On Hilbert's Inequality for Double Series and Its Applications

Abstract

This study shows that a refinement of the Hilbert inequality for double series can be established by introducing a real function 𝑢(𝑥) 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.

1. Introduction

Let {𝑎_𝑛} and {𝑏_𝑛} be two sequences of complex numbers. If 𝜆=0,1, then |^∞∞_{𝑚=1−𝜆𝑛=1−𝜆}𝑎_𝑚𝑏_𝑛𝑚+𝑛+𝜆|≤𝜋(^∞_{𝑛=1−𝜆}|𝑎_𝑛|²)^1/2(^∞_{𝑛=1−𝜆}|𝑏_𝑛|²)^1/2|_{∞∞𝑚=1−𝜆𝑛=1−𝜆𝑚≠𝑛},𝑎_𝑚𝑏_𝑛𝑚−𝑛|≤𝜋(^∞_{𝑚=1−𝜆}|𝑎_𝑛|²)^1/2(^∞_{𝑛=1−𝜆}|𝑏_𝑛|²)^1/2, (1.1)(1.2) 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.), that is, |^∞∞_{𝑚=1−𝜆𝑛=1−𝜆}𝑎_𝑚𝑏_𝑛|𝑚+𝑛+𝜆²+|^∞∞_{𝑚=1−𝜆𝑛=1−𝜆𝑚≠𝑛}𝑎_𝑚𝑏_𝑛|𝑚−𝑛²≤𝜋^2∞_{𝑛=1−𝜆}|𝑎_𝑛|^2∞_{𝑛=1−𝜆}|𝑏_𝑛|². (1.3) 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: 𝑉(𝑎,𝑏)=^∞∞_{𝑚=0𝑛=0𝑢(𝑚)≠𝑣(𝑛)}𝑎_𝑚𝑏_𝑛,𝑈𝑢(𝑚)−𝑢(𝑛)_𝑘(𝑎,𝑏)=^∞∞_𝑚=0𝑛=0𝑎_𝑚𝑏_𝑛(𝑢(𝑚)+𝑢(𝑛))^𝑘,𝑘=1,2,‖𝑥‖²=^∞_𝑛=0|𝑥_𝑛|²,𝑇_𝑘(𝑥)=^∞_𝑛=0|𝑥_𝑛|𝑢^𝑘(𝑛),𝑘=1,2,(𝑓,𝑔)=₀^2𝜋𝑓(𝑡)𝑔(𝑡)𝑑𝑡,‖𝛼‖²=₀^2𝜋|𝛼|²𝑑𝑡,where𝛼=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∈𝑁₀,𝜆 is an integer or 0<𝜆<1.

2. Lemmas

In order to prove our assertions, we need the following lemmas.

Lemma 2.1. If both ∑^∞_𝑛=0𝑎_𝑛 and ∑^∞_𝑛=0𝑏_𝑛 are absolute convergent, then (i)∑^∞_𝑚=0∑^∞_𝑛=0𝑎_𝑚𝑏_𝑛 is absolute convergent,(ii)‖𝑎‖² and ‖𝑏‖² are convergent.

The proof of it has been given in the paper [2]. Hence, it is omitted here.

Lemma 2.2. Let 𝑓,𝑔,ℎ∈𝐿²[0,2𝜋], and let ℎ be a variable unit-vector. Then, |(𝑓,𝑔)|²≤‖𝑓‖²‖𝑔‖²(1−(|(𝑓,ℎ)|−‖𝑓‖|(𝑔,ℎ)|)‖𝑔‖²). (2.1) In particular, when ℎ is orthogonal to 𝑓, we have |(𝑓,𝑔)|²≤‖𝑓‖²‖𝑔‖²(1−(|(𝑔,ℎ)|)‖𝑔‖²); (2.2) and the equality in (2.2) holds if and only if 𝑓,𝑔 and ℎ are linearly dependent.

The proof of these results has been given in [5, 6].

