$\textbf{Definition:}$ 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 $x_i+x_j\in(a,b)$, $1 \leq i, j \leq n$.

There is a proposition in many research papers, but I can not find any clear proof of it.

$\textbf{Proposition:}$ 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 $x_i\in(a,b)$, $1 \leq i \leq n$.

Can Someone give me its formal proof?

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