On holomorphic reflexivity conditions for complex Lie groups

We consider Akbarov's holomorphic version of the non-commutative Pontryagin duality for a complex Lie group. prove, under assumption that $G$ is Stein group with finitely many components, (1) topological Hopf algebra functions on holomorphically reflexive if and only linear; (2) dual cocommutative exponential analytic functional reflexive. give counterexample, which shows first criterion cannot be extended to case infinitely components. Nevertheless, we conjecture that, in general, question can solved terms Banach-algebra linearity $G$.