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In [[mathematics]], the '''Littlewood subordination theorem''', proved by [[J. E. Littlewood]] in 1925, is a theorem in [[operator theory]] and [[complex analysis]]. It states that any [[Holomorphic function|holomorphic]] [[univalent function|univalent]] self-mapping of the [[unit disk]] in the [[complex numbers]] that fixes 0 induces a [[contraction operator|contractive]] [[composition operator]] on various [[function space]]s of holomorphic functions on the disk. These spaces include the [[Hardy space]]s, the [[Bergman space]]s and [[Dirichlet space]].
==Subordination theorem==
Let ''h'' be a holomorphic univalent mapping of the unit disk ''D'' into itself such that ''h''(0) = 0. Then the
:<math>C_h(f) = f\circ h</math>
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:<math> \|f\|_{A^p}^p = {1\over \pi} \iint_D |f(z)|^p\, dx\,dy</math>
:<math> \|f\|_{\mathcal D}^2 = {1\over \pi} \iint_D |f^\prime(z)|^2\, dx\,dy= {1\over 4 \pi} \iint_D |\partial_x f|^2 + |\partial_y f|^2\, dx\,dy</math>
==Littlewood's inequalities==
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This inequality also holds for 0 < ''p'' < 1, although in this case there is no operator interpretation.
==Proofs==
===Case ''p'' = 2===
To prove the result for ''H''<sup>2</sup> it suffices to show that for ''f'' a polynomial<ref>{{harvnb|Nikolski|2002|
:<math>\displaystyle{\|C_h f\|^2 \le \|f\|^2,}</math>
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Since ''U''*''f'' has degree less than ''f'', it follows by induction that
:<math>\|C_h U^*f\|^2 \le \
and hence
:<math>\|C_h f\|^2 \le \|f\|^2.</math>
The same method of proof works for ''A''<sup>2</sup> and <math>\mathcal D.</math>
===General Hardy spaces===
If ''f'' is in Hardy space ''H''<sup>''p''</sup>, then it has a [[Hardy space#Factorization into inner and outer functions (Beurling)|factorization]]<ref>{{harvnb|Nikolski|2002|p=57}}</ref>
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:<math> \|C_h f\|_{H^p} \le \|(C_hf_i) (C_h f_o)\|_{H^p} \le \|C_h f_o\|_{H^p} \le \|C_h f_o^{p/2}\|_{H^2}^{2/p} \le \|f\|_{H^p}.</math>
===Inequalities===
Taking 0 < ''r'' < 1, Littlewood's inequalities follow by applying the Hardy space inequalities to the function
:<math> f_r(z)=f(rz).</math>
The inequalities can also be deduced, following {{harvtxt|Riesz|1925}}, using [[subharmonic function]]s.<ref>{{harvnb|Duren|1970}}</ref><ref>{{harvnb|Shapiro|1993|p=19}}</ref> The inequaties in turn immediately imply the subordination theorem for general Bergman spaces.
==Notes==
{{reflist|2}}
==References==
*{{citation|last=Duren|first= P. L.|title=Theory of H <sup>p</sup> spaces|series=Pure and Applied Mathematics|volume= 38|
publisher= Academic Press|year=1970}}
*{{citation|last=Littlewood|first=J. E.|title=On inequalities in the theory of functions|journal=Proc.
*{{citation|last=Nikolski|first=N. K.|title=Operators, functions, and systems: an easy reading. Vol. 1. Hardy, Hankel, and Toeplitz|series= Mathematical Surveys and Monographs|volume= 92|publisher= American Mathematical Society|year= 2002|
*{{citation|first=F.|last=Riesz|title=Sur une inégalite de M. Littlewood dans la théorie des fonctions|journal= Proc. London Math. Soc.|volume= 23|year=1925|pages= 36–39|doi=10.1112/plms/s2-23.1.1-s }}
*{{citation|last=Shapiro|first=J. H.|title=Composition operators and classical function theory|series=Universitext: Tracts in Mathematics|publisher= Springer-Verlag|year= 1993|
[[Category:Operator theory]]
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