Lobachevskii Journal of Mathematics Vol. 11, 2002, 7-12 \begin{matrix}c \\ ◯ \end{matrix}F. G. Avkhadiev, K.-J. Wirths

F. G. Avkhadiev, K.-J. Wirths

ON THE COEFFICIENT MULTIPLIERS THEOREM OF HARDY AND LITTLEWOOD

Abstract. Let a_{n}(f) be the Taylor coefficients of a holomorphic function f which belongs to the Hardy space H^{p}, 0 < p < 1. We prove the estimate C(p) ≤ πe^{p}/[p(1 − p)] in the Hardy-Littlewood inequality

∑_{n=0}^{∞} \frac{|a_{n}(f)|^{p}}{(n + 1)^{2−p}} ≤ C(p)(∥f∥_{p})^{p}.

We also give explicit estimates for sums ∑ |a_{n}(f)λ_{n}|^{s} in the mixed norm space H(1, s, β). In this way we obtain a new version of some results by Blasco and by Jevtic̆ and Pavlovic̆.

1. Introduction

In the present paper we consider functions

f(z) = ∑_{n=0}^{∞} a_{n}(f)z^{n}, |z| < 1,

holomorphic in the unit disk D. Let ∥f∥_{p} denote the Hardy norm of f ∈ H^{p} defined by

∥f∥_{p} = sup{M_{p}(f, r) : 0 < r < 1},

where

M_{p}(f, r) = (\frac{1}{2π} ∫_{0}^{2π} |f(re^{iθ})|^{p}dθ)^{1/p} , 0 < p < ∞.

A function f holomorphic in D belongs to the mixed norm space H(p, q, α) by definition if and only if

M_{p,q,α}(f) = (∫_{0}^{1}(1 − r)^{qα−1}M_{p}^{q}(f, r)dr)^{1/q} < ∞,  (1)

where 0 < p, q, α < ∞.

Conditions of type (1) were also considered by Hardy and Littlewood and especially by T. M. Flett (compare [6]). The space H(p, q, α) is frequently considered in papers on coefficient multipliers. Concerning the vast literature on this theme we refer to the paper [5] by P. L. Duren and A. L. Shields, to the survey article [3] by D. M. Campbell and R. J. Leach as well as to the paper [2] by O. Blasco and the recent paper [7] by M. Jevtic̆ and M. Pavlovic̆.

The first author was supported by the Deutsche Forschungsgemeinschaft and by the Russian Fund of Basic Research, grant 02-01-00168 .

We are concerned with the following two results of Hardy and Littlewood (see [4]). Let f ∈ H^{p}. If 0 < p < 2, then there exists a constant C(p), dependent only on p , such that

∑_{n=0}^{∞} \frac{|a_{n}(f)|^{p}}{(n + 1)^{2−p}} ≤ C(p) (∥f∥_{p})^{p} .  (2)

If q > p > 0, then there exists a constant C(p, q), dependent only on p and q, such that

∫_{0}^{1}(1 − r)^{−p/q}M_{q}^{p}(f, r)dr ≤ C(p, q) (∥f∥_{p})^{p} .  (3)

In [1] we proved the asymptotically sharp estimate

C(p, q) ≤ \frac{π}{1 − p/q} .  (4)

The purpose of the present paper is to obtain an explicit estimate of C(p) in the case 0 < p < 1. We will show that it is possible to do this using the estimate (4) and a refined version of some proofs given by P. L. Duren and A. L. Shields in their basic paper [5] on coefficient multipliers. Namely, we prove an explicit estimate for the sum

∑_{n=0}^{∞} |a_{n}(f)λ_{n}|^{s}

provided β > 0, 0 < s < ∞, f ∈ H(1, s, β) and

Λ_{s,β} = sup_{N≥0} \frac{∑_{n=0}^{N}(n + 1)^{s(1+β)}|λ_{n}|^{s}}{(N + 1)^{s}} < ∞.  (5)

Using this technical result we also give a new version of some results by O. Blasco [2] and by M. Jevtic̆ and M. Pavlovic̆ [7] on coefficient multipliers in H(p, q, α) for 0 < p ≤ 1, 0 < α < ∞ and 0 < q < ∞.

2. Results

The explicit form of (2) for 0 < p < 1 is given by the following theorem.

Theorem 1. If f ∈ H^{p}, 0 < p < 1, then

∑_{n=0}^{∞} \frac{|a_{n}(f)|^{p}}{(n + 1)^{2−p}} ≤ \frac{πe^{p}}{p(1 − p)} (∥f∥_{p})^{p} .  (6)

Our main technical result is the following assertion.

