AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2014/260326 260326 Research Article Double Sequence Spaces by Means of Orlicz Functions http://orcid.org/0000-0001-7780-8137 Alotaibi Abdullah 1 Mursaleen M. 2 http://orcid.org/0000-0002-2611-3391 Raj Kuldip 3 Kocinac Ljubisa 1 Department of Mathematics, Faculty of Science, King Abdulaziz University P.O. Box 80203, Jeddah 21589 Saudi Arabia kau.edu.sa 2 Department of Mathematics, Aligarh Muslim University, Aligarh 202002 India amu.ac.in 3 School of Mathematics, Shri Mata Vaishno Devi University, Katra, Jammu and Kashmir 182320 India smvdu.net.in 2014 1882014 2014 08 04 2014 23 07 2014 18 8 2014 2014 Copyright © 2014 Abdullah Alotaibi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We define some classes of double entire and analytic sequences by means of Orlicz functions. We study some relevant algebraic and topological properties. Further some inclusion relations among the classes are also examined.

1. Introduction and Preliminaries

Of the definitions of convergence commonly employed for double series, only that due to Pringsheim permits a series to converge conditionally. Therefore, in spite of any disadvantages which it may possess, this definition is better adapted than others to the study of many problems in double sequences and series. Chief among the reasons why the theory of double sequences, under the Pringsheim definition of convergence, present difficulties not encountered in the theory of simple sequences, is the fact that a double sequence (xij) may converge without xij being a bounded function of i and j. Thus it is not surprising that many authors in dealing with the convergence of double sequences should have restricted themselves to the class of bounded sequences or in dealing with the summability of double series to the class of series for which the function whose limit is the sum of the series is a bounded function of i and j. Without such a restriction, peculiar things may sometimes happen; for example, a double power series may converge with partial sum (Sij) unbounded at a place exterior to its associated circles of convergence. Nevertheless there are problems in the theory of double sequences and series where this restriction of boundedness as it has been applied is considerably more stringent than need be. The initial works on double sequences are found in Bromwich . Later on, it was studied by Hardy , Móricz , Móricz and Rhoades , Basarir and Sonalcan , and many others. Hardy  introduced the notion of regular convergence for double sequences. Mursaleen and Mohiuddine [6, 7] have characterized four dimensional matrix transformations between double sequences x=(xm,n). A good account of the study of double sequences can be found in most recent monograph by Mursaleen and Mohiuddine . More recently, Altay and Başar  have defined the spaces BS, BS(t), CSp, CSbp, CSr, and BV of double sequences consisting of all double series whose sequence of partial sums is in the spaces Mu, Mu(t), Cp, Cbp, Cr, and Lu, respectively, and also examined some properties of these sequence spaces and determined the α-duals of the spaces BS, BV, and CSbp and the β(v)-duals of the spaces CSbp and CSr of double series. Now, recently, Başar and Sever  have introduced the Banach space Lq of double sequences corresponding to the well-known space lq of single sequences and examined some properties of the space Lq. By the convergence of a double sequence we mean the convergence in the Pringsheim sense; that is, a double sequence x=(xk,l) has Pringsheim limit L (denoted by P-limx=L) provided that given ϵ>0 there exists nN such that |xk,l-L|<ϵ whenever k,l>n; see . We will write more briefly as P-convergent. The double sequence x=(xk,l) is bounded if there exists a positive number M such that |xk,l|<M for all k and l.

An Orlicz function M:[0,)[0,) is a continuous, nondecreasing, and convex function such that M(0)=0, M(x)>0 for x>0 and M(x) as x. Lindenstrauss and Tzafriri  used the idea of Orlicz function to define the following sequence space. Let w be the space of all real or complex sequences x=(xk); then (1)lM={xw:k=1M(|xk|ρ)<,  for  some  ρ>0} which is called as an Orlicz sequence space. Also lM is a Banach space with the norm (2)x=inf{ρ>0:k=1M(|xk|ρ)1}. Also, it was shown in  that every Orlicz sequence space lM contains a subspace isomorphic to lp  (p1). The Δ2- condition is equivalent to M(Lx)LM(x), for all L with 0<L<1. An Orlicz function M can always be represented in the following integral form: (3)M(x)=0xη(t)dt, where η is known as the kernel of M and is right differentiable for t0, η(0)=0, and η(t)>0; η is nondecreasing and η(t) as t.