Define a function by 𝑟(𝑥)=𝜋𝑈₁(𝑥)sin2𝜆𝜋−𝑈₂(𝑥)sin²𝜆𝜋. (2.3)

Lemma 2.3. Let {𝑎_𝑛} and {𝑏_𝑛} be two sequences of complex numbers. If 𝜆 is an integer or 1/2≤𝜆<1, then 𝑟(𝑎)𝑟(𝑏)≥0. (2.4)

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 𝑈₁(𝑥)=¹₀|^∞_𝑚=0𝑥_𝑚𝑡^{𝑢(𝑚)−1/2}|²𝑈𝑑𝑡>0,₂(𝑥)=¹₀𝑑𝑠𝑠^𝑠₀|^∞_𝑛=0𝑥_𝑛𝑡^{𝑢(𝑛)−1/2}|²𝑑𝑡>0. (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.

Lemma 2.4. Let 𝑓(𝑧)=∑^∞_𝑛=0𝑎_𝑛𝑧^{𝑢(𝑛)−1/2}. If 𝑓(𝑧) is analytic in the unit-disc |𝑧|<1, then |12𝜋^𝜋_−𝜋𝑡|𝑓(−𝑒^𝑖𝑡)|²𝑑𝑡|=|𝑉(𝑎)|. (2.6)

Proof. |^𝜋_−𝜋𝑡|𝑓(−𝑒^𝑖𝑡)|²𝑑𝑡|=|^𝜋_−𝜋𝑡𝑓(−𝑒^𝑖𝑡)𝑓(−𝑒^𝑖𝑡)𝑑𝑡|=|^𝜋_−𝜋𝑡^∞∞_𝑚=0𝑛=0𝑎_𝑚𝑎_𝑛(cos(𝜋+𝑡)+𝑖sin(𝜋+𝑡))^{𝑢(𝑚)−1/2}(cos(𝜋−𝑡)+𝑖sin(𝜋−𝑡))^{𝑢(𝑛)−1/2}𝑑𝑡|=2𝜋|(^∞∞_{𝑚=0𝑛=0𝑢(𝑚)≠𝑢(𝑛)}𝑎_𝑚𝑎_𝑛𝑢(𝑚)−𝑢(𝑛))sin𝜆𝜋+𝑖(^∞∞_{𝑚=0𝑛=0𝑢(𝑚)≠𝑢(𝑛)}𝑎_𝑚𝑎_𝑛𝑢(𝑚)−𝑢(𝑛))cos𝜆𝜋|=2𝜋|𝑉(𝑎)|. (2.7) Thereby, the relation (2.6) holds.

3. Theorems and Their Corollaries

In order to prove our assertions, we need also to introduce the following functions: 𝑠₁√(𝑥)=2|𝑇₁(𝑥)cos𝜆𝜋−(1/2𝜋)𝑇₂(𝑥)sin𝜆𝜋|(𝜋²‖𝑥‖²−𝑟(𝑥))^1/2,𝑠₂√(𝑥)=2|𝑇₁(𝑥)sin𝜆𝜋−(1/𝜋)𝑇₂(𝑥)sin²(𝜆𝜋/2)|(𝜋²‖𝑥‖²+𝑟(𝑥))^1/2. (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=0𝑎_𝑛 and ∑^∞_𝑛=0𝑏_𝑛 be absolute convergent. Then,

(i) if 𝜆 is an integer, then |𝑈₁(𝑎,𝑏)|²+|𝑉(𝑎,𝑏)|²≤(𝜋²‖𝑎‖²‖𝑏‖²)(1−𝑅²), (3.2) where 𝑅²=1𝜋²𝑇{(¹(𝑎)−𝑇‖𝑎‖²(𝑏))𝜋‖𝑏‖²𝑇+(¹(𝑏)−𝑇‖𝑏‖²(𝑎))𝜋‖𝑎‖²1}if𝜆isodd,𝜋²𝑇((¹(𝑎))‖𝑎‖²𝑇+(¹(𝑏))‖𝑏‖²)if𝜆iseven; (3.3)

(ii) if 0<𝜆<1, then |𝑈₁1(𝑎,𝑏)cos2𝜆𝜋−𝑈2𝜋²(𝑎,𝑏)sin2𝜆𝜋|²+|𝑉(𝑎,𝑏)|²≤{𝜋²‖𝑎‖²‖𝑏‖²−1𝜋²𝑟(𝑎)𝑟(𝑏)}(1−𝑅²), (3.4) where 𝑅²=min{(𝑠₁(𝑎)−𝑠₂(𝑏))²,(𝑠₁(𝑏)−𝑠₂(𝑎))²}, 𝑠_𝑖(𝑥)(𝑖=1,2) is defined by (3.1). In particular, when 1/2≤𝜆<1, we have 𝑟(𝑎)𝑟(𝑏)>0.

