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Let A\ denote the unit disc and O9A denote the unit circle both in the complex plane. Define the Doob's class D(p), 0 1. Recently the author and Rung [6] proved a conjecture of Doob made in 1935 by showing that the norm llf su = sup (1 _ 1z12)1f'(z)1 > 2sinl(p) 0 <(p) ? 7r p/2. ZEA eO(p) We then conjecture that the result should be true if the arc rf is replaced by a finite union of arcs whose total length is at least p. In this paper, we answer this problem. It turns out to be surprising that the answer depends on the connectivity of the union, namely, the answer is no for the disconnected case, but yes for the connected one.

A problem on the Bloch norm of functions in Doob’s class