Let X be a linear metric space. A function p:XR is called paranorm, if

p(x)0, for all xX,

p(-x)=p(x), for all xX,

p(x+y)p(x)+p(y), for all x,yX,

if (λn) is a sequence of scalars with λnλ    as    n and (xn) is a sequence of vectors with p(xn-x)0    as    n, then p(λnxn-λx)0    as    n.

A paranorm p for which p(x)=0 implies x=0 is called total paranorm and the pair (X,p) is called a total paranormed space. It is well known that the metric of any linear metric space is given by some total paranorm (see , Theorem 10.4.2, p.183). For more details about sequence spaces see [13, 1521].

A complex sequence, whose kth term is xk, is denoted by {xk}. Let φ be the set of all finite sequences. A sequence x={xk} is said to be analytic if supk|xk|1/k<. The vector space of all analytic sequences will be denoted by Λ. A sequence x is called entire sequence if  limk|xk|1/k=0. The vector space of all entire sequences will be denoted by Γ.

The notion of difference sequence spaces was introduced by Kızmaz , who studied the difference sequence spaces l(Δ), c(Δ), and co(Δ). The notion was further generalized by Et and Çolak  by introducing the spaces l(Δn), c(Δn), and co(Δn).

Let r, s be nonnegative integers; then for Z=l,c,co we have sequence spaces (4)Z(Δsr)={x=(xk)w:(Δsrxk)Z}, where Δsrx=(Δsrxk)=(Δsr-1xk-Δsr-1xk+1) and Δs0xk=xk for all kN, which is equivalent to the following binomial representation: (5)Δsrxk=v=0r(-1)v(rv)xk+sv. Taking s=1, we get the spaces which were studied by Et and Çolak . Taking r=s=1, we get the spaces which were introduced and studied by Kızmaz .

Let w′′ denote the space of all complex double sequences x=(xk,l). The space consisting of all those sequences x in w′′ such that Mk,l(|xk,l|1/k+l/ρ)0  as  k,l for some arbitrary fixed ρ>0 is denoted by ΓM2 and is known as double Orlicz space of entire sequences. The space ΓM2 is a metric space with the metric d(x,y)=supk,lMk,l(|xk,l-yk,l|1/k+l/ρ) for all x={xk,l} and y={yk,l} in ΓM2.

The space consisting of all those sequences x in w′′ such that (supk,l(Mk,l(|xk,l|1/k+l/ρ)))< for some arbitrarily fixed ρ>0 is denoted by ΛM2 and is known as double Orlicz space of analytic sequences.

A double sequence space E is said to be solid or normal if (αk,lxk,l)E whenever (xk,l)E and for all sequences of scalars (αk,l) with |αk,l|1 (see ).

The following inequality will be used throughout the paper. Let p=(pk,l) be a double sequence of positive real numbers with 0<pk,lsupk,l=H and let K=max{1,2H-1}. Then for the factorable sequences {ak,l} and {bk,l} in the complex plane, we have (6)|ak,l+bk,l|pk,lK(|ak,l|pk,l+|bk,l|pk,l). Let M=(Mk,l) be a sequence of Orlicz functions, let p=(pk,l) be a bounded sequence of positive real numbers, let u=(uk,l) be a sequence of strictly positive real numbers, and let X be locally convex Hausdorff topological linear space whose topology is determined by a set of continuous seminorms q. The symbols   Λ2(X), Γ2(X) denote the space of all double analytic and double entire sequences, respectively, defined over X. In this paper we define the following sequence spaces: (7)ΛM2(Δsr,u,p,q)={[Mk,l(q(|Δsrxk,l|1/k+lρ))]pk,lxΛ2(X):supm,n1mnfdfdf×k,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/k+lρ))]pk,lgfg<,for  some  ρ>0[Mk,l(q(|Δsrxk,l|1/k+lρ))]pk,lsupm,n1mn},(8)ΓM2(Δsr,u,p,q)={[Mk,l(q(|Δsrxk,l|1/k+lρ))]pk,lxΓ2(X):1mngfffffd×k,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/k+lρ))]pk,ldsdds0  asm,n,for  someρ>0[Mk,l(q(|Δsrxk,l|1/k+lρ))]pk,l}. The main aim of this paper is to introduce some double entire sequence spaces ΛM2(Δsr,u,p,q) and ΓM2(Δsr,u,p,q) defined by a sequence of Orlicz functions and study some topological properties and inclusion relation between these spaces.

