On a New Space 𝑚²(𝑀,𝐴,𝜙,𝑝) of Double Sequences

Abstract

We introduce a new space 𝑚²(𝑀,𝐴,𝜙,𝑝) of double sequences related to 𝑝-absolute convergent double sequence space, combining an Orlicz function and an infinity double matrix. We study some properties of 𝑚²(𝑀,𝐴,𝜙,𝑝) and obtain some inclusion relations involving 𝑚²(𝑀,𝐴,𝜙,𝑝).

1. Introduction

Throughout this work, 𝑁 and 𝐶 denote the set of positive integers and complex numbers, respectively. A complex double sequence is a function 𝑥 from 𝑁×𝑁 into 𝐶 and briefly denoted by {𝑥_𝑘,𝑙}. Throughout this work, 𝑤 and 𝑤² denote the spaces of single complex sequences and double complex sequences, respectively. If, for all 𝜀>0, there is 𝑛_𝜀∈𝑁 such that ‖𝑥_𝑘,𝑙−𝑎‖_𝑋<𝜀 where 𝑘>𝑛_𝜀 and 𝑙>𝑛_𝜀, then a double sequence {𝑥_𝑘,𝑙} is said to be converging (in terms of Pringsheim) to 𝑎∈𝐶. A real double sequence {𝑥_𝑘,𝑙} is nondecreasing, if 𝑥_𝑘,𝑙≤𝑥_𝑝,𝑞 for (𝑘,𝑙)<(𝑝,𝑞). A double series is infinity sum ∑^∞_𝑘,𝑙=1𝑥_𝑘,𝑙, and its convergence implies the convergence by |⋅| of partial sums sequence {𝑆_𝑛,𝑚}, where 𝑆_𝑛,𝑚=∑^𝑛_𝑘=1∑^𝑚_𝑙=1𝑥_𝑘,𝑙 (see [1–4]).

A double sequence space 𝐸 is said to be solid if {𝑥_𝑘,𝑙𝑦_𝑘,𝑙}∈𝐸 for all double sequences {𝑦_𝑘,𝑙} of scalars such that |𝑦_𝑘,𝑙|≤1 for all 𝑘,𝑙∈𝑁 whenever {𝑥_𝑘,𝑙}∈𝐸.

Now let ℘_𝑠 be a family of subsets 𝜎 having most elements 𝑠 in 𝑁. Also ℘_𝑠,𝑡 denotes the class of subsets 𝜎=𝜎₁×𝜎₂ in 𝑁×𝑁 such that the element numbers of 𝜎₁ and 𝜎₂ are most 𝑠 and 𝑡, respectively. Besides {𝜙_𝑘,𝑙} is taken as a nondecreasing double sequence of the positive real numbers such that 𝑘𝜙_𝑘+1,𝑙≤(𝑘+1)𝜙_𝑘,𝑙,𝑙𝜙_𝑘,𝑙+1≤(𝑙+1)𝜙_𝑘,𝑙. (1) An Orlicz function is a function 𝑀∶[0,∞)→[0,∞) which is continuous, nondecreasing, and convex with 𝑀(0)=0, 𝑀(𝑥)>0 for 𝑥>0, and 𝑀(𝑥)→∞ as 𝑥→∞.

An Orlicz function 𝑀 can always be represented in the following integral form: 𝑀(𝑥)=∫^𝑥₀𝜂(𝑡)𝑑𝑡, where 𝜂 is known as the kernel of 𝑀, is right differentiable for 𝑡≥0, 𝜂(0)=0, 𝜂(𝑡)>0 for 𝑡>0, 𝜂 is nondecreasing, and 𝜂(𝑡)→∞ as 𝑡→∞.

An Orlicz function is said to be satisfied Δ₂-condition for all values of 𝑢, if there exists a constant 𝑇>0 such that 𝑀(2𝑢)≤𝑇𝑀(𝑢) for all 𝑢≥0.

Lindenstrauss and Tzafriri [5] used the idea of Orlicz functions to construct Orlicz sequence space ℓ_𝑀=𝑥=𝑥_𝑘∈𝑤∶_𝑘≥1𝑀𝑥_𝑘𝜌<∞forsome𝜌>0. (2) The sequence space ℓ_𝑀 is a Banach space according to the norm defined by ‖𝑥‖=inf𝜌>0∶_𝑘≥1𝑀𝑥_𝑘𝜌≤1. (3) This space (ℓ_𝑀,‖⋅‖) is called an Orlicz sequence space. The space ℓ_𝑀 is closely related to the space ℓ_𝑝, which is an Orlicz sequence space with 𝑀(𝑥)=𝑥^𝑝 for 1≤𝑝<∞.

