A note on Söderlind's conjecture

Let f be a Lipschitz operator from a path-connected set D⊆C into C, with the lub-Lipschitz constant L(f) and the so-called "Gerschgorim range radius" r(f) subordinate to a given vector norm ∥·∥ of C. In 1986 [Numer. Math. 50 (1986) 27], Söderlind's conjectured that if r(f)<L(f), then there exists a new vector norm ∥·∥* of C such that the induced lub-Lipschitz constant L*(f)r(f). In this paper, we affirmatively prove Söderlind's conjecture for several class of Lipschitz operators f, whilst we construct a counterexample to disprove Söderlind's conjecture.