Theorem 2. Let Λ_{s,β} be defined by (5), where (λ_{n}) is a complex sequence. If f ∈ H(1, s, β), β > 0 and 0 < s < ∞ then

∑_{n=0}^{∞} |a_{n}(f)λ_{n}|^{s} ≤ se^{s}(1 + β)Λ_{s,β}M_{1,s,β}^{s}(f).  (7)

Using Theorem 2 and some results by T. M. Flett [6], we obtain

Theorem 3. Suppose that 0 < p < 1, 0 < α < ∞ and 0 < q ≤ s < ∞. Then

∑_{n=0}^{∞} |a_{n}(f)λ_{n}|^{s} < ∞

for all f ∈ H(p, q, α) if and only if

∑_{n=0}^{N} n^{s(α+1/p)}|λ_{n}|^{s} = O(N^{s}).  (8)

It is easy to show that Theorem 3 is equivalent to the known results from [2] and [7]. To do this we write (8) in the form

∑_{n=0}^{N} |μ_{n}|^{s} = O(N^{s}),

where μ_{n} = n^{α+1/p}λ_{n}. One has

sup_{N≥1} \frac{∑_{n=1}^{N} |μ_{n}|^{s}}{N^{s}} ≤ \frac{2^{s}}{1 − 2^{−s}} sup_{N≥1} ∑_{n=2^{N−1}}^{2^{N} −1} |\frac{μ_{n}}{n} |^{s}

and

sup_{N≥1} ∑_{n=2^{N−1}}^{2^{N} −1} |\frac{μ_{n}}{n} |^{s} ≤ 2^{s} sup_{N≥1} \frac{∑_{n=1}^{N} |μ_{n}|^{s}}{N^{s}} .

Hence, (8) is equivalent to the condition

sup_{N≥1} ∑_{n=2^{N−1}}^{2^{N} −1} |\frac{μ_{n}}{n} |^{s} < ∞,

which means

(μ_{n}/n) = (n^{α−1+1/p}λ_{n}) ∈ l(s, ∞)

by definition of the sequence space l(s, ∞) (see, for instance, [2] and [7]). We note that a look on the proof of Theorem 4.3 in [2] reveals a misprint during the use of the so-called duality principle. This misprint in turn causes a slight error in the formulation of the theorem. It is easy to see that the complicated value of r in the theorem has to be replaced by s. Concerning this, see also [7].

3. Proofs

Proof of Theorem 2 Denote

m(r) = se^{s}(1 + β)(1 − r)^{s(1+β)−1}M_{1}^{s}(f, r).

Taking

1 − r_{n} = \frac{1}{n + 1}

and using the Cauchy estimate

|a_{n}(f)| ≤ \frac{1}{2πr^{n}} ∫_{0}^{2π} |f(re^{iθ})|dθ, 0 < r < 1,

we obtain

\frac{|a_{n}(f)|^{s}}{(n + 1)^{s(1+β)}} ≤ \frac{(1 − r_{n})^{s(1+β)}M_{1}^{s}(f, r_{n})}{r_{n}^{ns}} ≤ ∫_{r_{n}}^{1} m(r)dr  (9)

since

\frac{1}{r_{n}^{ns}} = (1 + \frac{1}{n})^{ns} ≤ e^{s}

and M(f, r) is a nondecreasing function of r.

Let S_{n} be the sum

∑_{k=0}^{n}(k + 1)^{s(1+β)}|λ_{k}|^{s}.

Using (9) and summing by the Abel transform, we get

∑_{n=0}^{∞} |a_{n}(f)λ_{n}|^{s} ≤ ∑_{n=0}^{∞}(n + 1)^{s(1+β)}|λ_{n}|^{s} ∫_{r_{n}}^{1} m(r)dr =

S_{0} ∫_{0}^{1} m(r)dr + ∑_{n=1}^{∞}(S_{n} − S_{n−1}) ∫_{r_{n}}^{1} m(r)dr = ∑_{n=0}^{∞} S_{n} ∫_{r_{n}}^{r_{n+1}} m(r)dr.

Since

S_{n} ≤ (n + 1)^{s}Λ_{s,β} = \frac{Λ_{s,β}}{(1 − r_{n})^{s}}

and the function (1 − r)^{−s} is increasing in (0, 1), one obtains

∑_{n=0}^{∞} |a_{n}(f)λ_{n}|^{s} ≤ Λ_{s,β} ∑_{n=0}^{∞} ∫_{r_{n}}^{r_{n+1}}(1 − r)^{−s}m(r)dr =

se^{s}(1 + β)Λ_{s,β}M_{1,s,β}^{s}(f).

This completes the proof of Theorem 2.

Proof of Theorem 1 If f ∈ H^{p}, 0 < p < 1, then f ∈ H(1, p, 1/p − 1) according to definition (1) and Hardy-Littlewood’s inequality (3). Using (3) and (4), one has

M_{1,p,1/p−1}^{p}(f) ≤ \frac{π}{1 − p} (∥f∥_{p})^{p} .