The double sequence spaces in the various forms defined by Orlicz functions were introduced and studied by Khan and Tabassum in [6–12] and by Khan et al. in [13].

The space 𝑚(𝜙), introduced by Sargent in [14], is in the form 𝑚𝜙=𝑥=𝑥_𝑘∈𝑤∶‖𝑥‖_𝑚(𝜙)=sup_{𝑠≥1,𝜎∈℘𝑠}1𝜙_{𝑠𝑘∈𝜎}𝑥_𝑘<∞. (4) Sargent studied some properties of this space and examined relationship between this space and 𝑙_𝑝-space. Similar sequence classes were studied by many mathematicians using Orlicz functions (see [15–17]).

Later on, this space was investigated from sequence space point of view by Rath [18], Rath and Tripathy [19], Tripathy and Sen [20], Tripathy and Mahanta [17], and others. Recently Altun and Bilgin [15] introduced and studied the following sequence space 𝑚(𝑀,𝐴,𝜙,𝑝).

Let 𝐴=(𝑎_𝑖𝑘) be an infinite matrix of complex numbers, 𝑀 an Orlicz space, and 𝑝=(𝑝_𝑖) a bounded sequence of positive real numbers such that 0<ℎ=inf𝑝_𝑖≤𝑝_𝑖≤𝐻=sup𝑝_𝑖<∞. Then the space 𝑚(𝑀,𝐴,𝜙,𝑝) is defined by 𝑚²=𝑥𝑀,𝐴,𝜙,𝑝_𝑘∈𝑤∶sup_{𝑠≥1,𝜎∈℘𝑠}1𝜙_{𝑠𝑘∈𝜎}𝑀𝐴_𝑖(𝑥)𝜌^𝑝𝑖<∞forsome𝜌>0, (5) where 𝐴𝑥=(𝐴_𝑖(𝑥)) if 𝐴_𝑖(𝑥)=∑^∞_𝑘=1𝑎_𝑖𝑘𝑥_𝑘 converges for each 𝑖.

Let 𝑥={𝑥_𝑘,𝑙} be a double sequence. A set 𝑆(𝑥) is defined by 𝑆(𝑥)=𝑥_{𝜋1(𝑘),𝜋2(𝑘)}∶𝜋₁and𝜋₂arepermutationsof𝑁. (6)

If 𝑆(𝑥)⊆𝐸 for all 𝑥∈𝐸, then 𝐸 is said to be symmetric.

In this work, we introduce the following sequence space.

Let 𝐴=(𝑎_{𝑖𝑗𝑘𝑙}) be an infinite double matrix of complex numbers, 𝑀 an Orlicz function and 𝑝=(𝑝_𝑖𝑗) bounded double sequence of positive real numbers such that 0<ℎ=inf𝑝_𝑖𝑗≤𝑝_𝑖≤𝐻=sup𝑝_𝑖𝑗<∞. Then the space 𝑚²(𝑀,𝐴,𝜙,𝑝) is defined by 𝑚²=𝑥𝑀,𝐴,𝜙,𝑝𝑥=_𝑘𝑙∈𝑤²1∶sup𝜙_{𝑠𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗(𝑥)𝜌^𝑝𝑖𝑗∶(𝑠,𝑡)≥(1,1),𝜎₁×𝜎₂∈℘_𝑠𝑡<∞,forsome𝜌>0, (7) where 𝐴𝑥=(𝐴_𝑖𝑗(𝑥)) if 𝐴_𝑖𝑗(𝑥)=∑^∞_𝑘,𝑙=1𝑎_{𝑖𝑗𝑘𝑙}𝑥_𝑘𝑙 converges for each (𝑖,𝑗)∈𝑁×𝑁.

