Lower bound of four-dimensional Hausdorff matrices

Abstract

Let $\mathsf{H}=(h_{nmjk})$ be a non-negative four-dimensional matrix. Denote by $L_{p}(\mathsf{H})$ the supremum of those ℓ satisfying the following inequality:

$$ { \Biggl( {\sum_{n = 0}^{\infty }{\sum _{m = 0}^{\infty } {{{ \Biggl( {\sum _{j = 0}^{\infty }{\sum_{k = 0}^{\infty }{{h_{nmjk}} {x_{j,k}}} } } \Biggr)}^{p}}} } } \Biggr)^{1/p}} \ge \ell { \Biggl( {\sum_{j = 0}^{\infty }{\sum _{k = 0}^{ \infty }{x_{j,k}^{p}} } } \Biggr)^{1/p}}, $$

where $x=(x_{j,k}) \in \mathcal{L}_{p}$ with $x_{j,k}\ge 0$. In this paper a Hardy type formula is established for $L_{p}(\mathsf{H}_{ \mu \times \lambda }^{t})$, where $0< p\le 1$ and $\mathsf{H}_{\mu \times \lambda }$ is a four-dimensional Hausdorff matrix. A similar result is also obtained for the case in which $\mathsf{H}_{\mu \times \lambda }$ is replaced by $\mathsf{H}_{\mu \times \lambda }^{t}$. As a consequence, we apply the results to some special four-dimensional Hausdorff matrices such as Cesàro, Euler, Hölder and Gamma matrices. Our results contain some generalizations of Copson’s discrete inequality.

1 Introduction

The space $\mathcal{L}_{p}$ of absolutely p-summable double sequences over the complex field

$$ \mathcal{L}_{p} = \Biggl\{ ( {{x_{n,m}}} ): \sum _{n= 0}^{\infty }{\sum_{m = 0}^{\infty }{{{ \vert {{x_{n,m}}} \vert }^{p}} < \infty } } \Biggr\} \quad (0< p< \infty ) $$

was introduced by Başar and Sever [1] for $1< p< \infty$. It is a complete sequence space with the $\|\cdot \|_{{\mathcal{L} _{p}}}$-norm

$$ { \Vert x \Vert _{{\mathcal{L}_{p}}}}: = { \Biggl( {\sum _{n = 0}^{\infty }{\sum_{m = 0}^{\infty }{{{ \vert {{x_{n,m}}} \vert }^{p}}} } } \Biggr)^{1/p}}, $$

and for $0< p<1$ the space ${\mathcal{L}_{p}}$ is a complete p-normed (cf. Yeşilkayagil and Başar [13, 14]) space with the p-norm

$$ |\!|\!|x|\!|\!|_{\mathcal{L}_{p}}={ \Vert x \Vert _{\mathcal{L}_{p}}^{p}}:= {\sum_{n= 0}^{\infty }{\sum _{m = 0}^{\infty }{{{ \vert {{x_{n,m}}} \vert }^{p}}} }}. $$

A four-dimensional infinite matrix $\mathsf{H}=(h_{nmjk})$ defines a matrix mapping from the double sequence space X into the double sequence space Y if, for every sequence $x=(x_{n,m})\in X$, the sequence $\mathsf{H}x=\{(\mathsf{H}x)_{n,m}\}$ is in Y, where

$$ ( {\mathsf{H}x} )_{n,m} =\sum_{j= 0}^{\infty } \sum_{k = 0}^{\infty }{h_{nmjk} x_{j,k} }\quad (n,m=0,1,\dots ). $$

For $p\in \mathbb{R} \backslash \{0\}$, the number $L_{p}(\mathsf{H})$ is defined as the supremum of those ℓ, which obey the following inequality:

$$\begin{aligned} \Vert {\mathsf{H}x} \Vert _{\mathcal{L}_{p}} \ge \ell \Vert x \Vert _{\mathcal{L}_{p} } , \end{aligned}$$

(1.1)

where $x\ge 0$, $x\in \mathcal{L}_{p}$.

For further details on the normed spaces of double sequences, four-dimensional matrices and the boundedness problems of matrix operators in normed (p-normed) sequence spaces, we refer the readers to the textbook [4] and the recent papers [7,8,9, 11].