Proof. Define two functions by 𝑓(𝑎,𝑡)=^∞_𝑚=0𝑎_𝑚√𝑡sin(𝑢(𝑚))𝑡,𝑔(𝑏,𝑡)=^∞_𝑛=0𝑏_𝑛√𝑡cos(𝑢(𝑛))𝑡,𝑡∈[0,2𝜋]. (3.5) Since both ∑^∞_𝑛=0𝑎_𝑛 and ∑^∞_𝑛=0𝑏_𝑛 are absolute convergent by Lemma 2.1, the double series ∑^∞_𝑚=0∑^∞_𝑛=0𝑎_𝑚𝑏_𝑛 is absolute convergent. 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 ‖𝑓‖²=𝜋²‖𝑎‖²−𝑟(𝑎),‖𝑔‖²=𝜋²‖𝑏‖²+𝑟(𝑏),|(𝑓,𝑔)|=|₀^2𝜋𝑓(𝑎,𝑡)𝑔(𝑏,𝑡)𝑑𝑡|=𝜋|𝑉(𝑎,𝑏)+(𝑈₁1(𝑎,𝑏)cos2𝜆𝜋−𝑈2𝜋²(𝑎,𝑏)sin2𝜆𝜋)|, (3.6) where 𝑟(𝑥) is a function defined by (2.3). By Lemma 2.2, we have |𝑉(𝑎,𝑏)+(𝑈₁1(𝑎,𝑏)cos2𝜆𝜋−𝑈2𝜋²(𝑎,𝑏)sin2𝜆𝜋)|²=1𝜋²|(𝑓,𝑔)|²≤1𝜋²‖𝑓‖²‖𝑔‖²=1(1−𝑟)𝜋²{𝜋²‖𝑎‖²−𝑟(𝑎)}{𝜋²‖𝑏‖²+𝑟(𝑏)}(1−𝑟), (3.7) where 𝑟=(|(𝑓,ℎ)|/‖𝑓‖−|(𝑔,ℎ)|/‖𝑔‖)², 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 ℎ₁=√2𝑡/2𝜋0𝑥0𝑓𝑓0𝑐 it is easy to deduce that ‖ℎ₁‖=1, and |(𝑓,ℎ₁)|=|√2(^∞_𝑚=0𝑎_𝑚𝑢(𝑚))|. (3.8) Since the series ∑^∞_𝑛=0𝑎_𝑛 is absolute convergent, it is justified that the complex number 𝑎_𝑛 is replaced by |𝑎_𝑛| in (3.8). Hence, we have |(𝑓,ℎ₁)|=√2𝑇₁(𝑎). (3.9) Similarly |(𝑔,ℎ₁√)|=2𝜋𝑇₂(𝑏)if𝜆isodd,0if𝜆iseven. (3.10)

We therefore obtain that 𝑟₁=(|(𝑓,ℎ₁)|−‖𝑓‖|(𝑔,ℎ₁)|)‖𝑔‖²=2𝜋²(𝑇₁(𝑎)−𝑇‖𝑎‖²(𝑏))𝜋‖𝑏‖²2if𝜆isodd,𝜋²(𝑇₁(𝑎))‖𝑎‖²if𝜆iseven. (3.11) Hence, the inequality (3.7) can be reduced to |𝑉(𝑎,𝑏)+𝑈₁(𝑎,𝑏)|²≤{𝜋²‖𝑎‖²‖𝑏‖²}(1−𝑟₁). (3.12)