2. Main Results Theorem 1.

Let M=(Mk,l) be a sequence of Orlicz functions, let p=(pk,l) be a bounded sequence of positive real numbers, and let u=(uk,l) be a sequence of strictly positive real numbers; then the spaces ΓM2(Δsr,u,p,q) and ΛM2(Δsr,u,p,q) are linear spaces over the field of complex numbers C.

Proof.

Let x,yΓM2(Δsr,u,p,q) and α,βC. In order to prove the result, we need to find some ρ3>0 such that (9)1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsr(αxk,l+βyk,l)|1/(k+l)ρ3))]pk,l0as  m,n. Since x,yΓM2(Δsr,u,p,q), there exist some positive ρ1 and ρ2 such that (10)1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ1))]pk,l0as  m,n,1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsryk,l|1/(k+l)ρ2))]pk,l0as  m,n. Since M=(Mk,l) is a nondecreasing convex function, q is a seminorm and Δsr is linear and so, by using inequality (6), we have (11)1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsr(αxk,l+βyk,l)|1/(k+l)ρ3))]pk,l1(mn)×k,l=1,1m,nuk,l[Mk,l(q(α1/(k+l)|Δsrxk,l|1/(k+l)ρ3dsddsdfdfdfdffdsdddddf+  β1/(k+l)|Δsryk,l|1/(k+l)ρ3))]pk,l. Take ρ3>0 such that 1/ρ3  =min{1/αρ1,1/βρ2}: (12)1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsr(αxk,l+βyk,l)|1/(k+l)ρ3))]pk,l1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ1fdsfdfdfdfdfddfdffdd+  |Δsryk,l|1/(k+l)ρ2))]pk,l1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ1))pk,lfsdfsdfsdfsdfsdfdsfff+uk,lMk,l(q(|Δsryk,l|1/(k+l)ρ2))pk,l]K1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ1))]pk,l+K1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsryk,l|1/(k+l)ρ2))]pk,l0asm,n. Hence (13)k,l=1,1m,nuk,l[Mk,l(q(|αΔsrxk,l+βΔsryk,l|1/(k+l)ρ3))]pk,l0fdsfdfsdfsdfsdfsdfsdfsdfsdfsdfdsfdsfsdfdffdas  m,n. This proves that ΓM2(Δsr,u,p,q) is a linear space. Similarly, we can prove ΛM2(Δsr,u,p,q) is a linear space.

Theorem 2.

Let M=(Mk,l) be a sequence of Orlicz functions, let p=(pk,l) be a bounded sequence of positive real numbers, and let u=(uk,l) be a sequence of strictly positive real numbers; then the space ΓM2(Δsr,u,p,q) is a paranormed space with paranorm defined by (14)gΔ(x)=inf{[Mk,l(q(|Δsrxk,l|1/(k+l)ρ))]pk,lρpm,n/H:dsdsdsdsupk,l1uk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ))]pk,lfdsfsdf1;ρ>0[Mk,l(q(|Δsrxk,l|1/(k+l)ρ))]pk,lρpm,n/H:}  , where H=max(1,supk,lpk,l).

Proof.

