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In this short note we calculate the Kobayashi pseudodistance for elementary Reinhardt domains in C2 They deliver us a number of examples giving a negative answer to a problem posed by S. Kobayashi. For a domain D C (Cn, w, z E D define (see [K] and [L]) kD(W, z) := inf{p(Al, A2): there is w E (9(E, D) such that w(A1) = w, W(A2) =Z} where p is the Poincare distance on the unit disk E in C. Let us also define kD as the largest pseudodistance smaller or equal than kD. We call kD the Lempert function and kD the Kobayashi pseudodistance of the domain D. It is well known (see [K] and [JP]) that if F: D --* G is a holomorphic covering (D and G are domains in :n), F(QCv) = w, then (1) kG(W, z) = inf{kD(iV,i)}, kG(W, z) = inf{kD(V, i)}, w, z E G, where the infimum, in both cases, is taken over all z e D such that F(z) = z. The problem whether the infimum is always attained was posed in [K], page 48 (in case of the Kobayashi pseudodistance). The analoguous problem concerning the Lempert function was posed in [JP]. Fix a = (a,,... , an) e (IR,)n. Let us denote D, := 1Z C Cn : Izlla I, .... . IZnlCtn 0 and a2 0, a2 2 )l/' d1 E. This together with the invariance of kE (= p) with respect to rotations finishes the proof for the Lempert function. Let us observe that the Lempert function in both cases satisfies the triangle inequality, therefore the Kobayashi pseudodistance coincides with the Lempert function. The proof of the case al ,i 2 1. Then w,z E D, Arg(s) = 0 whereas Arg(-s) = 7r and kDW (W, Z) = kDW (W, z) = 0 = inf{kE(sl?a2,sl+a2 exp((2k2 + 1)i7roa2)), k2 E Z}. The infimum is as in the proof of Theorem 1; it represents the infimum from the formula (1) for the holomorphic covering '1b. It is evident that the infimum is not attained by any k2 E Z. The similar procedure goes for the case ol, a2 < 0 and Oa being of irrational type. Remark 3. Let us notice here that it is possible to calculate the formulas for the Lempert function and the Kobayashi pseudodistance for all elementary Reinhardt domains (in arbitrary dimension). Nevertheless, since the idea is exactly the same as above, the obtained formulas deliver us no new phenomena, and the proof is more complicated, we skip it here.

On an example concerning the Kobayashi pseudodistance