Also, we introduce and investigate the following space: 𝑚²=𝑥𝑀,𝜙,𝑝𝑥=_𝑘𝑙∈𝑤²∶1sup𝜙_{𝑠𝑡𝑘∈𝜎1𝑙∈𝜎2}𝑀𝑥_𝑘,𝑙𝜌^𝑝𝑘𝑙∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠𝑡<∞forsome𝜌>0. (8)

In this work, we also use the following sequence spaces: 𝑙⁽²⁾𝑀,𝐴,𝜙,𝑝₀=𝑥=𝑥_𝑘,𝑙∈𝑤²∶^∞_𝑖,𝑗=1𝑀𝐴_𝑖𝑗(𝑥)𝜌^𝑝𝑖𝑗<∞forsome,𝑙𝜌>0₍₂₎𝑀,𝐴,𝜙,𝑝_∞=𝑥=𝑥_𝑘,𝑙∈𝑤²∶sup_{𝑖,𝑗∈𝑁×𝑁}𝑀𝐴_𝑖𝑗(𝑥)𝜌^𝑝𝑖𝑗<∞forsome𝜌>0. (9)

The following inequality will be used throughout this paper: |𝑎+𝑏|^𝑝𝑞𝑟≤max1,2^𝐻−1|𝑎|^𝑝𝑞𝑟+|𝑏|^𝑝𝑞𝑟, (10) where 𝑎,𝑏∈𝐶and𝐻=sup{𝑝_𝑞𝑟∶(𝑞,𝑟)∈𝑁×𝑁}.

2. Main Results

Definition 1. Let 𝐾⁽²⁾ be a set of increasing positive integer binaries, namely, (𝑘₁,𝑘₂)<(𝑙₁,𝑙₂) if and only if 𝑘₁<𝑙₁ and 𝑘₂<𝑙₂, and 𝐸 be a double sequence space. A 𝐾⁽²⁾-set space is a double sequence space, defined by 𝜆^𝐸_𝐾(2)=𝑥_{𝑘𝑚,𝑙𝑚}∈𝑤²∶𝑥_𝑘,𝑙∈𝐸,𝑘_𝑚,𝑙_𝑚∈𝐾⁽²⁾. (11) The canonical preimage of a double sequence {𝑥_{𝑘𝑚,𝑙𝑚}}∈𝜆^𝐸_𝐾(2) is a double sequence {𝑦_𝑘,𝑙}∈𝑤² with 𝑦_𝑘,𝑙=𝑥_𝑘,𝑙,(𝑘,𝑙)∈𝐾⁽²⁾0,(𝑘,𝑙)∉𝐾⁽²⁾. (12) The canonical preimage of a set space 𝜆^𝐸_𝐾(2) is a set of canonical preimages of all elements in 𝜆^𝐸_𝐾(2).

Definition 2. If a double sequence space 𝐸 contains the canonical preimages of all set spaces, then 𝐸 is said to be monotone.

The following lemma is an easy result of the definitions.

Lemma 3. If a double sequence space 𝐸 is solid, then 𝐸 is monotone.

Proposition 4. The space 𝑚²(𝑀,𝐴,𝜙,𝑝) is a 𝐶-linear space.

Proof. Let 𝑥={𝑥_𝑘,𝑙}, 𝑦={𝑦_𝑘,𝑙} be in 𝑚²(𝑀,𝐴,𝜙,𝑝) and 𝛼, 𝛽 in 𝐶⧵{0}. Then there exist positive numbers 𝜌₁ and 𝜌₂ such that 1sup𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗(𝑥)𝜌₁^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡1<∞,sup𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗𝑦𝜌₂^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡<∞. (13) Let 𝜌₃=max{2|𝛼|𝜌₁,2|𝛽|𝜌₂}. Using that 𝑀 is nondecreasing convex function, we have _{𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗𝛼𝑥+𝛽𝑦𝜌₃^𝑝𝑖𝑗≤_{𝑖∈𝜎1𝑗∈𝜎2}𝑀∑^∞_𝑘,𝑙=1𝛼𝑎_{𝑖𝑗𝑘𝑙}𝑥_𝑘,𝑙+𝛽𝑎_{𝑖𝑗𝑘𝑙}𝑦_𝑘,𝑙𝜌₃^𝑝𝑖𝑗≤_{𝑖∈𝜎1𝑗∈𝜎2}𝑀𝛼𝐴_𝑖𝑗(𝑥)2|𝛼|𝜌₁+𝛽𝐴_𝑖𝑗𝑦2𝛽𝜌₂^𝑝𝑖𝑗=_{𝑘∈𝜎1𝑙∈𝜎2}𝑀𝐴_𝑖𝑗(𝑥)2𝜌₁+𝐴_𝑖𝑗𝑦2𝜌₂^𝑝𝑖𝑗≤max1,2^𝐻−1_{𝑖∈𝜎1𝑖∈𝜎2}𝑀𝐴_𝑖𝑗(𝑥)𝜌₁^𝑝𝑖𝑗+_{𝑘∈𝜎1𝑙∈𝜎2}𝑀𝐴_𝑖𝑗𝑦𝜌₂^𝑝𝑖𝑗. (14) Thus we can write 1sup𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗𝛼𝑥+𝛽𝑦𝜌₃^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡≤max1,2^𝐻−1×1sup𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗(𝑥)𝜌₁^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡1+sup𝜙_{𝑠,𝑡𝑘∈𝜎1𝑙∈𝜎2}𝑀𝐴_𝑖𝑗𝑦𝜌₂^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡<∞. (15) This shows that {𝛼𝑥_𝑘,𝑙+𝛽𝑦_𝑘,𝑙}∈𝑚²(𝑀,𝐴,𝜙,𝑝). Hence 𝑚²(𝑀,𝐴,𝜙,𝑝) is a linear space.

