(Weak)m-extremals andm-geodesics

Three-point Nevanlinna–Pick problem in the polydisc

“…There is also another domain for which the equality of holomorphically invariant metrics is non-trivial, the tetrablock which, in turn, may be expressed as an image of the classical Cartan domain of the second type in C 2×2 under a proper holomorphic mapping of multiplicity 2 (see [1,12] for details). Thus, the properties of both special domains as well as behaviour of 3-extremals in the Euclidean unit ball (see [17,24]) and the fact the Coman conjecture remains true there suggest that a counterpart of Theorem 1 holds in a bigger class of domains containing among others classical Cartan domains (it is worth pointing out that the Coman conjecture was also proved for the unit ball in C n , see [11, 13]).…”

Nevanlinna-Pick interpolation problem in the ball

“…3-complex geodesics. It was proven in [24] (Theorem 5.8) that any 3-extremal in the unit ball is a complex 3-geodesic. We recall the reasoning that led to this result.…”

“…thus showing that such an f is a complex 3-geodesic. Later, Warszawski in [24] (Theorem 5.8) showed additionally that any 3-extremal f that is not a 2-extremal is actually equivalent with a 3-complex geodesic of the form λ → (am c (λ), √ 1 − a 2 m 2 c (λ)), a ∈ [0, 1). A more detailed result on 3-geodesity of 3-extremals (not being 2extremals) proven in [24] is presented in Proposition 3.…”

Coman conjecture for the bidisc

“…The notions of (weak) m-extremals and m-geodesics, which have clear origin in Nevanlinna-Pick problems for functions in the unit disk, have been recently introduced and studied in [1], [2], [12], [10] and [18]. It is worth recalling that the description of m-extremals in the unit disc is classical and well-known.…”