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We investigate the invariant metrics and complex geodesics in the universalTeichm\"{u}ller space and Teichm\"{u}ller space of the punctured disk usingMilin's coefficient inequalities. This technique allows us to establish thatall non-expanding invariant metrics in either of these spaces coincide with itsintrinsic Teichm\"{u}ller metric.  Other applications concern the variational theory for univalent functionswith quasiconformal extension. It turns out that geometric features caused bythe equality of metrics and connection with complex geodesics provide deepdistortion results for various classes of such functions and create newphenomena which do not appear in the classical geometric function theory.

Milin's coefficients, complex geometry of Teichm\"{u}ller spaces and variational calculus for univalent functions