Proposition 5. The space 𝑚²(𝑀,𝐴,𝜙,𝑝) is a paranormed space with the paranorm 𝑔(𝑥)=inf𝜌^{𝑝𝑞𝑟/𝐻}∶1sup𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗(𝑥)𝜌^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡≤1^1/𝐻,𝑞∈𝑁,𝑟∈𝑁. (16)

Proof. It is clear that 𝑔(𝑥)=𝑔(−𝑥) and 𝑔(𝑥)=0 if 𝑥=0. If there are 𝜌₁>0 and 𝜌₂>0 such that 1sup𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗(𝑥)𝜌₁^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡1≤1,sup𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗𝑦𝜌₂^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡≤1, (17) then 1sup𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗𝑥+𝑦𝜌₁+𝜌₂^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡1≤sup𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗(𝑥)𝜌₁+𝜌₂+𝐴_𝑖𝑗𝑦𝜌₁+𝜌₂^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡≤𝜌₁𝜌₁+𝜌₂^ℎ1sup𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗(𝑥)𝜌₁^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡+𝜌₂𝜌₁+𝜌₂^ℎ1sup𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗(𝑥)𝜌₁^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡, (18) where ℎ=inf𝑝_𝑖𝑗. This shows that 𝑔(𝑥+𝑦)≤𝑔(𝑥)+𝑔(𝑦). Using this triangle inequality we can write 𝑔𝜆^𝑛𝑥^𝑛−𝜆𝑥≤𝑔𝜆^𝑛𝑥^𝑛−𝜆^𝑛𝑥+𝑔𝜆^𝑛𝑥−𝜆𝑥. (19) Separately, we obtain 𝑔𝜆^𝑛𝑥^𝑛−𝜆^𝑛𝑥=inf𝜌^𝑝𝑞𝑟_𝑛^/𝐻∶1sup𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗𝜆^𝑛𝑥^𝑛−𝜆^𝑛𝑥𝜌_𝑛^𝑝𝑖𝑗∶(𝑠,𝑡)≥(1,1),𝜎₁×𝜎₂∈℘_𝑠,𝑡≤1^1/𝐻𝜌≤1,𝑞∈𝑁,𝑟∈𝑁=inf^𝑝𝑞𝑟_𝑛^/𝐻∶1sup𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗𝑥^𝑛−𝑥𝜌_𝑛/𝜆^{𝑛𝑝𝑖𝑗}∶(𝑠,𝑡)≥(1,1),𝜎₁×𝜎₂∈℘_𝑠,𝑡≤1^1/𝐻𝜆≤1,𝑞∈𝑁,𝑟∈𝑁=inf^𝑛𝜌_𝑛^{𝑝𝑞𝑟/𝐻}∶1sup𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗𝑥^𝑛−𝑥𝜌_𝑛^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡^1/𝐻𝜆≤1,𝑞∈𝑁,𝑟∈𝑁≤max^𝑛ℎ/𝐻,𝜆^𝑛.inf𝜆^𝑛𝜌_𝑛^{𝑝𝑞𝑟/𝐻}∶1sup𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗𝑥^𝑛−𝑥𝜌_𝑛^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡^1/𝐻𝜆≤1,𝑞∈𝑁,𝑟∈𝑁=max^𝑛ℎ/𝐻,𝜆^𝑛⋅𝑔𝑥^𝑛,𝑔𝜆−𝑥_𝑛𝜌𝑥−𝜆𝑥=inf^𝑝𝑞𝑟_𝑛^/𝐻∶1sup𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗𝜆^𝑛−𝜆𝑥𝜌_𝑛^𝑝𝑖𝑗∶(𝑠,𝑡)≥(1,1),𝜎₁×𝜎₂∈℘_𝑠,𝑡^1/𝐻𝜌≤1,𝑞∈𝑁,𝑟∈𝑁=inf^𝑝𝑞𝑟_𝑛^/𝐻∶1sup𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗(𝑥)𝜌_𝑛/𝜆^𝑛−𝜆^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡^1/𝐻𝜆≤1,𝑞∈𝑁,𝑟∈𝑁=inf^𝑛−𝜆𝜌_𝑛^{𝑝𝑞𝑟/𝐻}∶1sup𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗(𝑥)𝜌_𝑛^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡^1/𝐻𝜆≤1,𝑞∈𝑁,𝑟∈𝑁≤max^𝑛−𝜆^ℎ/𝐻,𝜆^𝑛−𝜆𝑔(𝑥). (20) Hence 𝑔(𝜆^𝑛𝑥^𝑛−𝜆𝑥)→0 where 𝜆^𝑛→𝜆 and 𝑥^𝑛→𝑥 as 𝑛→∞. Consequently 𝑔 is a paranom on 𝑚²(𝑀,𝐴,𝜙,𝑝).

