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We show that the tangential speed of a parabolic semigroup of holomorphicself-maps in the unit disc is asymptotically bounded from above by (1/2)logt,proving a conjecture by Bracci. In order to show the proof we need a result of"asymptotical monotonicity" of the tangential speed for proper pairs ofparabolic semigroups with positive hyperbolic step.

Asymptotic upper bound for tangential speed of parabolic semigroups of holomorphic self-maps in the unit disc