In this paper, we are going to prove that

$$ {L_{p}} \bigl( \mathsf{H}_{\mu \times \lambda }^{t} \bigr) = \int _{0}^{1} { \int _{0}^{1} {{{ ( {\alpha \beta } )}^{-1/p^{*}}}} }\,d\mu ( \alpha ) \times d\lambda ( \beta ) \quad ( {0 < p \le 1} ) $$

and

$$ {L_{p}} ( \mathsf{H}_{\mu \times \lambda } ) \ge { \int _{0}^{1} { \int _{0}^{1} { ( {\alpha \beta } )} ^{ - 1/p}}}\,d\mu ( \alpha ) \times d\lambda ( \beta ) \quad ( {0 < p \le 1} ), $$

where $p^{*}$ is the conjugate exponent of p and $\mathsf{H}_{ \mu \times \lambda }$ is the four-dimensional Hausdorff matrix. Here the transpose of H is denoted by $\mathsf{H}^{t}=(h_{nmjk}^{t})$, where $h_{nmjk}^{t}:=h_{jknm}$. As a consequence, we apply the results to some special four-dimensional Hausdorff matrices such as Cesàro, Euler Hölder and Gamma matrices.

2 Lower bounds for four-dimensional Hausdorff matrices

In this section we consider the four-dimensional Hausdorff matrix and its transpose as operators mapping the double sequence space $\mathcal{L}_{p}$ where $0< p\le 1$ to itself. Then we focus on the evaluation of $L_{p}(\mathsf{H}_{\mu \times \lambda }^{\mathsf{t}})$ and $L_{p}(\mathsf{H}_{\mu \times \lambda })$, where dμ and dλ are two Borel probability measures and $\mathsf{H}_{ \mu \times \lambda }=(h_{nmjk})$ is the four-dimensional Hausdorff matrix [5], defined by

$$h_{nmjk}={\int }_0^1{\int }_0^1\left(\begin{array}{c}n\\ j\end{array}\right)\left(\begin{array}{c}m\\ k\end{array}\right){\alpha }^j{\beta }^k{(1-\alpha )}^{n-j}{(1-\beta )}^{m-k}d\mu (\alpha )\times d\lambda (\beta ),$$

for $(n\ge j,m\ge k)$, and $h_{nmjk}=0$ for all $j>n$ or $k>m$. Clearly, we have

$$h_{nmjk}=\left(\begin{array}{c}n\\ j\end{array}\right)\left(\begin{array}{c}m\\ k\end{array}\right){\mathrm{\Delta }}_1^{n-j}{\mathrm{\Delta }}_2^{m-k}{\mu }_{j,k},$$

where

$$ {\mu _{j,k}} := \int _{0}^{1} { \int _{0}^{1} {{\alpha ^{j}} {\beta ^{k}}\,d\mu (\alpha ) \times d\lambda (\beta ) } } \quad (j,k = 0,1, \dots ) $$

and

$${\mathrm{\Delta }}_1^{n-j}{\mathrm{\Delta }}_2^{m-k}{\mu }_{j,k}=\sum _{s=0}^{n-j}\sum _{t=0}^{m-k}{(-1)}^{s+t}\left(\begin{array}{c}n-j\\ s\end{array}\right)\left(\begin{array}{c}m-k\\ t\end{array}\right){\mu }_{j+s,k+t}.$$

Here, we’ve listed some famous classes of four-dimensional Hausdorff matrices:

• 1. The choices $d\mu (\alpha ) = \eta (1 - \alpha )^{\eta - 1}\,d\alpha$ and $d\lambda (\beta ) = \gamma (1 - \beta )^{\gamma - 1}\,d\beta$ give the four-dimensional Cesàro matrix $\mathsf{C}( \eta , \gamma )$ of orders η and γ; see [6].

• 2. The choices $d\mu (\alpha ) = \vert {\log \alpha } \vert ^{\eta - 1} /\varGamma (\eta )\,d\alpha$ and $d\lambda (\beta ) = \vert {\log \beta } \vert ^{\gamma - 1} /\varGamma (\gamma )\,d\beta$ give the four-dimensional Hölder matrix $\mathsf{H}(\eta , \gamma )$ of orders η and γ.

