By M. Potápov - V. Alexándrov - P. Pasichenko

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**Extra info for Álgebra y Análisis de Funciones Elementales**

**Example text**

11) = 8k + o(~), where now h E 1i(8). 4, to conclude the proof it suffices to show that ~p = hp + o(~), £ = 0,1, ... , n - 1, and ~ = O(h). Note however that the second relation follows from the first, and from the fact proved earlier that h- p - ~_p = o(~) for £ = 1,2, ... , s. 9) by 8 we get n-1 Ch - L Ry£~p = R8k p=o and + o(~), n-1 8U~g - L p=o 8Xp~p - R8k = o(~), These two relations can be combined to yield n-1 8U~g - L p=o Since 8Xp~p - RYp~p = -Cp~p, n-1 8Xp~p - Ch + L p=o Ry£~p = o(~). we obtain n-1 Ch - L Cp~p p=o = 8U~g + o(~).

BERCOVICI f3 D. VorCULESCU 40 PROOF. Denote by u the Poisson integral of the function IIIP, and observe that Jensen's inequality implies that IF(z)IP ::; u(z), z E C+. We conclude that IG(z)IP ::; u(w(z)), z E C+. 1, u 0 w is the Poisson integral of a measure with total mass ::; J~oo II(t)IP dt = 1I111~, so that J~oo u(w(x + iy)) dx ::; Ilfll~, y > o. Therefore J~oo IG(x + iy)IP dx ::; IIIII~, y > 0, and this implies the result in view of the remarks preceding the statement of the proposition. 2 is that we also have (with the notation of the proposition) IIGyilp ::; IlFyllp for all y > o.

3. L is o:-Holder with constant c. LB3v is also o:-Holder with constant::::; c. PROOF. L(x - h) for all x.

### Álgebra y Análisis de Funciones Elementales by M. Potápov - V. Alexándrov - P. Pasichenko

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