Notice that 𝑈₁(𝑏,𝑎)=𝑈₁(𝑎,𝑏) and 𝑉(𝑏,𝑎)=−𝑉(𝑎,𝑏). If we still select the unit-vector ℎ₂=√2𝑡/2𝜋, then, interchanging 𝑎 and 𝑏 in (3.12), we have |−𝑉(𝑎,𝑏)+𝑈₁(𝑎,𝑏)|²≤(𝜋²‖𝑎‖²‖𝑏‖²)(1−𝑟₂), (3.13) where 𝑟₂ is defined by 𝑟₂=(|(𝑓,ℎ₂)|−‖𝑓‖|(𝑔,ℎ₂)|)‖𝑔‖²=2𝜋²(𝑇₁(𝑏)−𝑇‖𝑏‖²(𝑎))𝜋‖𝑎‖²2if𝜆isodd,𝜋²(𝑇₁(𝑏))‖𝑏‖²if𝜆iseven. (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 ℎ₁=√2𝑡/2𝜋. It is easy to deduce that |(𝑓,ℎ₁)|=√2|(^∞_𝑚=0𝑎_𝑚1𝑢(𝑚))cos𝜆𝜋−(2𝜋^∞_𝑚=0𝑎_𝑚(𝑢(𝑚))²)sin𝜆𝜋|,|(𝑔,ℎ₁)|=√2|(^∞_𝑛=0𝑏_𝑛1𝑢(𝑛))sin𝜆𝜋−𝜋(^∞_𝑛=0𝑏_𝑛(𝑢(𝑛))²)sin²𝜆𝜋2|. (3.15) Since the series ∑^∞_𝑛=0𝑎_𝑛 and ∑^∞_𝑛=0𝑏_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 𝑠₁(𝑥)=|(𝑓,ℎ₁)|/‖𝑓‖),𝑠₂(𝑥)=|(𝑔,ℎ₁)|/‖𝑔‖. By using (3.1), we find 𝑠₁(𝑎), 𝑠₂(𝑏). Let 𝑅²₁=(𝑠₁(𝑎)−𝑠₂(𝑏))². We obtain from (3.7) |𝑉(𝑎,𝑏)+(𝑈₁1(𝑎,𝑏)cos2𝜆𝜋−𝑈2𝜋²(𝑎,𝑏)sin2𝜆𝜋)|²≤{𝜋²‖𝑎‖²‖𝑏‖²−(‖𝑏‖²𝑟(𝑎)−‖𝑎‖²1𝑟(𝑏))−𝜋²𝑟(𝑎)𝑟(𝑏)}(1−𝑅²₁). (3.16)

Notice that 𝑈₁(𝑏,𝑎)=𝑈₁(𝑎,𝑏), 𝑈₂(𝑏,𝑎)=𝑈₂(𝑎,𝑏), and 𝑉(𝑏,𝑎)=−𝑉(𝑎,𝑏). If we still select the unit-vector ℎ₂=√2𝑡/2𝜋, then, interchanging 𝑎 and 𝑏 in (3.16), we have |−𝑉(𝑎,𝑏)+(𝑈₁1(𝑎,𝑏)cos2𝜆𝜋−2𝜋𝑈₂(𝑎,𝑏)sin2𝜆𝜋)|²≤{𝜋²‖𝑎‖²‖𝑏‖²−(‖𝑎‖²𝑟(𝑏)−‖𝑏‖²1𝑟(𝑎))−𝜋²𝑟(𝑏)𝑟(𝑎)}(1−𝑅²₂), (3.17) where 𝑅²₂=(𝑠₁(𝑏)−𝑠₂(𝑎))². Let 𝑅²=min{𝑅²₁,𝑅²₂}. 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.