Proposition 6. The class 𝑚²(𝑀,𝜙,𝑝) of double sequences is solid.

Proof. Let 𝛼={𝛼_𝑘,𝑙} be a double sequence of scalars such that |𝛼_𝑘,𝑙|≤1 and 𝑦={𝑦_𝑘,𝑙}∈𝑚²(𝑀,𝜙,𝑝). Then we can write 1sup𝜙_{𝑠,𝑡𝑘∈𝜎1𝑙∈𝜎2}𝑀𝛼_𝑘,𝑙𝑥_𝑘,𝑙𝜌^𝑝𝑘𝑙∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡1≤sup𝜙_{𝑠,𝑡𝑘∈𝜎1𝑙∈𝜎2}𝑀𝛼_𝑘,𝑙𝑦_𝑘,𝑙𝜌^𝑝𝑘𝑙∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡1≤sup𝜙_{𝑠,𝑡𝑘∈𝜎1𝑙∈𝜎2}𝑀𝑦_𝑘,𝑙𝜌^𝑝𝑘𝑙∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡. (21)
This implies that {𝛼_𝑘,𝑙𝑦_𝑘,𝑙}∈𝑚²(𝑀,𝜙,𝑝), and hence the class 𝑚²(𝑀,𝜙,𝑝) is solid.

Corollary 7. The space 𝑚²(𝑀,𝜙,𝑝) is monotone.

Theorem 8. Let 𝜓 be another double sequence like 𝜙. Then 𝑚²(𝑀,𝐴,𝜙,𝑝)⊆𝑚²(𝑀,𝐴,𝜓,𝑝) if and only if sup_{(𝑠,𝑡)≥(1,1)}(𝜙_𝑠,𝑡/𝜓_𝑠,𝑡)<∞.