• 3. The choices $d\mu (\alpha ) = \eta \alpha ^{\eta - 1}\,d\alpha$ and $d\lambda (\beta ) = \gamma \beta ^{\gamma - 1}\,d\beta$ give the four-dimensional Gamma matrix $\boldsymbol{\varGamma }( {\eta ,\gamma } )$ of orders η and γ.

The four-dimensional Cesàro, Hölder and Gamma matrices have non-negative entries whenever $\eta >0$ and $\gamma >0$.

We are going to exhibit a Hardy type formula for $L_{p}( \mathsf{H}_{\mu \times \lambda }^{\mathsf{t}})$ and $L_{p}( \mathsf{H}_{\mu \times \lambda })$, where $0< p\le 1$. Then we apply our results to the four-dimensional Cesàro, Hölder and Gamma matrices. First, we state and prove the following generalization of Proposition 7.4 in [2], which plays an essential role in the rest of the paper.

Lemma 2.1

Let $0< p<1$, and suppose that H is a non-negative four-dimensional matrix. If

$$ \mathop{\sup } _{n,m} \sum_{j= 0}^{\infty }{ \sum_{k = 0}^{\infty }{{h_{nmjk}}} } = \lambda >0 $$

and

$$ \mathop{\inf } _{j,k} \sum_{n = 0}^{\infty }{ \sum_{m = 0}^{\infty }{{h_{nmjk}}} } = \gamma , $$

then (1.1) holds with

$$ \ell \geq \lambda ^{\frac{1}{p^{*}}}\gamma ^{\frac{1}{p}}. $$

Proof

The proof will be a modification of that given by Bennett. By Hölder’s inequality, we have

$$\begin{aligned} \sum_{j = 0}^{\infty }{\sum _{k = 0}^{\infty }{{h_{nmjk}}x _{j,k}^{p}} } =& \sum_{j = 0}^{\infty }{\sum _{k = 0}^{\infty }{h_{nmjk} ^{1 - p}{{ ( {{h_{nmjk}} {x_{j,k}}} )}^{p}}} } \\ \le& { \Biggl( {\sum_{j = 0}^{\infty }{\sum _{k = 0}^{\infty } {{h_{nmjk}}} } } \Biggr)^{1 - p}} { \Biggl( {\sum_{j = 0}^{ \infty }{ \sum_{k = 0}^{\infty }{{h_{nmjk}} {x_{j,k}}} } } \Biggr) ^{p}} \\ \le& {\lambda ^{1 - p}} { \Biggl( {\sum_{j = 0}^{\infty }{ \sum_{k = 0}^{\infty }{{h_{nmjk}} {x_{j,k}}} } } \Biggr)^{p}}. \end{aligned}$$

Therefore

$$\begin{aligned} {\lambda ^{1 - p}}\sum_{n = 0}^{\infty }{ \sum_{m = 0} ^{\infty }{{{ \Biggl( {\sum _{j = 0}^{\infty }{\sum_{k = 0} ^{\infty }{{h_{nmjk}} {x_{j,k}}} } } \Biggr)}^{p}}} } \ge& \sum_{j = 0}^{\infty }{\sum _{k = 0}^{\infty }{x_{j,k} ^{p}\sum _{n = 0}^{\infty }{\sum_{m = 0}^{\infty }{{h _{nmjk}}} } } } \\ \ge& \gamma \sum_{j = 0}^{\infty }{\sum _{k = 0}^{\infty } {x_{j,k}^{p}} }, \end{aligned}$$

which is equivalent to the inequality stated in the lemma. □

For $r,s\geq 0$, let $\mathsf{E}(r,s) =(e_{nmjk}^{r,s})$ denote the four-dimensional Euler matrix, defined by

$$e_{nmjk}^{r,s}=\{\begin{array}{cc}\left(\begin{array}{c}n\\ j\end{array}\right)\left(\begin{array}{c}m\\ k\end{array}\right)r^j{s}^k{(1-r)}^{n-j}{(1-s)}^{m-k}\hfill & (0\le j\le n,0\le k\le m),\hfill \\ 0\hfill & (j>n\text{ or }k>m),\hfill \end{array}$$

for all $n,m,j,k\in \mathbb{N}\cup \{0\}$. Clearly, this matrix is the four-dimensional Hausdorff matrix with $d\mu (\alpha ) =\mbox{point}$ evaluation at $\alpha =r$ and $d\lambda (\beta ) =\mbox{point}$ evaluation at $\beta =s$ and has non-negative entries whenever $0\le r\le 1$ and $0\le s\le 1$. For this matrix we have the following lemma, which extends [12, Lemma 3.2].

