Sandwiching the Riemann hypothesis

We consider a system of three analytic functions, two of which are known to have all their zeros on the critical line $\Re (s)=\sigma=1/2$. We construct inequalities which constrain the third function, $\xi(s)$, on $\Im(s)=0$ to lie between the other two functions, in a sandwich structure. We investigate what can be said about the location of zeros and radius of convergence of expansions of $\xi(s)$, with promising results.