime}{p}}$$ which implies the inclusion in the wrong direction. What is wrong in my proof and what should be the right proof.
The function $\varphi (x) =|x|^{\frac{p}{p' }}$ is concave continous and $\varphi (0)=0,$ so for any $k\in \mathbb{N} $ we obtain that $$\left(\sum_{j=1}^k \eta_j |a_j|^{p'}\right)^{\frac{p}{p'}} =\varphi \left(\sum_{j=1}^k \eta_j |a_j|^{p'}\right)=\varphi \left(\sum_{j=1}^k \eta_j |a_j|^{p'}+\left(\sum_{j=1}^k \eta_j \right)\cdot 0\right)\geq \sum_{j=1}^k \eta_j \varphi \left(|a_j|^{p'}\right) +0=\sum_{j=1}^k \eta_j |a_j|^{p}$$ and hence $$\left(\sum_{j=1}^k \eta_j |a_j|^{p'}\right)^{\frac{1}{p'}} \geq \left(\sum_{j=1}^k \eta_j |a_j|^{p}\right)^{\frac{1}{p}}.$$ Now by the continuity of power functions we have $$\left(\sum_{j=1}^{\infty}\eta_j |a_j|^{p'}\right)^{\frac{1}{p'}} =\lim_{k\to\infty}\left(\sum_{j=1}^k\eta_j |a_j|^{p'}\right)^{\frac{1}{p'}}\geq \lim_{k\to\infty}\left(\sum_{j=1}^k \eta_j |a_j|^{p}\right)^{\frac{1}{p}}=\left(\sum_{j=1}^{\infty}\eta_j |a_j|^{p}\right)^{\frac{1}{p}}.$$