Exponential convexity and total positivity

Class of exponentially convex functions is a sub-class of convex functions on a given interval $(a, b)$. For exponentially convex function $f(x)$ S. N. Bernstein's integral representation holds. A condition for $f(x)$, providing the kernel $K(x, y)=f(x+y)$ to be totally positive is given. New examples of totally positive kernels are obtained. For example the kernel $(x+y)^{-\alpha}$ is totally positive for any $\alpha > 0$.