Corollary 3.2 . Let ∑^∞_𝑛=0𝑎_𝑛 be absolute convergent. Then, (i)if 𝜆 is an integer, then |𝑈₁(𝑎)|²+|𝑉(𝑎)|²≤(𝜋²‖𝑎‖⁴)(1−𝑅²), (3.18) where 𝑅²=2𝜋²‖𝑎‖²(𝑇₁𝑇(𝑎)−²(𝑎)𝜋)²2𝑖𝑓𝜆𝑖𝑠𝑜𝑑𝑑,𝜋²‖𝑎‖²(𝑇₁(𝑎))²𝑖𝑓𝜆𝑖𝑠𝑒𝑣𝑒𝑛; (3.19)(ii)if 0<𝜆<1, then |𝑈₁1(𝑎)cos2𝜆𝜋−𝑈2𝜋²(𝑎)sin2𝜆𝜋|²+|𝑉(𝑎)|²≤{𝜋²‖𝑎‖⁴−1𝜋²𝑟²(𝑎)}(1−𝑅²), (3.20) where 𝑅²=(𝑠₁(𝑎)−𝑠₂(𝑎))²,𝑠_𝑖(𝑥)(𝑖=1,2) is defined by (3.1). In particular, when 𝑢(𝑛)=𝑛+𝜆/2, according to (3.2), one obtains a refinement of (1.3) immediately.

Corollary 3.3. If 𝜆=0,1, then |∑^∞∞_{𝑚=1−𝜆}∑_{𝑛=1−𝜆}𝑎_𝑚𝑏_𝑛|𝑚+𝑛+𝜆²+|∑^∞∞_{𝑚=1−𝜆}∑_{𝑛=1−𝜆𝑚≠𝑛}𝑎_𝑚𝑏_𝑛|𝑚−𝑛²≤(𝜋∑^2∞_{𝑛=1−𝜆}|𝑎_𝑛|∑^2∞_{𝑛=1−𝜆}|𝑏_𝑛|²)(1−𝑅²), (3.21) where 𝑅²=1𝜋²𝑇{(¹(𝑎)−𝑇‖𝑎‖²(𝑏))𝜋‖𝑏‖²𝑇+(¹(𝑏)−𝑇‖𝑏‖²(𝑎))𝜋‖𝑎‖²1}𝑖𝑓𝜆=1,𝜋²𝑇((¹(𝑎))‖𝑎‖²𝑇+(¹(𝑏))‖𝑏‖²)𝑖𝑓𝜆=0, (3.22) where 𝑇_𝑘(𝑥)=^∞_{𝑛=1−𝜆}|𝑥_𝑛|(𝑛+𝜆/2)^𝑘,𝑘=1,2. (3.23)

Corollary 3.4. If 𝑢(𝑛)=𝑛+1/4, then |(^∞∞_𝑚=0𝑛=0𝑎_𝑚𝑏_𝑛𝑚+𝑛+1/2)|²+|^∞∞_{𝑚=0𝑛=0,𝑚≠𝑛}𝑎_𝑚𝑏_𝑛|𝑚−𝑛²≤{𝜋²‖𝑎‖²‖𝑏‖²−1𝜋²𝑟(𝑎)𝑟(𝑏)}(1−𝑅²), (3.24) where 𝑟(𝑎)𝑟(𝑏)>0,𝑅²=min{(𝑠₁(𝑎)−𝑠₂(𝑏))²,(𝑠₁(𝑏)−𝑠₂(𝑎))²𝑠},₁√(𝑥)=|−(1/2𝜋)𝑇₂(𝑥)|(𝜋²‖𝑥‖²−𝑟(𝑥))^1/2,𝑠₂√(𝑥)=2|𝑇₁(𝑥)−(1/2𝜋)𝑇₂(𝑥)|(𝜋²‖𝑥‖²+𝑟(𝑥))^1/2,𝑇_𝑘(𝑥)=^∞_𝑛=0|𝑥_𝑛|(𝑛+1/4)^𝑘,𝑘=1,2. (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 |(^∞∞_𝑚=0𝑛=0𝑎_𝑚𝑏_𝑛𝑚+𝑛+1/2)|²+|^∞∞_{𝑚=0𝑛=0,𝑚≠𝑛}𝑎_𝑚𝑏_𝑛|𝑚−𝑛²<𝜋²‖𝑎‖²‖𝑏‖². (3.26) The inequalities (3.24) and (3.26) are refinements of the Hilbert-Ingham inequality |^∞∞_𝑚=0𝑛=0𝑎_𝑚𝑏_𝑛𝑚+𝑛+1/2|≤𝜋‖𝑎‖‖𝑏‖. (3.27)