Clearly gΔ(x)0,gΔ(x)=gΔ(-x) and gΔ(θ¯)=0, where θ is the zero sequence of X. For (xk,l),(yk,l)ΓM2(Δsr,u,p,q), there exist ρ1,ρ2>0 such that (15)supk,l1uk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ1))]pk,l1,supk,l1uk,l[Mk,l(q(|Δsryk,l|1/(k+l)ρ2))]pk,l1. Suppose that ρ=ρ1+ρ2; then (16)supk,l1uk,l[Mk,l(q(|Δsr(xk,l+yk,l)|1/(k+l)ρ))]pk,l(ρ1ρ1+ρ2)supk,l1uk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ1))]pk,l+(ρ2ρ1+ρ2)×supk,l1uk,l[Mk,l(q(|Δsryk,l|1/(k+l)ρ2))]pk,l1. Hence (17)gΔ(x+y)inf{[Mk,l(q(|Δsrxk,l|1/(k+l)ρ1+ρ2))]pk,l(ρ1+ρ2)pm,n/H:dasdsdsdFsupk,l1uk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ1+ρ2))]pk,ldsdsdsds1,ρ1,ρ2>0,m,nN[Mk,l(q(|Δsrxk,l|1/(k+l)ρ1+ρ2))]pk,l(ρ2)pm,n/H}inf{[Mk,l(q(|Δsryk,l|1/(k+l)ρ2))]pk,l(ρ1)pm,n/H:dsdsdsdsFsupk,l1uk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ1))]pk,lfsdfsdfsdf1,ρ1>0,m,nN(ρ1)pm,n/H[Mk,l(q(|Δsryk,l|1/(k+l)ρ2))]pk,l}+inf{[Mk,l(q(|Δsryk,l|1/(k+l)ρ2))]pk,l(ρ2)pm,n/H:dsadsdsdsdfsupk,l1uk,l[Mk,l(q(|Δsryk,l|1/(k+l)ρ2))]pk,lfdsfdffsdffds1,ρ2>0,m,nN(ρ2)pm,n/H[Mk,l(q(|Δsryk,l|1/(k+l)ρ2))]pk,l}. Thus we have gΔ(x+y)gΔ(x)+gΔ(y). Hence gΔ satisfies the triangle inequality. Now, (18)gΔ(λx)=inf{[Mk,l(q(|Δsryk,l|1/(k+l)ρ2))]pk,l(ρ)pm,n/H:fdfdfffsupk,l1uk,l[Mk,l(q(|λΔsrxk,l|1/(k+l)ρ))]pk,ldsaddsf1,ρ>0,m,nN[Mk,l(q(|Δsryk,l|1/(k+l)ρ2))]pk,l(ρ)pm,n/H}=inf{[Mk,l(q(|Δsryk,l|1/(k+l)ρ2))]pk,l(r|λ|)pm,n/H:dfdfdfsupk,l1uk,l[Mk,l(q(|Δsrxk,l|1/(k+l)r))]pk,lfdfdffdf1,r>0,m,nN[Mk,l(q(|Δsryk,l|1/(k+l)ρ2))]pk,l(ρ)pm,n/H}, where r=ρ/|λ|. Hence ΓM2(Δsr,u,p,q) is a paranormed space.

Theorem 3.

If M=(Mk,l) and M′′=(Mk,l′′) are two sequences of Orlicz functions, then (19)ΓM2(Δsr,u,p,q)ΓM′′2(Δsr,u,p,q)ΓM+M′′2(Δsr,u,p,q).

Proof.

Let xΓM2(Δsr,u,p,q)ΓM′′2(Δsr,u,p,q). Then there exist ρ1 and ρ2 such that (20)1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ1))]pk,l0as  m,n,1mnk,l=1,1m,nuk,l[Mk,l′′(q(|Δsrxk,l|1/(k+l)ρ2))]pk,l0as  m,n. Let ρ=min(1/ρ1  ,1/ρ2  ). Then we have (21)1mnk,l=1,1m,nuk,l[(Mk,l+Mk,l′′)(q(|Δsrxk,l|1/(k+l)ρ))]pk,lK[1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ1))]pk,l]+K[1mnk,l=1,1m,nuk,l[Mk,l′′(q(|Δsrxk,l|1/(k+l)ρ2))]pk,l],0as    m,n by (20). Therefore, xΓM+M′′2(Δsr,u,p,q).

Theorem 4.

Let r1. Then we have the following inclusions:

ΓM2(Δsr-1,u,p,q)ΓM2(Δsr,u,p,q),

ΛM2(Δsr-1,u,p,q)ΛM2(Δsr,u,p,q).

Proof.

Let xΓM2(Δsr-1,u,p,q). Then we have (22)1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsr-1xk,l|1/(k+l)ρ))]pk,l0fsdfdsfsdfsdfdsfsdfsfas  m,n,for  some  ρ>0. Since M=(Mk,l) is nondecreasing convex function and q is a seminorm, we have (23)1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ))]pk,l1mn×k,l=1,1m,nuk,lfsdfsd×[Mk,l(q(|Δsr-1xk,l-Δsr-1xk+1,l+1|1/(k+l)ρ))]pk,lK{1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsr-1xk,l|1/(k+l)ρ))]pk,lfdsfdfdff+1mnfdsfdfdff×k,l=1,1m,nuk,lfdsfdsfdsfdsfd×[Mk,l(q(|Δsr-1xk+1,l+1|1/(k+l)ρ))]pk,l}0as  m,n. Therefore, (1/mn)k,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)/ρ))]pk,l0 as m,n. Hence, xΓM2(Δsr,u,p,q). This completes the proof of (i). Similarly, we can prove (ii).