Lemma 2.2

Let $0< p\le 1$, $\frac{1}{p}+\frac{1}{p^{*}}=1$ and $\mathsf{E}(r,s)$ be the four-dimensional Euler matrix of order r, s where $0\le r,s \le 1$. Then

$$ L_{p} \bigl(\mathsf{E}^{t}(r,s) \bigr)\geq (rs )^{-1/p ^{*}}. $$

Proof

We have

$$\sum _{j=\ell }^{\mathrm{\infty }}\sum _{k=t}^{\mathrm{\infty }}\left(\begin{array}{c}j\\ \ell \end{array}\right)\left(\begin{array}{c}k\\ t\end{array}\right)r^{\ell }{s}^t{(1-r)}^{j-\ell }{(1-s)}^{k-t}=\frac{1}{rs}(\ell ,t=0,1,2,\dots )$$

and

$$\sum _{\ell =0}^j\sum _{t=0}^k\left(\begin{array}{c}j\\ \ell \end{array}\right)\left(\begin{array}{c}k\\ t\end{array}\right)r^{\ell }{s}^t{(1-r)}^{j-\ell }{(1-s)}^{k-t}=1(j,k=0,1,2,\dots ).$$

Thus, for $0 < p < 1$, applying Lemma 2.1 when $\lambda =\frac{1}{rs}$ and $\gamma =1$, we infer that $L_{p} (\mathsf{E}^{t}(r,s) ) \geq (rs )^{-1/p^{*}}$. For $p = 1$, it follows from the Fubini’s theorem that

$$\begin{aligned} { \bigl\Vert {\mathsf{E}^{t}(r,s) x} \bigr\Vert _{\mathcal{L}_{1}}} =& \sum_{n = 0}^{\infty }{\sum _{m = 0}^{\infty }{ \Biggl( {\sum _{j = 0}^{\infty }{\sum_{k = 0}^{\infty }{{{ \bigl( {e_{nmjk}^{r,s}} \bigr)}^{t}} {x_{j,k}}} } } \Biggr)} } \\ =& \sum_{j = 0}^{\infty }{\sum _{k = 0}^{\infty }{ \Biggl( {\sum _{n = 0}^{\infty }{\sum_{m = 0}^{\infty }{e_{jknm} ^{r,s}} } } \Biggr){x_{j,k}}} } = { \Vert x \Vert _{\mathcal{L} _{1}}}, \end{aligned}$$

which gives the desired inequality. This completes the proof. □

Now, we come to the evaluation of $L_{p}( \mathsf{H}_{\mu \times \lambda }^{\mathsf{t}})$.

Theorem 2.3

Let $0< p\le 1$, $\frac{1}{p}+\frac{1}{p^{*}}=1$ and $\mathsf{H}_{\mu \times \lambda }$ be the four-dimensional Hausdorff matrix. Then

$$ L_{p} \bigl(\mathsf{H}_{\mu \times \lambda }^{\mathsf{t}} \bigr) \geq \int _{0}^{1} { \int _{0}^{1} {{{ ( {\alpha \beta } )}^{ - 1/{p^{*} }}}\,d\mu ( \alpha ) \times d\lambda ( \beta )} }. $$