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

Theorem 3.5. With the assumptions as Theorem 3.1, if 𝜆=1/4, then |1𝑈2𝜋²(𝑎,𝑏)|²+|𝑉(𝑎,𝑏)|²≤{𝜋²‖𝑎‖²‖𝑏‖²−1𝜋²𝑟(𝑎)𝑟(𝑏)}(1−𝑅²), (3.28) where 𝑅²=min{(𝑠₁(𝑎)−𝑠₂(𝑏))²,(𝑠₁(𝑏)−𝑠₂(𝑎))²𝑠},₁(𝑥)=|𝑇₁(𝑥)−(1/2𝜋)𝑇₂(𝑥)|(𝜋²‖𝑥‖²−𝑟(𝑥))^1/2,𝑠₂(𝑥)=|𝑇₁(𝑥)−(1/2𝜋)𝑇₂(𝑥)(√2−1)|(𝜋²‖𝑥‖²+𝑟(𝑥))^1/2,𝑇_𝑘(𝑥)=^∞_𝑛=0|𝑥_𝑛|(𝑛+1/8)^𝑘,𝑘=1,2. (3.29)

4. Applications to 𝐻_𝑃 Function

Let 𝑓(𝑧) be analytic in the unit-disc |𝑧|<1. If 𝑓(𝑧)=∑^∞_𝑛=0𝑎_𝑛𝑧^𝑛∈𝐻_𝑝 with 𝑝>0, then ¹₀|𝑓(𝑡)|^𝑝1𝑑𝑡≤2₀^2𝜋|𝑓(𝑒^𝑖𝑡)|^𝑝𝑑𝑡, (4.1) where the coefficient 1/2 is the best possible. It is called the Fejer-Riesz inequality in 𝐻_𝑝 function [7].

We will give both an extension and a refinement of (4.1) in what follows.

Theorem 4.1. Let 𝑓(𝑧)=∑^∞_𝑛=0𝑎_𝑛𝑧^{𝑢(𝑛)−1/2}∈𝐻_𝑝, where 𝑝>0 and 𝑢(𝑛)=𝑍_𝑛+(𝜆/2)(𝑍_𝑛∈𝑍,𝜆∈𝑁₀). If 𝑓(𝑧) is analytic in the unit-disc |𝑧|<1, then |∫¹₀|𝑓(𝑡)|^𝑝𝑑𝑡|²1+|∫2𝜋^𝜋−𝜋𝑡|𝑓(−𝑒^𝑖𝑡)|^𝑝𝑑𝑡|²1≤(2∫₀^2𝜋|𝑓(𝑒^𝑖𝑡)|^𝑝𝑑𝑡)²(1−𝑅²), (4.2) where 𝑅²>0.

