My works lead to the true of the following inequality: For any $p>0$, there exist a constant $C_p>0$ which depends only on $p$, such that for any nonnegative sequence $(x_k)_{k\ge1}$ and for any nonnegative measurable functions sequence $(f_k)$ on measurable space $(X,\mu)$, with $\mu(X)<\infty$, we have $\left(\sum\limits_{k=1}^\infty x_k^p\left(\int_Xf_k(t)d\mu\right)^p\right)^{1/p}\leq C_p\int_X\left(\sum\limits_{k=1}^\infty x_k^pf_k(t)^p\right)^{1/p}d\mu$.
When $p=1$, we could chose $C_p=1$. When $p>1$, this looks like the Minkowski inequality. For $0<p<1$, I have no idea.
Does anyone help me to give a proof of it, or find out a counter example? (for all three case)
For $p\geq 1$, we can choose $C_p=1$, which is a variant of the Minkowski integral inequality. You can prove it using the characterization of $\ell^p$ norm of $x=(x_k)_{k\geq 1}$, $$ \|\ell\|_{\ell^p} = \sup_{ y\in \ell^{p'}} \sum_{k=1}^\infty x_ky_k, $$ where $p'=p/(p-1)$ is the exponent conjugate to $p$. Then
\begin{align} \left(\sum\limits_{k=1}^\infty x_k^p\left(\int_Xf_k(t)d\mu\right)^p\right)^{1/p} &= \sup_{ y\in \ell^{p'}} \sum_{k=1}^\infty \int_X x_ky_kf_k(t) d\mu \cr & \leq \int_X \left[\sup_{ y\in \ell^{p'}}\sum_{k=1}^\infty x_ky_kf_k(t)\right] d\mu\cr & \leq \int_X\left(\sum\limits_{k=1}^\infty x_k^pf_k(t)^p\right)^{1/p}d\mu. \end{align}
For $p \in (0,1)$, the inequality is actually reversed, $$\int_X\left(\sum\limits_{k=1}^\infty x_k^pf_k(t)^p\right)^{1/p}d\mu \leq \left(\sum\limits_{k=1}^\infty x_k^p\left(\int_Xf_k(t)d\mu\right)^p\right)^{1/p}. $$ You can rearrange the exponents, so that it is reduced to another variant of Minkowski inequality for $q=1/p >1$, that is $$ \left(\int_X\left(\sum\limits_{k=1}^\infty z_kh_k(t)\right)^{q}d\mu \right)^{1/q} \leq \sum\limits_{k=1}^\infty z_k\left(\int_Xh_k(t)^qd\mu\right)^{1/q} $$ by the substitution $z_k=x_k^p, h_k(t)=f_k(t)^p$.