Strongly invariant means on commutative hypergroups

We introduce and study strongly invariant means m on commutative hypergroups, m(T_xφ · ψ) = m(φ · T_x̃ ψ), x ∈ K, φ ψ ∈ L^∞(K). We show that the existence of such means is equivalent to a strong Reiter condition. For polynomial hypergroups we derive a growth condition for the Haar weights which is equivalent to the existence of strongly invariant means. We apply this characterization to show that there are commutative hypergroups which do not possess strongly invariant means.