Proof. At first, we prove the theorem for case 𝑝=2. Let 𝑓(𝑧)=∑^∞_𝑚=0𝑎_𝑚𝑧^{𝑢(𝑚)−1/2}. It is easy to deduce that ∫¹₀|𝑓(𝑡)|²𝑑𝑡=∑^∞∞_𝑚=0∑_{𝑛=0𝑢(𝑚)+𝑢(𝑛)≠0}𝑎_𝑚𝑎_𝑛𝑢(𝑚)+𝑢(𝑛)=𝑈₁1(𝑎),∫2𝜋^02𝜋|𝑓(𝑒^𝑖𝑡)|²𝑑𝑡=‖𝑎‖². (4.3) By Lemma 2.4, we have |1∫2𝜋^𝜋−𝜋𝑡|𝑓(−𝑒^𝑖𝑡)|²𝑑𝑡|=|^∞∞_{𝑚=0𝑛=0𝑢(𝑚)≠𝑢(𝑛)}𝑎_𝑚𝑎_𝑛𝑢(𝑚)−𝑢(𝑛)|=|𝑉(𝑎)|. (4.4) Since the series ∑^∞_𝑛=0𝑎_𝑛 is absolutely convergent, it is justified that the complex number 𝑎_𝑛 is replaced by |𝑎_𝑛|.
According (3.18), we have |∫¹₀|𝑓(𝑡)|²𝑑𝑡|²1+|∫2𝜋^𝜋−𝜋𝑡|𝑓(−𝑒^𝑖𝑡)|²𝑑𝑡|²1≤(2∫₀^2𝜋|𝑓(𝑒^𝑖𝑡)|²𝑑𝑡)²(1−𝑅²), (4.5) where 𝑅² is defined by (3.19). It is easy to deduce that 2𝜋²‖𝑎‖²=4𝜋(₀^2𝜋|𝑓(𝑒^𝑖𝑡)|²𝑑𝑡)⁻¹,𝑇₁(𝑎)=¹₀𝑡^−1/2𝑇|𝑓(𝑡)|𝑑𝑡,₂(𝑎)=¹₀𝑑𝑠𝑠^𝑠₀𝑡^−1/2|𝑓(𝑡)|𝑑𝑡. (4.6) Because 𝑢(𝑛)=𝑍_𝑛+𝜆/2≠1/𝜋, 𝑇₁(𝑎)−𝑇₂(𝑎)/𝜋≠0. It shows that 𝑅²>0. 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.
Let 𝐹(𝑧)=(𝐺(𝑧))^𝑝/2∈𝐻₂. According to the above result for 𝑝=2, we have ¹₀|𝑓(𝑡)|^𝑝𝑑𝑡|²1+|2𝜋^𝜋_−𝜋𝑡|𝑓(−𝑒^𝑖𝑡)|^𝑝𝑑𝑡|²=|¹₀|𝐹(𝑡)|²𝑑𝑡|²1+|2𝜋^𝜋_−𝜋𝑡|𝐹(−𝑒^𝑖𝑡)|²𝑑𝑡|²1≤{2₀^2𝜋|𝐹(𝑒^𝑖𝑡)|²𝑑𝑡}²(1−𝑅²_𝐹)1={2₀^2𝜋|𝐺(𝑒^𝑖𝑡)|^𝑝𝑑𝑡}²(1−𝑅²_𝐺)1={2₀^2𝜋|𝑓(𝑒^𝑖𝑡)|^𝑝𝑑𝑡}²(1−𝑅²). (4.7) Based on the case for 𝑝=2, we have 𝑅²_𝐹>0. Hence, 𝑅²>0. The proof of Theorem 4.1 is completed.
Let 𝑓(𝑧)=^∞_𝑚=0𝑐_𝑚𝑧^𝑚∈𝐻₁. (4.8) Then, ^∞_𝑛=0|𝑐_𝑛|≤1𝑛+12^𝜋_−𝜋|𝑓(𝑒^𝑖𝑡)|𝑑𝑡. (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.

Theorem 4.2. Let 𝑓(𝑧)=∑^∞_𝑚=0𝑐_𝑚𝑧^𝑢(𝑚) be analytic in the unit-disc |𝑧|<1, where 𝑢(𝑚)=𝑍_𝑚+𝜆/2 (with 𝑍_𝑚∈𝑍 and 𝜆∈𝑁₀) and 𝑓∈𝐻₁. Then, (^∞_𝑛=0|𝑐_𝑛|)𝑢(𝑛)²1+|2𝜋^𝜋_−𝜋𝑡|𝑓(−𝑒^𝑖𝑡)|²𝑑𝑡|²1≤(2^𝜋_−𝜋|𝑓(𝑒^𝑖𝑡)|𝑑𝑡)²(1−𝑅²), (4.10) where 𝑅²=4𝜋(¹₀𝑡⁻¹1|𝑓(𝑡)|𝑑𝑡−𝜋¹₀𝑑𝑠𝑠^𝑠₀𝑡⁻¹|𝑓(𝑡)|𝑑𝑡)²(₀^2𝜋|𝑓(𝑒^𝑖𝑡)|𝑑𝑡)⁻¹4𝑖𝑓𝜆isodd,𝜋(¹₀𝑡⁻¹|𝑓(𝑡)|𝑑𝑡)²(₀^2𝜋|𝑓(𝑒^𝑖𝑡)|𝑑𝑡)⁻¹𝑖𝑓𝜆iseven. (4.11)

