We consider certain functions meromorphic in the plane that are of bounded type in suitable half planes. LetZjbe a sequence in a half plane not satisfying a Blaschke condition there. We deduce that conditions of being positive semi-definite on matrices whose elements depend onzjand the values that such a functionftakes at thezj, determinefuniquely. This can be used to study the properties offfurther. In particular, we give a function theoretic derivation for the fact that a certain set of inequalities is equivalent to Riemann's hypothesis. We also comment on iteration theoretic methods for studying such problems.