Proof. Let 𝐾=sup_{(𝑠,𝑡)≥(1,1)}(𝜙_𝑠,𝑡/𝜓_𝑠,𝑡)<∞. Then 𝜙_𝑠,𝑡≤𝐾⋅𝜓_𝑠,𝑡 for all (𝑠,𝑡)≥(1,1). If 𝑥={𝑥_𝑘,𝑙}∈𝑚²(𝑀,𝐴,𝜙,𝑝), then 1sup𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗(𝑥)𝜌^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡<∞, (22) for some 𝜌>0. Thus 1sup𝐾𝜓_{𝑠,𝑡𝑘∈𝜎1𝑙∈𝜎2}𝑀𝐴_𝑖𝑗(𝑥)𝜌^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡<∞, (23) for some 𝜌>0, and hence 𝑥={𝑥_𝑘,𝑙}∈𝑚²(𝑀,𝐴,𝜓,𝑝). This shows that 𝑚²(𝑀,𝐴,𝜙,𝑝)⊆𝑚²(𝑀,𝐴,𝜓,𝑝).
Conversely, let 𝑚²(𝑀,𝐴,𝜙,𝑝)⊆𝑚²(𝑀,𝐴,𝜓,𝑝). We say 𝛼_𝑠,𝑡=𝜙_𝑠,𝑡/𝜓_𝑠,𝑡 for all (𝑠,𝑡)≥(1,1) and suppose sup_{(𝑠,𝑡)≥(1,1)}𝛼_𝑠,𝑡=∞. Then there exists a subsequence {𝛼_{𝑠𝑖,𝑡𝑖}} of {𝛼_𝑠,𝑡} such that lim_𝑖→∞𝛼_{𝑠𝑖,𝑡𝑖}=∞. If 𝑥={𝑥_𝑘,𝑙}∈𝑚²(𝑀,𝐴,𝜙,𝑝), then we have 1sup𝜓_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗(𝑥)𝜌^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡=sup𝛼_𝑠,𝑡1𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗(𝑥)𝜌^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡≥sup_𝑚≥1𝛼_{𝑠𝑚,𝑡𝑚}1⋅sup𝜙_{𝑠𝑚,𝑡𝑚𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗(𝑥)𝜌^𝑝𝑖𝑗∶𝜎₁×𝜎₂∈℘_{𝑠𝑚,𝑡𝑚}=∞. (24)
This is a contradiction as 𝑥={𝑥_𝑘,𝑙}∉𝑚²(𝑀,𝐴,𝜙,𝑝). This completes the proof.

Corollary 9. 𝑚²(𝑀,𝐴,𝜙,𝑝)=𝑚²(𝑀,𝐴,𝜓,𝑝) if and only if sup_{(𝑠,𝑡)≥(1,1)}𝛼_𝑠,𝑡<∞ and sup_{(𝑠,𝑡)≥(1,1)}𝛼⁻¹_𝑠,𝑡<∞.

Theorem 10. Let 𝑀, 𝑀₁, and 𝑀₂ be Orlicz functions satisfying Δ₂-condition. Then (a)𝑚²(𝑀₁,𝐴,𝜙,𝑝)⊆𝑚²(𝑀∘𝑀₁,𝐴,𝜙,𝑝),(b)𝑚²(𝑀₁,𝐴,𝜙,𝑝)∩𝑚²(𝑀₂,𝐴,𝜙,𝑝)⊆𝑚²(𝑀₁+𝑀₂,𝐴,𝜙,𝑝).

