$\textbf{Definition:}$ 1. A function $h : (a,b)\rightarrow\mathbb{R}$ is exponentially convex if it is continuous and $$\sum _{i, j=1}^n\xi_i\xi_jh(x_i+x_j)\geq 0,$$ for all $n\in\mathbb{N}$ and all choices of $\xi_i,x_i\in\mathbb{R}$, $i = 1,\ldots ,n$, such that $x_i+x_j\in(a,b)$, $1 \leq i, j \leq n$.
$\textbf{Proposition:}$ Let $h:(a,b)\rightarrow\mathbb{R}$. The following propositions are equivalent.
(i) $h$ is exponentially convex.
(ii)$ h$ is continuous and $$\sum _{i, j=1}^n\xi_i\xi_jh(\frac{x_i+x_j}{2})\geq 0,$$ for all $n\in\mathbb{N}$ and all choices of $\xi_i,x_i\in\mathbb{R}$, $i = 1, \ldots,n$, such that $x_i\in(a,b)$, $1 \leq i \leq n$.
The proof of above stated proposition is shown in this thread. This proposition is further followed by a corollary.
$\textbf{Corollary:}$ If $h$ is exponentially convex then $$det\big[h\big(\frac{x_i+x_j}{2}\big)\big]_{i,j=1}^n\geq 0.$$ for all $n\in\mathbb{N}$ and $x_i\in(a,b)$, $1 \leq i \leq n$.
Can Someone help me to prove it?
