By El-Fallah O., Kellay K., Mashreghi J., Ransford T.

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**Example text**

Ii) Let f := Cg + η(1/3). Show that f ∈ D and that Re f (z) ≥ η(dist(z, E)) (z ∈ D). 3. 15) is valid on all of D, not just the boundary. 5 Exponentially tangential approach regions In this section we shall consider limits along certain tangential approach regions. We have already seen one result of this kind. 3, we showed that, if f ∈ D, then for almost every ζ ∈ T we have f (z) → f ∗ (ζ) as z → ζ in the oricyclic region |z − ζ| < κ(1 − |z|)1/2 . Using the techniques developed in this chapter, we can now prove the following much stronger version of this result, in which the approach region is exponentially tangential.

Hence, if n ≥ m, then Kνm dνn = 1/cm = Kνm dνm , and consequently hn − hm , hm = 0. 4 thus applies. Now hn 2 = IK (νn ) − log 2 = 1/cn − log 2. 4, 2 converges in D. This in turn implies that n cn hn converges in D, n hn / h whence also n cn fνn . Thus f ∈ D, as claimed. Next, we show that | Im f | < on D. 13). Finally we prove that lim inf z→ζ Re f (z) ≥ η(d(ζ, E)) for all ζ ∈ T. This is obvious if d(ζ, E)) ≥ δn0 , since Re f ≥ n0 everywhere in D. So assume that d(ζ, E) < δn0 , and let N be the integer such that δN+1 ≤ d(ζ, E) < δN .

3). 1, can be strengthened a little further. 2. Let f ∈ D. 6) where A is an absolute constant. The strategy of the proof is the same as for the weak-type inequality. We first establish the following inequality for the Cauchy transform. 5 Let g ∈ L2 (A). 7) where A is an absolute constant. 6. We shall prove that, if g ∈ L2 (A), then ∞ cK (Cg > t) dt2 ≤ 8 g 0 2 L2 (A) . 8) If so, then, since c is majorized by a multiple of cK , the same inequality holds with cK by c, and 8 replaced by another absolute constant.