Proof. (a) Let 𝑥={𝑥_𝑘,𝑙}∈𝑚²(𝑀₁,𝐴,𝜙,𝑝). Then there exists 𝜌>0 such that 1sup𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀₁𝐴_𝑖𝑗(𝑥)𝜌^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡<∞. (25)By the continuity of 𝑀, we select a number 𝛿 with 0<𝛿<1 such that 𝑀(𝑡)<𝜀, whenever 0≤𝑡<𝛿, for arbitrary 0<𝜀<1. Now let 𝑦_𝑖,𝑗=𝑀₁(|𝐴_𝑖𝑗(𝑥)|/𝜌). We can write _{𝑖∈𝜎1𝑗∈𝜎2}𝑀𝑦_𝑖,𝑗^𝑝𝑖𝑗=&_{𝑦𝑖,𝑗≤𝛿}𝑀𝑦_𝑖,𝑗^𝑝𝑖𝑗+_{𝑦𝑖,𝑗>𝛿}𝑀𝑦_𝑖,𝑗^𝑝𝑖𝑗. (26) By the properties of 𝑀, we have _{𝑦𝑖,𝑗≤𝛿}𝑀𝑦_𝑖,𝑗^𝑝𝑖𝑗≤max1,𝑀(1)^𝐻_{𝑦𝑖,𝑗≤𝛿}𝑦_𝑖,𝑗^𝑝𝑖𝑗. (27)
Again we can write 𝑀𝑦_𝑖,𝑗𝑦<𝑀1+^𝑖,𝑗𝛿<12𝑀(2)+12𝑀2𝑦_𝑖,𝑗𝛿, (28) for 𝑦_𝑖,𝑗>𝛿. If we use that 𝑀 satisfies Δ₂-condition, then we find 𝑀𝑦_𝑖,𝑗<12𝑇𝑦_𝑖,𝑗𝛿𝑀(2)+12𝑇𝑦_𝑖,𝑗𝛿𝑀(2)𝑦=𝑇^𝑖,𝑗𝛿𝑀(2), (29) and so _{𝑦𝑖,𝑗>𝛿}𝑀𝑦_𝑖,𝑗^𝑝𝑖𝑗𝑇≤max1,𝛿𝑀(2)^𝐻_{𝑦𝑖,𝑗>𝛿}𝑦_𝑖,𝑗. (30) Hence we get 1sup𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀𝑀₁𝐴_𝑖𝑗(𝑥)𝜌^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡≤max1,𝑀(1)^𝐻1×sup𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑦_𝑖,𝑗^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡𝑇+max1,𝛿𝑀(2)^𝐻1×sup𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑦_𝑖,𝑗^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡. (31)
Finally, we have {𝑥_𝑘,𝑙}∈𝑚²(𝑀∘𝑀₁,𝜙), and hence 𝑚²(𝑀₁,𝜙)⊆𝑚²(𝑀∘𝑀₁,𝜙).(b) Let {𝑥_𝑘,𝑙}∈𝑚²(𝑀₁,𝐴,𝜙,𝑝)∩𝑚²(𝑀₂,𝐴,𝜙,𝑝). Then there exists 𝜌>0 such that 1sup𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀₁𝐴_𝑖𝑗(𝑥)𝜌^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡1<∞,sup𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀₂𝐴_𝑖𝑗(𝑥)𝜌^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡<∞. (32)By the inequality _{𝑖∈𝜎1𝑗∈𝜎2}𝑀₁+𝑀₂𝐴_𝑖𝑗(𝑥)𝜌^𝑝𝑖𝑗≤max1,2^𝐻−1_{𝑖∈𝜎1𝑗∈𝜎2}𝑀₁𝐴_𝑖𝑗(𝑥)𝜌^𝑝𝑖𝑗+max1,2^𝐻−1_{𝑖∈𝜎1𝑗∈𝜎2}𝑀₂𝐴_𝑖𝑗(𝑥)𝜌^𝑝𝑖𝑗, (33) we have 𝑚²(𝑀₁,𝐴,𝜙,𝑝)∩𝑚²(𝑀₂,𝐴,𝜙,𝑝)⊆𝑚²(𝑀₁+𝑀₂,𝐴,𝜙,𝑝).

Theorem 11. (a)𝑙⁽²⁾(𝑀,𝐴,𝜙,𝑝)⊆𝑚²(𝑀,𝐴,𝜙,𝑝)⊆𝑙⁽²⁾(𝑀,𝐴,𝜙,∞). (b)𝑚²(𝑀,𝐴,𝜙,𝑝)=𝑙⁽²⁾(𝑀,𝐴,𝜙,𝑝) if and only if sup_{(𝑠,𝑡)≥(1,1)}𝜙_𝑠,𝑡<∞.(c)𝑚²(𝑀,𝐴,𝜙,𝑝)=𝑙⁽²⁾(𝑀,𝐴,𝜙,∞) if and only if sup_{(𝑠,𝑡)≥(1,1)}(𝜙_𝑠,𝑡/𝑠.𝑡)>0.