Proof

Let $x=(x_{n,m})$ be a non-negative double sequence in $\mathcal{L} _{p}$. Since

$$ { \mathsf{H}_{\mu \times \lambda }^{\mathsf{t}}} x = \int _{0}^{1} { \int _{0} ^{1} \mathsf{E}^{\mathsf{t}}(\alpha , \beta )x\,d\mu (\alpha ) \times d\lambda (\beta ) }, $$

applying Minkowski’s inequality and Lemma 2.2, we have

$$\begin{aligned} \bigl\Vert \mathsf{H}_{\mu \times \lambda }^{\mathsf{t}} x \bigr\Vert _{{\mathcal{L} _{p}}} =& { \biggl\Vert { \int _{0}^{1} { \int _{0}^{1} { {\mathsf{E}^{\mathsf{t}}(\alpha , \beta )x}\,d\mu (\alpha ) \times d\lambda (\beta ) } } } \biggr\Vert _{ {\mathcal{L}_{p}}}} \\ \geq& \int _{0}^{1} { \int _{0}^{1} {{{ \bigl\Vert { \mathsf{E}^{\mathsf{t}}(\alpha , \beta )x} \bigr\Vert }_{{\mathcal{L}_{p}}}}\,d\mu ( \alpha ) \times d \lambda (\beta ) } } \\ \geq& \biggl( { \int _{0}^{1} { \int _{0}^{1} {{{ ( {\alpha \beta } )} ^{ - 1/p^{*}}}\,d\mu (\alpha ) \times d\lambda (\beta )} } } \biggr) { \Vert x \Vert _{{\mathcal{L}_{p}}}}. \end{aligned}$$

This is equivalent to the inequality stated in the theorem. □

Remark 2.4

By choosing $d\mu (\alpha )=d\alpha$ and $d\lambda (\beta )=d\beta$, we have

$$ \int _{0}^{1} { \int _{0}^{1} {{{ ( {\alpha \beta } )}^{ - 1/p ^{*}}}\,d\mu ( \alpha ) \times d\lambda ( \beta ) } }=p^{2}. $$

Therefore, the inequality in Theorem 2.3 reduces to the following generalization of Copson’s inequality to double series:

$$ \sum_{n = 0}^{\infty }{\sum _{m = 0}^{\infty }{ \Biggl( {\sum _{j = n}^{\infty }{\sum_{k = m}^{\infty }{ \frac{ {{x_{j,k}}}}{{ ( {j + 1} ) ( {k + 1} )}}} } } \Biggr)} } \ge {p^{2p}}\sum _{j = 0}^{\infty }{\sum_{k = 0}^{\infty }{x_{j,k}^{p}} }, $$

in which the constant $p^{2}$ is the best possible [10]. Accordingly, the value

$$ \int _{0}^{1} { \int _{0}^{1} {{{ ( {\alpha \beta } )}^{ - 1/p ^{*}}}\,d\mu ( \alpha ) \times d\lambda ( \beta ) } } $$

in Theorem 2.3 is the best possible, which is equal to the lower bound of the four-dimensional Hausdorff matrix as operator mapping ${\mathcal{L}_{p}}$ to itself, i.e.,

$$ L_{p} \bigl(\mathsf{H}_{\mu \times \lambda }^{\mathsf{t}} \bigr)= \int _{0}^{1} { \int _{0}^{1} {{{ ( {\alpha \beta } )}^{ - 1/{p^{*} }}}\,d\mu ( \alpha ) \times d\lambda ( \beta )} }. $$

Let $\mathsf{E}(\eta ,\gamma )$, $\mathsf{C}(\eta , \gamma )$, $\mathsf{H}(\eta , \gamma )$ and $\boldsymbol{\varGamma }( {\eta , \gamma } )$ be the four-dimensional Euler, Cesàro, Hölder and Gamma matrices of order η and γ, respectively. Applying Theorem 2.3 together with Remark 2.4 to these matrices, we have the following corollary.

Corollary 2.5

Let $0< p\le 1$ and $\frac{1}{p}+\frac{1}{p^{*}}=1$. Then for $\eta >0$ and $\gamma >0$, we have

• 1. $L_{p} (\mathsf{E}^{\mathsf{t}}(\eta ,\gamma ) )= (\eta \gamma )^{-1/p^{*}}$, $\eta \le 1$, $\gamma \le 1$;

• 2. $L_{p} (\mathsf{C}^{\mathsf{t}}(\eta , \gamma ) )=\frac{ {\varGamma ( {\eta + 1} ){\varGamma ^{2}} ( {\frac{1}{p}} ) \varGamma ( {\gamma + 1} )}}{{\varGamma ( {\eta + \frac{1}{p}} )\varGamma ( {\gamma + \frac{1}{p}} )}}$;

