Given $T:\mathbb{R}^{2}\to \mathbb{R}$ s.t.
\begin{eqnarray*} |T(x)|\leq \sum_{k=1}^{2}|x_{k}|\,and\,|T(x)-T(y)|\leq \sum_{k=1}^{2}|x_{k}-y_{k}| \end{eqnarray*}
for all $x=(x_{1},x_{2}),\,y=(y_{1},y_{2})\in \mathbb{R}^{2}$
Assumption1: For all $i \in \{1,2\}$, $\lambda_{i}\geq0$.
Assumption2: For all $i,j \in \{1,2\}$, $a_{ij}\geq0$, $a_{ij}=a_{ji}$ and $\lambda_{j}\geq\sum_{i=1}^{2}a_{ij}$
We define $b_{ij}=\lambda_{i}\delta_{ij}-a_{ij}$ ($\delta_{ij}$ is Kronecker delta)
I want to show the following inequality:
\begin{eqnarray*} (\sum_{i,j=1}^{2}T(\alpha_{i1},\alpha_{i2})T(\alpha_{j1},\alpha_{j2})b_{ij})^{1/2}\leq\sum_{k=1}^{2}(\sum_{i,j=1}^{2}\alpha_{ki}\alpha_{kj}b_{ij})^{1/2} \end{eqnarray*}
Where $\alpha_{ij}\in \mathbb{R}$
(We may assume $\sum_{i,j=1}^{2}T(\alpha_{i1},\alpha_{i2})T(\alpha_{j1}\alpha_{j2})b_{ij} , \sum_{i,j=1}^{2}\alpha_{ki}\alpha_{kj}b_{ij}\geq0$)
My process:
Step1: Deduce the follwing identity
\begin{eqnarray*} \sum_{i,j=1}^{2}z_{i}z_{j}b_{ij}=\sum_{i<j}a_{ij}(z_{i}-z_{j})^{2}+\sum_{j=1}^{2}m_{j}z_{j}^{2} \end{eqnarray*}
$m_{j}=\lambda_{j}-\sum_{i=1}^{2}a_{ij}$($\geq 0$) by assumption1
From Step1 and assumption of $T$
\begin{eqnarray*} \sum_{i,j=1}^{2}T(\alpha_{i1},\alpha_{i2})T(\alpha_{j1},\alpha_{j2})b_{ij}&=&\sum_{i<j}a_{ij}\left|T(\alpha_{i1},\alpha_{i2})-T(\alpha_{j1},\alpha_{j2})\right|^{2}+\sum_{j=1}^{2}m_{j}\left|T(\alpha_{j1},\alpha_{j2})\right|^{2}\\ &\leq&\sum_{i<j}a_{ij}\left(\sum_{k=1}^{2}|\alpha_{ik}-\alpha_{jk}|\right)^{2}+\sum_{j=1}^{2}m_{j}\left(\sum_{k=1}^{2}|\alpha_{jk}|\right)^{2}\\ &=&\sum_{i<j}a_{ij}\left(\sum_{k=1}^{2}|\alpha_{ik}-\alpha_{jk}|\right)^{2}+\sum_{j=1}^{2}(\lambda_{j}-\sum_{i=1}^{2}a_{ij})\left(\sum_{k=1}^{2}|\alpha_{jk}|\right)^{2}\\ &=&\sum_{i<j}a_{ij}\left(\sum_{k=1}^{2}|\alpha_{ik}-\alpha_{jk}|\right)^{2}+\sum_{j=1}^{2}(\sum_{i=1}^{2}(\lambda_{j}\delta_{ij}-a_{ij}))\left(\sum_{k=1}^{2}|\alpha_{jk}|\right)^{2}\\ &=&\sum_{i<j}a_{ij}\left(\sum_{k=1}^{2}|\alpha_{ik}-\alpha_{jk}|\right)^{2}+\sum_{j=1}^{2}\sum_{i=1}^{2}b_{ij}\left(\sum_{k=1}^{2}|\alpha_{jk}|\right)^{2}\\ \end{eqnarray*}
How do I apply Minkowski's Inequality? Please tell me.
