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 $\mathrm{\Re }(s)=\sigma =1\mathrm{/}2$. We construct inequalities which constrain the third function, $\xi (s)$, on $\mathrm{\Im }(s)=0$ 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.