• 3. $L_{p} (\mathsf{H}^{\mathsf{t}}(\eta , \gamma ) )=\frac{1}{ {\varGamma ( \eta )\varGamma ( \gamma )}}\int _{0} ^{1} {\int _{0}^{1} {{{ ( {\alpha \beta } )}^{1/{p^{*}}}}| \log \alpha {|^{\eta - 1}}|\log \beta {|^{\gamma - 1}}\,d\alpha \times d\beta } }$;

• 4. $L_{p} (\boldsymbol{\varGamma }^{\mathsf{t}} ( {\eta ,\gamma } ) )=\frac{{{p^{2}}\eta \gamma }}{{ ( {p\eta - p + 1} ) ( {p\gamma - p + 1} )}}$, $\eta > \frac{1}{ {{p^{*}}}}$, $\gamma > \frac{1}{{{p^{*}}}}$.

In the rest of our paper, we consider the four-dimensional Hausdorff matrices as operators mapping ${\mathcal{L}_{p}}$ to itself and establish a Hardy type formula as a lower estimate for ${L_{p}} ( \mathsf{H}_{\mu \times \lambda } )$. First we state the following lemma as an extension of [3, Lemma 2.2] to the four-dimensional case. Its proof can be easily adapted from that of Lemma 2.2.

Lemma 2.6

Let $0< p\le 1$, $\frac{1}{p}+\frac{1}{p^{*}}=1$ and $\mathsf{E}(r,s)$ be the four-dimensional Euler matrix of order r, s where $0\le r,s \le 1$. Then $L_{p} (\mathsf{E}(r,s) )\geq (rs ) ^{-1/p}$.

For $x\geq 0$, ${ \mathsf{H}_{\mu \times \lambda } }x = \int _{0}^{1} {\int _{0}^{1} {{\mathsf{E}(\alpha ,\beta )x}\,d\mu (\alpha ) \times d \lambda (\beta ) } }$. Hence Lemma 2.6 enables us to estimate the value of ${L_{p}} ( \mathsf{H}_{\mu \times \lambda } )$. The details are given below.

Theorem 2.7

Let $0< p\le 1$, and $\mathsf{H}_{\mu \times \lambda }$ be the four-dimensional Hausdorff matrix. Then

$$ {L_{p}} ( \mathsf{H}_{\mu \times \lambda } ) \ge { \int _{0}^{1} { \int _{0}^{1} { ( {\alpha \beta } )} ^{ - 1/p}}}\,d\mu ( \alpha ) \times d\mu ( \beta ). $$

Proof

The proof can be easily adapted from that of Theorem 2.3, and so is omitted. □

Applying Theorem 2.7 to the four-dimensional Cesàro, Hölder and Gamma matrices, we have the following corollary.

Corollary 2.8

Let $0< p\le 1$. Then for $\eta >0$ and $\gamma >0$, we have

• 1. $L_{p} (\mathsf{C}(\eta , \gamma ) )=\infty$;

• 2. $L_{p} (\mathsf{H}(\eta , \gamma ) )=\infty$;

• 3. $L_{p} (\boldsymbol{\varGamma }( {\eta ,\gamma } ) )= \infty$ (${\eta \leqslant \frac{1}{p},\gamma \leqslant \frac{1}{p}}$);

• 4. $L_{p} (\boldsymbol{\varGamma }( {\eta ,\gamma } ) )\geq \frac{{\eta \gamma }}{{ ( {\eta - \frac{1}{p}} ) ( {\gamma - \frac{1}{p}} )}}$ (${\eta > \frac{1}{p},\gamma > \frac{1}{p}}$).

3 Conclusions

In this paper, we consider the four-dimensional Hausdorff matrix and its transpose as operators on the double sequence space $\mathcal{L}_{p}$ and establish a Hardy type formula for their lower bounds. Then we apply the results to some special classes of four dimensional Hausdorff matrices such as Cesàro, Euler, Hölder and Gamma matrices. Our results contain some generalizations of Copson’s discrete inequality.