# An extension of the Hardy-Littlewood inequality

An extension of the Hardy-Littlewood inequality

MATEMATICKI VESNIK

6(19)(34), 1982

M. Mateljevic

M. Pavlovic

II AN EXTENSION OF THE HARDY-LITTLEWOOD INEQUALITY

1. I NTRODUCT ION

A function f regular in ~={z:lzl

56 H. Mat el j evic and M. Pavl ovi c .

Holland a nd Twome y [9) have a l so i ntroduced t h e s pace

t;. I f p < ~ then I i c onsi s t s of a l l f r egu l a r i n A f or which ( 1 . 2 )

hol d s , while fEl : P i f a nd o n l y i f t k l ak l 2 < ~ . Thus 11=A w i s t he

Di r ichlet space . I n [1 01 t h e space DP i s defined i n a simi lar way

b y ( 1 .4). The s pace s AP , t ffi and DP (wit h o b v i ous norms )are Bana c h

s p aces i f p ~l a nd (non- l o c a l ly convex ) F- s p a c e s i f o

An Extens i on of the Hardy- Li t t lewood I ne guality 57

D ( b S) cD ( p ~ q ) s i n c e Sn( q lf)~ S n (s ~f) . On the other han d if

! ( z)=r.b a 2 " wher e III 1l = zn U - p +p/ s )

n

n

t hen fED(p ~q ) . wh i l e ftf. D ( l ~ 8 ) . Thus o ur resu l t i mprov e s the Ho l l a nd ­

Twomey theorem.

Note that a l l inclus i ons whic h oc cur i n The or em 1 a r e prope

r . For e xamp l e , i f

2 2 "

! (z) =r. n- z

the n fEli "" and f\$ D (p ~q) f or zcpc» a nd p-! +q - l

M. Mat el jevi c and M. Pavlovi c

A proof ma y be f o un d in ( 111 •

(Hardy- Litt l ewood) . I f fe H r ~ O < P < A < ~ , then

t (l -P ) -P /AMA (r ~f )P dr < ~ .

Fo r t h e proof see [5 ~ Th e o rem 5 . 11J .

LErt1l'-1A 3 (Lr l s l zl s l aol ) • Let (an ) ~ n>-:l , be a seque nc e o f non-negat i ve

numbers , p> o ~ a>o . Then the f ollowing a s sertions a r e equ i val ent:

( a)

(b)

(e)

1 a -I ~ n p

f (l - x) ( E a x ) dx

An Ex'tension of the Hardy-Littlewood Inequality

and 2.

(b) Let fEH P, 1~p

An Ex'tension of the Hardy-Littlewood Inequality

and 2.

(b) Let fEH P, 1~p

__._~_An_E_·x_t_e_n_s!~mof the IIar~IJittlewo()d.lnequa~,...Y

",;;.,;;;;o 61

REF ERE NeE S

1. R.Askey, "L P behaviour of power series with positive

coefficients ", Proc.Amer.Math.Soc,19(1968), 303-305.

2. R.Askey and R.P.Boas, Jr., "Some integrability theorems

for power series with positive coefficient.s II ,Mathematical

Essays dedicated to A.J.Macintyre (Ohio University Press, 1970).

3. J.M.Anderson and A.L.Shields, "Coefficients multipliers

of Bloch functions", Trans.Anler.Math.Soc.,224 (1976) ,255-265.

4. A.Baernstein, "Analytic functions of bounded mean

oscillation", Aspects of Contemporary Complex Analysis edited by

D.A.Brannan and j.G.Clunie (Academic Press, 1980).

5. P.L.Duren, Theory of H P spaces (Academic Press,1970).

6. G.B.Hardy and J.E.Littlewood, "Some new properties of

Fourier constants", Math.Ann. 97(1926), 159-209.

7. G.H.Hardy and J.E.Littlewood, "Some properties of

fractional i~tegrals II", Math.Z., 34(1931), 403-4~9.

8. F.Holland and J.B.Twomey, "On Hardy classes and the

area function", J.London Math.Soc., 17(1978), 275-283.

9. F.Holland and J.B.TwomeYi "Conditions for membership

of Hardy spaces", Aspects of Contemporary Complex Analysis edited

by D.A.Drannan and J.G.Clunie (Academic Press, 1980).

10. M.Mateljevi6 and M.Pavlovi6, "LP behaviour of power

series with positive coefficients and Hardy spaces" (to appear).

11. A.Zygmund, Trigonometric series (Cambridge Dniv.Press,

London and New York, 1959).

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