Abstract. We use the approximation properties of the Faber polynomials to obtain some direct theorems of the polynomial approximation in Smirnov-Orlicz classes.

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Let G be a finite domain with z 0 2 G and bounded by a Jordan curve L :¼ @G. The Bieberbach polynomials n , n ¼ 1, 2, . . . , associated with the pair (G, z 0 ) can be used to approximate the conformal mapping 0 from G to D(0, r 0 ) :¼ {w : jwj < r 0 } normalized by 0 ðz 0 Þ ¼ 0, 0 0 ðz 0 Þ ¼ 1. In this work we define a new subclass of smooth Jordan curves and give the estimates of k 0 À n k G :¼ sup z2G j 0 ðzÞ À n ðzÞj in accordance with the parameters defining this subclass.

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