On Hilbert's Inequality for Double Series and Its Applications

Recommended by Feng Qi This study shows that a refinement of the Hilbert inequality for double series can be established by introducing a real function ux and a parameter λ. In particular, some sharp results of the classical Hilbert inequality are obtained by means of a sharpening of the Cauchy inequality. As applications, some refinements of both the Fejer-Riesz inequality and Hardy inequality in H p function are given.


Introduction
Let {a n } and {b n } be two sequences of complex numbers.If λ 0, 1, then where the constant factor π is the best possible.It is all known that the inequalities 1.1 and 1.2 are called the Hilbert theorem for double series.The two forms 1.1 and 1.2 on the Hilbert inequality were combined into one similar form in some papers e.g., 1, 2 , etc. , International Journal of Mathematics and Mathematical Sciences that is, Recently, the various extensions on 1.1 appeared in some papers e.g., 3, 4 , etc. .They focalize on changing the denominator of the function of the left-hand side of 1.1 .Such as the denominator m n λ is replaced by α m β n μ in 3 , and the denominator m n λ is replaced by mu m nv n μ in 4 .Some new results in these papers were yielded.The inequality 1.3 is obviously a significant refinement of 1.1 and 1.2 .However, both extensions and refinements on 1.2 and 1.3 do not commonly appear in previous papers.The main purpose of the present paper is to establish both an extension and a significant refinement on 1.3 .Explicitly, let u x > 0x ∈ 0, ∞ be a real function and let lim x→∞ u x ∞.If the denominator m n λ of the first term of the left-hand side of 1.3 is replaced by u m u n , and the denominator m − n of the second term of the left-hand side of 1.3 is replaced by u m − u n , then a new inequality established is significant in theory and applications; and as applications, we will give both extensions and refinements on Fejer-Riesz's inequality and Hardy's inequality.For convenience, we introduce some notations and functions as follows: |α| 2 dt, where α f, g, h.

1.4
In particular, when b a, the notations U k a, a and V a, a are denoted, respectively, by U k a and V a .Throughout this paper, we will frequently use these notations, and we stipulate that Z denotes integer set and that u n Z n λ/2, where Z n ∈ Z, n ∈ N 0 , λ is an integer or 0 < λ < 1.

Lemmas
In order to prove our assertions, we need the following lemmas.The proof of it has been given in the paper 2 .Hence, it is omitted here.Lemma 2.2.Let f, g, h ∈ L 2 0, 2π , and let h be a variable unit-vector.Then,

2.1
In particular, when h is orthogonal to f, we have Proof.When λ is an integer, it is clear that r a r b 0. So we consider only the case for 1/2 ≤ λ < 1.It is easy to deduce that

2.5
When 1/2 ≤ λ < 1, it follows from 2.3 that r x < 0 for any x∈ C. Hence, we have r a r b >0.
International Journal of Mathematics and Mathematical Sciences Thereby, the relation 2.6 holds.

Theorems and their corollaries
In order to prove our assertions, we need also to introduce the following functions:

3.1
Theorem 3.1.Let r x be a function defined by 2.3 , let {a n } and {b n } be two nonzero sequences of complex numbers, and let both ∞ n 0 a n and ∞ n 0 b n be absolute convergent.Then, 1, 2 is defined by 3.1 .In particular, when 1/2 ≤ λ < 1, we have r a r b > 0.

3.5
Since both ∞ n 0 a n and ∞ n 0 b n are absolute convergent by Lemma 2.1, the double series Accordingly, f a, t g b, t is uniformly convergent in the interval 0, 2π .Thereby, the interchange in order of summation and integration can be made.In what follows, we stipulate that the interchanges in order of summation and integration are justified.It is easy to deduce that where r x is a function defined by 2.3 .By Lemma 2.2, we have h is a variable unit-vector, it can be properly chosen in accordance with our requirement.i When λ is an integer, it is known from 2.3 that r x 0. We select h 1 √ 2t/2π it is easy to deduce that h 1 1, and Since the series ∞ n 0 a n is absolute convergent, it is justified that the complex number a n is replaced by |a n | in 3.8 .Hence, we have

International Journal of Mathematics and Mathematical Sciences
Similarly

3.10
We therefore obtain that if λ is even.

3.11
Hence, the inequality 3.7 can be reduced to , then, interchanging a and b in 3.12 , we have where r 2 is defined by

3.14
Adding 3.12 and 3.13 , then the inequality 3.2 follows after simplifications.ii When 0 < λ < 1, we firstly consider h in 3.7 .We still select unit-vector

3.15
Since the series ∞ n 0 a n and ∞ n 0 b n are absolute convergent, it is justified that the complex numbers a n and b n are replaced, respectively, by |a n | and |b n | in the above relations.Let s 1 x f, h 1 / f , s 2 x g, h 1 / g .By using 3.1 , we find s 1 a ,

3.16
Notice that U 1 b, a U 1 a, b , U 2 b, a U 2 a, b , and V b, a −V a, b .If we still select the unit-vector h 2 √ 2t/2π, then, interchanging a and b in 3.16 , we have 3.17 where Adding 3.16 and 3.17 , the inequality 3.4 can be gotten after simplifications.In particular, when 1/2 ≤ λ < 1, by Lemma 2.3, we have r a r b ≥ 0. The proof of Theorem 3.1 is completed.
In particular, when u n n λ/2, according to 3.2 , one obtains a refinement of 1.3 immediately.

3.25
Since λ 1/2, it is known from Lemma 2.3 that r a r b > 0.
If r a r b and R in 3.24 are replaced by zero, then the inequality 3.24 can be reduced to

3.26
The inequalities 3.24 and 3.26 are refinements of the Hilbert-Ingham inequality ≤ π a b .

3.27
One has yet a new inequality according to Theorem 3.1 ii .

Applications to H P function
where R 2 > 0.
Proof.At first, we prove the theorem for case

International Journal of Mathematics and Mathematical Sciences
By Lemma 2.4, we have Since the series ∞ n 0 a n is absolutely convergent, it is justified that the complex number a n is replaced by |a n |.
According 3.18 , we have where R 2 is defined by 3.19 .It is easy to deduce that 2 Let F z G z p/2 ∈ H 2 .According to the above result for p 2, we have

4.7
Based on the case for p 2, we have R 2 F > 0. Hence, R 2 > 0. The proof of Theorem 4.1 is completed. Let 4.9 It is called the Hardy inequality in H p function 7 .
We will give both an extension and a refinement of 4.9 as follows. where if λ is even.

4.11
Proof.By Blaschke decomposition theorem, we have where B z is Blaschke function,

4.13
Owing to f z f 1 z f 2 z , it holds that By Lemma 2.4, we find that

4.17
These show that the inequality 4.10 is valid.

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Lemma 2 . 1 .m 0 ∞n
If both ∞ n 0 a n and ∞ n 0 b n are absolute convergent, then i ∞ 0 a m b n is absolute convergent, ii a 2 and b 2 are convergent.
Hence, the inequality 4.2 is valid when p 2. If p / 2, then by the Blaschke decomposition theorem, it holds that f z B z G z , where B z is Blaschke function and G z / 0, |B z | ≤ 1 in |z| < 1 and |B e it | 1.

Theorem 4 . 2 .
Let f z ∞ m 0 c m z u m be analytic in the unit-disc |z| < 1, where u m Z m λ/2 (with Z m ∈ Z and λ ∈ N 0 ) and f ∈ H 1 .Then,

First
Round of ReviewsOctober 1, 2009 15 International Journal of Mathematics and Mathematical Sciences It follows from 4.13 , 4.14 , 4.15 , and Corollary 3.2 that