Using this as well as (7) for s = p, β = 1/p − 1 and λ_{n} = (n + 1)^{p−2}, we get

∑_{n=0}^{∞} \frac{|a_{n}(f)|^{p}}{(n + 1)^{2−p}} ≤ \frac{πe^{p}}{1 − p}Λ (∥f∥_{p})^{p} ,

where

Λ = sup_{n≥0} \frac{1}{(n + 1)^{p}} ∑_{k=0}^{n} \frac{1}{(k + 1)^{1−p}} ≤ \frac{1}{p}.

These two inequalities complete the proof of Theorem 1.

Proof of Theorem 3 The sufficiency of condition (8) is a consequence of Theorem 2 and of the following assertions: Let 0 < p ≤ 1, 0 < q, α < ∞ and β = α + 1/p − 1. If f ∈ H(p, q, α) then

(a) f ∈ H(1, q, β),
(b) (1 − r)^{β}M_{1}(f, r) = o(1) (r → 1 − 0)
(c) f ∈ H(1, s, β) for s ≥ q.

These may be proved as follows (a): The embedding

H(p, q, α) ⊂ H(1, q, β)

for 0 < p ≤ 1 is due to T. M. Flett (compare [6], Lemma 4). It follows from the Hardy and Littlewood inequality that

M_{1}(f, r) ≤ C(1 − r)^{1−1/p}M_{p}(f, r), 0 < r < 1,  (10)

where C is a constant. (b): T. M. Flett (compare [6], Lemma 3) also proved

(1 − r)^{α}M_{p}(f, r) = o(1) (r → 1 − 0).  (11)

From (10) and (11) we obtain (b). (c): If we apply (a) and (b) together with the elementary inequality

∫_{0}^{1}(1 − r)^{sβ−1}M_{1}^{s}(f, r) dr ≤

sup_{0≤r<1} ((1 − r)^{β}M_{1}(f, r))^{s−q} ∫_{0}^{1}(1 − r)^{qβ−1}M_{1}^{q}(f, r) dr

to a function f ∈ H(p, q, α), we immediately get the assertion.

The necessity of condition (8) may be proved in the usual way (compare [5]). If

∑_{n=0}^{∞} |a_{n}(f)λ_{n}|^{s} < ∞

for all f ∈ H(p, q, α) then by an application of the closed graph theorem we get the existence of a constant C such that

(∑_{n=0}^{∞} |a_{n}(f)λ_{n}|^{s})^{1/s} ≤ C M_{p,q,α}(f)

for all f ∈ H(p, q, α) and all sequences (λ_{n}), which are multipliers from the space H(p, q, α) to the space ℓ^{s}.Choosing

f_{t}(z) = (1 − tz)^{−α−1−1/p} = ∑_{n=0}^{∞} a_{n}(t)z^{n}

we have

|a_{n}(t)| = O(n^{α+1/p}t^{n}),
M_{p}^{p}(f_{t}, r) = O ((1 − tr)^{p(1+α)})

and

(M_{p,q,α}(f_{t}))^{q} = O (∫_{0}^{1} \frac{(1 − r)^{qα−1}}{(1 − r)^{qα+q}} dr) = O((1 − t)^{−q}).

For references concerning these formulae compare for instance [2], [4] and [5]. Terminating the above series at n = N and taking 1 − t = 1/n we get the assertion.

12 F. G. AVKHADIEV, K.-J. WIRTHS

References

[1] F. G. Avkhadiev and K.-J. Wirths, Asymptotically sharp bounds in the Hardy-Littlewood inequalities on mean values of analytic functions, Bull. London Math. Soc. 33 (2001), 695-700.

[2] O. Blanco, Multipliers on spaces of analytic functions, Can. J. Math. 47 (1995),44-64.

[3] D. M. Campbell and R. J. Leach, A Survey of H^{p} Multipliers as related to classical Function Theorie, Complex Var. 3 (1984), 85-111.

[4] P. L. Duren,Theorie of H^{p} spaces. Academic Press, New York /San Francisko /London, 1970.

[5] P. L. Duren and A. L. Shields, Coefficient Multipliers of H^{p} and B^{p} spaces, Pacific J. Math. 32 (1970), 69-78.

[6] T. M. Flett, The dual of an inequality of Hardy and Littlewood and some related Inequalities, J. Math. Anal. Appl. 38 (1972), 746-765.

[7] M. Jevtic̆ and M. Pavlovic̆, Coefficient multipliers on spaces of analytic functions, Acta Sci. Math. (Szeged) 64 (1998), 531-545.

F. G. Avkhadiev, Chebotarev Research Institute, Kazan State University R-420008, Kazan, Russia E-mail address: favhadiev@ksu.ru K.- J. Wirths, Institut für Analysis, Technische Universität, D-38106 Braunschweig, Germany E-mail address: kjwirths@tu-bs.de

Received November 26, 2002