I want to show when ${G(A,B)=\frac{1}{m}\sum_{i=1}^m}\frac{1}{\frac{1}{n}\sum_{j=1}^{n}e^{-\vert{a_i-b_j}\vert}}$, ${G(A,C)\leq G(A,B) \cdot G(B,C)}$.
I think from $\vert a_i-b_k+b_k-c_j \vert \leq \vert a_i - b_k \vert + \vert b_k-c_j\vert$,
$e^{\vert a_i-b_k+b_k-c_j \vert} \leq e^{\vert a_i - b_k \vert} \cdot e^{\vert b_k - c_j \vert} $.
how can i show from ${\frac{1}{e^{-\vert a_i-b_k+b_k-c_j \vert}}} \leq \frac{1}{e^{-\vert a_i - b_k \vert }} \cdot \frac {1}{e^{-\vert b_k - c_j \vert}} $ to ${G(A,C)\leq G(A,B) \cdot G(B,C)} ?$
Is this right approach to this problem?