Abstract
The purpose of this paper is to prove some facts about integral means
of (d21dz2)(log\f(z)Iz])—or equivalently/"//, for/ in a class of starlike mappings of
a "singular" nature. In particular it is noted that the Koebe function is not extremal for
the Hardy means Mp(r,f"/f)
for functions in this class.
Results & Lemmas (6)
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THEOREM 1.
THEOREM 1. Suppose/ G SQ with /i(0) = limr_i argf(rew), and 0 < p < 1. 77*é?« £>!/" € ^ # F ^ € ^ IfS'^ G TV f/œw Dj^ G M arcd */S^ £ N,…
THEOREM 1. Suppose/ G SQ with /i(0) = limr_i argf(rew), and 0 < p < 1. 77*é?« £>!/" € ^ # F ^ € ^ IfS'^ G TV f/œw Dj^ G M arcd */S^ £ N, then Djf £ HP for all p>0. PROOF. With/ and p as above, we have that /(z) = ,exp{i/^logr-i— «/„(,)), https://doi.org/10.4153/CMB-1993-007-1 Published online by Cambridge University Press
Corollary 4
Corollary 4, p. 118, S'^/S^ G N+, hence in N. Finally, if ^ £ W, then by the same corollary, S'^/S^ £ 7V+, hence S'^/S^ £ Hp for all/? > 0.…
Corollary 4, p. 118, S'^/S^ G N+, hence in N. Finally, if ^ £ W, then by the same corollary, S'^/S^ £ 7V+, hence S'^/S^ £ Hp for all/? > 0. The proof is complete. • For purposes of orientation let us note the following: first, for all/ G S*, \ogf(z)/z G flpoo #p- N o w iff € 5*, D[f = \ $" ^f^ where \i is the boundary argument function for/, where // need not be singular [14], pp. 209-210; thus D}f G C\P<\ Hp- Recall that by a theorem of Hardy and Littlewood ([6] p. 88) g' G Hp, 0 < p < 1 implie$
COROLLARY 1.
COROLLARY 1. Suppose that f G SQ, fi is continuous and uj^ t) = 0(ta), for some a, 0 < a < 1. Then Djf (£ H^. Furthermore, this is best…
COROLLARY 1. Suppose that f G SQ, fi is continuous and uj^{t) = 0(ta), for some a, 0 < a < 1. Then Djf (£ H^. Furthermore, this is best possible in the sense that for each a, 0 < a < 1, there is an fa G SQ with argument function fia such that uj,a(t) = Oif) andf'a G Wforallp < ^ . PROOF. Suppose/ G SQ with /i continuous and u^{f) = 0(ta), for some a, 0 < a < 1. Then if 5^ is the associated singular inner function, we have by [1] p. 341 that S^ ^ /J2=§, so by our Theorem 1 the first statement fol
COROLLARY 2.
COROLLARY 2. Suppose/ G S tfftd d/i is supported on E, where [0,27r] — E — Uk(ak,bk), andpiO) = limr_^i arg/(r^). Lef 0 < 7 < 1. 77ien «)…
COROLLARY 2. Suppose/ G S$ tfftd d/i is supported on E, where [0,27r] — E — Uk(ak,bk), andpiO) = limr_^i arg/(r^). Lef 0 < 7 < 1. 77ien «) #"E* \h - a*|7 < oo, then Djf G 7/1?. fcJ //E* |&* - flikl log | ^ < oo, rten D£/ G M PROOF. Both of these results follow immediately from [5] Theorem 1, p. 284 and our Theorem 1. • The corollary may be viewed as saying that the faster the lengths of intervals of con- stant boundary argument for/ go to zero, the better the behavior of Djf. Finally, we turn to$
COROLLARY 3.
COROLLARY 3. Suppose/ G SQ and p,(0) = limr_+i arg/(rel9) is purely atomic with jumps nXk at ak. Let 0 < 7 < 1/2. Then i/T,k^l < °°> we…
COROLLARY 3. Suppose/ G SQ and p,(0) = limr_+i arg/(rel9) is purely atomic with jumps nXk at ak. Let 0 < 7 < 1/2. Then i/T,k^l < °°> we nave ^if ^ HP> for a^ p<l/2. PROOF. This follows from [4], see also [1] p. 346. • This corollary may be viewed as saying that if the wedge arguments in the image of / go to zero "faster than 1/fc2", then the behavior of Djf is the best possible over the class SQ. It is perhaps interesting in this regard that ([14] p. 211 ) if a is the largest wedge argument, the
THEOREM 2.
THEOREM 2. Suppose f G SQ. Then for any (3 > 0 there is a constant C = C(f3,f) such that J 6: ^ argf(reie) > /3j > Cy/ — r as r —• 1. If in…
THEOREM 2. Suppose f G SQ. Then for any (3 > 0 there is a constant C = C(f3,f) such that J 6 : ^ argf(reie) > /3j\ > Cy/\ — r as r —• 1. If in addition f has continuous boundary argument p with modulus of continuity uj^i) = 0(ta) for some a, 0 < a < 1, then for any (3 there is a constant C(f, /3) such that : ^ arg f(rei9)>f3 > C ( l - r ) ^ , asr- PROOF. This is [2] Theorems 4 and 5; see also [1] for further details on the calcula- tion of the quantity 8(r). m REFERENCES 1. P. Ahern, The Mean Mo