Proof. (a) Let 𝑥={𝑥_𝑘,𝑙}∈𝑙⁽²⁾(𝑀,𝐴,𝜙,𝑝), and let a set 𝐵 be defined as follows: 1𝐵=𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗(𝑥)𝜌^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡. (34)
Since {𝜙_𝑘,𝑙} is a nondecreasing double sequence, {1/𝜙_𝑘,𝑙} is a nonincreasing double sequence. So we obtain 1𝜙^∞_{1,1𝑖,𝑗=1}𝑀𝐴_𝑖𝑗(𝑥)𝜌^𝑝𝑖𝑗≥𝑏, (35) for all 𝑏∈𝐵 and hence 𝐴∞>𝑀^𝑖𝑗(𝑥)𝜌^𝑝𝑖𝑗≥𝜙_1,1⋅sup𝐵. (36) Thus we have 𝑙⁽²⁾(𝑀,𝐴,𝜙,𝑝)⊆𝑚²(𝑀,𝐴,𝜙,𝑝).
Conversely if 𝑥={𝑥_𝑘,𝑙}∈𝑚²(𝑀,𝐴,𝜙,𝑝), then it is clear that 1sup𝐵≥𝜙_1,1𝑀𝐴_𝑖𝑗(𝑥)𝜌^𝑝𝑖𝑗, (37) for all (𝑖,𝑗)∈𝑁×𝑁, and hence 1sup𝐵≥𝜙_1,1sup_{𝑖,𝑗∈𝑁×𝑁}𝑀𝐴_𝑖𝑗(𝑥)𝜌^𝑝𝑖𝑗. (38) This shows that if 𝑥={𝑥_𝑘,𝑙}∈𝑚²(𝑀,𝐴,𝜙,𝑝), then {𝑥_𝑘,𝑙}∈𝑙⁽²⁾(𝑀,𝐴,𝜙,∞). Thus we have 𝑚²(𝑀,𝐴,𝜙,𝑝)⊆𝑙⁽²⁾(𝑀,𝐴,𝜙,∞).(b) It is clear that 𝑚²(𝑀,𝐴,𝜓,𝑝)=𝑙⁽²⁾(𝑀,𝐴,𝜙,∞) where 𝜓_𝑠,𝑡=1 for all (𝑠,𝑡)∈𝑁×𝑁. Then we can write sup_{(𝑠,𝑡)≤(1,1)}𝜙_𝑠,𝑡=sup_{(𝑠,𝑡)≤(1,1)}(𝜙_𝑠,𝑡/𝜓_𝑠,𝑡)<∞. By Theorem 8, we have 𝑚²(𝑀,𝐴,𝜙,𝑝)⊆𝑚²(𝑀,𝐴,𝜓,𝑝), and according to alternative (a) 𝑚²𝑀,𝐴,𝜙,𝑝=𝑙⁽²⁾𝑀,𝐴,𝜙,𝑝. (39)(c) Firstly we show that 𝑚²(𝑀,𝐴,𝜓,𝑝)=𝑙⁽²⁾(𝑀,𝐴,𝜙,∞) if 𝜓_𝑠,𝑡=𝑠.𝑡 for all (𝑠,𝑡)∈𝑁×𝑁. Let 𝑥={𝑥_𝑘,𝑙}∈𝑙⁽²⁾(𝑀,𝐴,𝜙,∞). Then for (𝑠,𝑡)≥(1,1) and 𝜎₁×𝜎₂∈℘_𝑠,𝑡, we can find some 𝜌>0 such that 1𝑠.𝑡_{𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗(𝑥)𝜌^𝑝𝑖𝑗≤1𝑠.𝑡𝑠.𝑡.sup_{𝑖,𝑗∈𝑁×𝑁}𝑀𝐴_𝑖𝑗(𝑥)𝜌^𝑝𝑖𝑗<∞. (40)This gives the inclusion 𝑙⁽²⁾(𝑀,𝐴,𝜙,∞)⊆𝑚²(𝑀,𝐴,𝜓,𝑝). Conversely let 𝑥={𝑥_𝑘,𝑙}∈𝑚²(𝑀,𝐴,𝜓,𝑝). Then for (𝑖,𝑗)∈𝑁×𝑁, we can find some 𝜌>0 such that 𝑀𝐴_𝑖𝑗(𝑥)𝜌^𝑝𝑖𝑗=1𝐴𝑠.𝑡𝑠.𝑡.𝑀^𝑖𝑗(𝑥)𝜌^𝑝𝑖𝑗1≤sup𝜙_{𝑠,𝑡𝑖∈𝜎1𝑗∈𝜎2}𝑀𝐴_𝑖𝑗(𝑥)𝜌^𝑝𝑖𝑗∶()≥()𝑠,𝑡1,1,𝜎₁×𝜎₂∈℘_𝑠,𝑡<∞. (41) This shows that 𝑚²(𝑀,𝐴,𝜓,𝑝)⊆𝑙⁽²⁾(𝑀,𝐴,𝜙,∞). By Theorem 8 and alternative (a), we can write 𝑚²(𝑀,𝐴,𝜙,𝑝)=𝑙⁽²⁾(𝑀,𝐴,𝜙,∞) if and only if sup_{(𝑠,𝑡)≥(1,1)}(𝑠.𝑡/𝜙_𝑠,𝑡)<∞. This completes the proof.