Proof. By Blaschke decomposition theorem, we have 𝑓(𝑧)=𝐵(𝑧)𝐺(𝑧)=𝐵(𝑧)𝐺^1/2(𝑧)𝐺^1/2(𝑧)=𝑓₁(𝑧)𝑓₂(𝑧), (4.12) where 𝐵(𝑧) is Blaschke function, 𝑓₁,𝑓₂∈𝐻₂. Let 𝑓₁(𝑧)=𝐵(𝑧)𝐺^1/2(𝑧)=∑^∞_𝑚=0𝑎_𝑚𝑧^𝑢(𝑚),𝑓₂(𝑧)=𝐺^1/2(𝑧)=∑^∞_𝑛=0𝑏_𝑛𝑧^𝑢(𝑛). It is easy deduce that ‖𝑎‖²=‖𝑏‖²=^∞_𝑚=0|𝑎_𝑚|²=^∞_𝑛=0|𝑏_𝑛|²=12𝜋^𝜋_−𝜋|𝐺^1/2(𝑒^𝑖𝑡)|²=1𝑑𝑡2𝜋^𝜋_−𝜋|𝐵(𝑒^𝑖𝑡)𝐺^1/2(𝑒^𝑖𝑡)|²1𝑑𝑡=2𝜋₀^2𝜋|𝑓(𝑒^𝑖𝑡)|𝑑𝑡. (4.13) Owing to 𝑓(𝑧)=𝑓₁(𝑧)𝑓₂(𝑧), it holds that ∫¹₀𝑡⁻¹|𝑓(𝑡)|𝑑𝑡=∫¹₀𝑡⁻¹|𝑓₁(𝑡)|²𝑑𝑡. It is easy to deduce that ^∞_𝑛=0|𝑐_𝑛|≤𝑢(𝑛)^∞_{𝑛=0𝑟+𝑠=𝑛}|𝑎_𝑟||𝑎_𝑠|=𝑢(𝑟)+𝑢(𝑠)^∞∞_𝑚=0𝑛=0|𝑎_𝑚||𝑎_𝑛|.𝑢(𝑚)+𝑢(𝑛) (4.14) By Lemma 2.4, we find that |12𝜋^𝜋_−𝜋𝑡|𝑓(−𝑒^𝑖𝑡1)|𝑑𝑡|=|2𝜋^𝜋_−𝜋𝑡|𝑓₁(−𝑒^𝑖𝑡)|²𝑑𝑡|=|^∞∞_{𝑚=0𝑛=0𝑢(𝑚)≠𝑢(𝑛)}|𝑎_𝑚||𝑎_𝑛|𝑢(𝑚)−𝑢(𝑛)|. (4.15) It follows from (4.13), (4.14), (4.15), and Corollary 3.2 that (^∞_𝑛=0|𝑐_𝑛|)𝑢(𝑛)²1+|2𝜋^𝜋_−𝜋𝑡|𝑓(−𝑒^𝑖𝑡)|𝑑𝑡|²1≤(2₀^2𝜋|𝑓(𝑒^𝑖𝑡)|𝑑𝑡)²(1−𝑅²), (4.16) where 𝑅² is defined by (3.19). It is easy to deduce that 2𝜋²‖𝑎‖²=4𝜋(₀^2𝜋|𝑓(𝑒^𝑖𝑡)|²𝑑𝑡)⁻¹,𝑇₁(𝑎)=¹₀𝑡⁻¹|𝑓₁(𝑡)|²𝑑𝑡=¹₀𝑡⁻¹𝑇|𝑓(𝑡)|𝑑𝑡,₂(𝑎)=¹₀𝑑𝑠𝑠^𝑠₀𝑡⁻¹|𝑓₁(𝑡)|²𝑑𝑡=¹₀𝑑𝑠𝑠^𝑠₀𝑡⁻¹|𝑓(𝑡)|𝑑𝑡. (4.17) These show that the inequality (4.10) is valid.

Acknowledgment

The research is supported by the Scientific Research Fund of Hunan Provincial Education Department (no. 06C657).