Given a quasi-symmetric self-homeomorphism h of the unit circle S l , let Q(h) be the set of all quasiconformal mappings with the boundary correspondence h. In this paper, it is shown that there exists certain quasi-symmetric homeomorphism h, such that Q(h) satisfies either of the conditions, (1) Q{h) admits a quasiconformal mapping that is both uniquely locallyextremal and uniquely extremal-non-decreasable instead of being uniquely extremal; (2) Q(h) contains infinitely many quasiconformal

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... ings each of which has an extremal non-decreasable dilatation. An infinitesimal version of this result is also obtained.