Theorem 5.

Let 0pk,ltk,l and let {tk,l/pk,l} be bounded. Then ΓM2(Δsr,u,t,q)ΓM2(Δsr,u,p,q).

Proof.

Let xΓM2(Δsr,u,t,q). Then (24)1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ))]tk,l0fgdsgdfgsdfggfgsfdgsdgdfgsdgsdffgas  m,n. Let wk,l=(1/mn)k,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)/ρ))]qk,l and λk,l=pk,l/tk,l  . Since pk,ltk,l, we have 0λk,l1. Take 0<λ<λk,l. Define (25)uk,l={wk,lif    wk,l10if    wk,l<1,vk,l={0if    wk,l1wk,lif    wk,l<1, where wk,l=uk,l+vk,l and wk,lλk,l=uk,lλk,l+vk,lλk,l. It follows that uk,lλk,luk,lwk,land vk,lλk,lvk,lλ. Since wk,lλk=uk,lλk+vk,lλk, then wk,lλk,lwk,l+vk,lλ. Thus, (26)1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ))tk,l]λk,l1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ))]tk,l1mn×k,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ))tk,l]pk,l/tk,l  1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsm,nxk,l|1/(k+l)ρ))]tk,l1mn×k,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ))]pk,l1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ))]tk,l0        as  m,n(by(24)). Therefore, (27)1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ))]pk,l0fdsfsdfdsfsdfdsfdfsdfsdfdsfsdfdfffas  m,n. Hence xΓM2(Δsr,u,p,q). From (24), we get ΓM2(Δsr,u,t,q)ΓM2(Δsr,u,p,q).

Theorem 6.

(i) Let 0<infpk,lpk,l1. Then ΓM2(Δsr,u,p,q)ΓM2(Δsr,u,q). (ii) Let 1pk,lsuppk,l<. Then ΓM2(Δsr,u,q)ΓM2(Δsr,u,p,q).

Proof.

(i) Let xΓM2(Δsr,u,p,q). Then (28)1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ))]pk,l0fdsfsdfffffffffffffffffffffffffffffffffffffdfdas  m,n. Since 0<infpk,lpk,l1, (29)1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ))]1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ))]pk,l0as  m,n. From (28) and (29) it follows that xΓM2(Δsr,u,q). Thus ΓM2(Δsr,u,p,q)ΓM2(Δsr,u,q).

(ii) Let pk,l1 for each k,l and suppk,l< and let xΓM2(Δsr,u,q). Then (30)1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ))]0as  m,n. Since 1pk,lsuppk,l<, we have (31)1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ))]pk,l1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ))]0as  m,n. This implies that xΓM2(Δsr,u,p,q). Therefore, ΓM2(Δsr,u,q)ΓM2(Δsr,u,p,q).

Theorem 7.

Suppose (1/mn)k,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)/ρ))]pk,l|xk,l|1/k+l; then Γ2ΓM2(Δsr,u,p,q).

Proof.

Let xΓ2. Then we have (32)|xk,l|1/k+l0as  k,l. But (1/mn)k,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)/ρ))]pk,l|xk,l|1/k+l, by our assumption, implies that (33)1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ))]pk,l0fsdfsdfdsfsdfsdfsdfdsfsdffdas  m,n  by  (32). Then xΓM2(Δsr,u,p,q) and Γ2ΓM2(Δsr,u,p,q).

Theorem 8.

Γ M 2 ( Δ s r , u , p , q ) is solid.

Proof.

Let (xk,l)ΓM2(Δsr,u,p,q); then (34)1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ))]pk,l0fdsfdsfsdfsdfdffdas  m,n,for  some  ρ>0. Let (αk,l) be a double sequence of scalars such that |αk,l|1 for all k,lN×N. Then we have (35)1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsrαk,lxk,l|1/(k+l)ρ))]pk,l1mnk,l=1,1m,nuk,l[Mk,l(q(|Δsrxk,l|1/(k+l)ρ))]pk,l0gdfgsdgdfsgdfgsdgsdfgsdfgsdfgdsfgsdfas  m,n and this completes the proof.

Corollary 9.

Γ M 2 ( Δ s r , u , p , q ) is monotone.

Proof.

It is obvious.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.

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