# The double growth rate of Orlicz sequence spaces.

[section]1. Introduction

Throughout w, x and A denote the classes of all, gai and analytic scalar valued single sequences, respectively. We write [w.sup.2] for the set of all complex sequences ([x.sub.mn]), where m, n [member of] [??], the set of positive integers. Then, [w.sup.2] is a linear space under the Coordinatewise addition and scalar multiplication.

Some initial work on double sequence spaces were found in Bromwich [4]. Later on, they were investigated by Hardy [13], Moricz [19], Moricz and Rhoades [20], Basarir and Solankan [3], Tripathy [37], Tripathy and Dutta ([41],[42]), Tripathy and Sarma ([43],[44],[45],[46]), Tripathy and Sen [48], Turkmenoglu [49], and many others.

Let us define the following sets of double sequences:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where t = ([t.sub.mn]) is the sequence of positive reals for all m, n [member of] n and P-[lim.sub.m,n [right arrow][infinity]] denotes the limit in the Pringsheim's sense. In the case [t.sub.mn] = 1 for all m, n [member of] [??]; [M.sub.u] (t), [C.sub.p](t), [C.sub.0p](t), [L.sub.u] (t), [C.sub.bp](t) and [C.sub.0bp](t) reduce to the sets [M.sub.u], [C.sub.p], [C.sub.0p], [L.sub.u], [C.sub.bp] and [C.sub.0bp], respectively. Now, we may summarize the knowledge given in some document related to the double sequence spaces. Gokhan and Colak [11,12] have proved that [M.sub.u] (t) and Cp (t), [C.sub.bp] (t) are complete paranormed spaces of double sequences and obtained the [alpha]-, [beta], [gamma]-duals of the spaces [M.sub.u] (t) and [C.sub.bp] (t). Quite recently, in her PhD thesis, Zelter [50] has essentially studied both the theory of topological double sequence spaces and the theory of summability of double sequences. Mursaleen and Edely [21] and Tripathy [35] have recently introduced the statistical convergence and Cauchy for double sequences independently and given the relation between statistical convergent and strongly Cesaro summable double sequences. Later, Mursaleen [22] and Mursaleen and Edely [23] have defined the almost strong regularity of matrices for double sequences and applied these matrices to establish a core theorem and introduced the M-core for double sequences and determined those four dimensional matrices transforming every bounded double sequences x = ([x.sub.jk]) into one whose core is a subset of the M-core of x. More recently, Altay and Basar [1] have defined the spaces BS, BS (t), [CS.sub.p], [CS.sub.bp], [CS.sub.r] and BV of double sequences consisting of all double series whose sequence of partial sums are in the spaces [M.sub.u], [M.sub.u](t), [C.sub.p], [C.sub.bp], [C.sub.r] and [L.sub.u], respectively, and have also examined some properties of those sequence spaces and determined the a-duals of the spaces BS, BV, [CS.sub.bp] and the [beta] ([??})-duals of the spaces [CS.sub.bp] and [CS.sub.r] of double series. Quite recently Basar and Sever [5] have introduced the Banach space [L.sub.q] of double sequences corresponding to the well-known space [l.sub.q] of single sequences and have examined some properties of the space [L.sub.q]. Quite recently Subramanian and Misra ([29],[30]>[33]) have studied the space [s.sup.2.sub.M] (p, q, u) and the generalized gai of double sequences and have proved some inclusion relations.

We need the following inequality in the sequel of the paper. For a, b > 0 and 0 < p < 1, we have

[(a + b).sup.p] [less than or equal to] [a.sup.p] + [b.sup.p]. (1)

The double series [[SIGMA].sup.[infinity].sub.m,n=1] [x.sub.mn] is called convergent if and only if the double sequence ([s.sub.mn]) is convergent, where [s.sub.mn] = [[SIGMA].sup.m,n.sub.i,j=1] [x.sub.ij] (m, n [member of] [??]).

A sequence x = ([x.sub.mn]) is said to be double analytic if [sup.sub.mn] [[absolute value of [x.sub.mn]].sup.1/m+n] < [infinity]. The vector space of all double analytic sequences will be denoted by [[LAMBDA].sup.2]. A sequence x = ([x.sub.mn]) is called double entire sequence if [[absolute value of [x.sub.mn]].sup.1/m+n] [right arrow] 0 as m,n [right arrow] [infinity]. The double entire sequences will be denoted by [[GAMMA].sup.2]. A sequence x = ([x.sub.mn]) is called double gai sequence if ((m + n)! [[absolute value of [x.sub.mn]]).sup.1/m+n] [right arrow] 0 as m,n [right arrow] [infinity]. The double gai sequences will be denoted by [chi square]. Let [phi] denote the set of all finite sequences.

Consider a double sequence x = ([x.sub.ij]). The [(m, n).sup.th] section [x.sup.[m,n]] of the sequence is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all m, n [member of] [??]; where [[??].sub.ij] denotes the double sequence whose only non-zero term is 1/(i+j)! in the [(i,j).sup.th] place for each i, j [member of] [??].

An FK-space (or a metric space) X is said to have AK property if ([[??].sub.mn]) is a Schauder basis for X or equivalently [x.sup.[m,n]] [right arrow] x.

An FDK-space is a double sequence space endowed with a complete metrizable; locally convex topology under which the coordinate mappings x = ([x.sub.k]) [right arrow] ([x.sub.mn]) (m, n [member of] [??]) are also continuous.

Orlicz [25] used the idea of Orlicz function to construct the space ([L.sup.M]). Lindenstrauss and Tzafriri [16] investigated Orlicz sequence spaces in more detail, and they proved that every Orlicz sequence space [l.sub.M] contains a subspace isomorphic to [l.sub.p] (1 [less than or equal to] p < [infinity]). subsequently, different classes of sequence spaces were defined by Parashar and Choudhary [26], Mursaleen et al. [23], Tripathy et al. [36], Rao and Subramanian [6], and many others. The Orlicz sequence spaces are the special cases of Orlicz spaces studied in [25].

Recalling [14] and [25], an Orlicz function is a function M : [0, [infinity]) [right arrow] [0, [infinity]) which is continuous, non-decreasing, and convex with M (0) = 0, M (x) > 0, for x > 0 and M (x) [right arrow] [infinity] as x [right arrow] [infinity]. If convexity of Orlicz function M is replaced by subadditivity of M, then this function is called modulus function, defined by Nakano [24] and further discussed by Ruckle [27] and Maddox [18], Tripathy and Chandra [40] and many others.

An Orlicz function M is said to satisfy the [[DELTA].sub.2]-condition for all values of u if there exists a constant K > 0 such that M (2u) [less than or equal to] KM (u) (u [greater than or equal to] 0). The [[DELTA].sub.2]-condition is equivalent to M (Lu) [less than or equal to] KLM (u), for all values of u and for L > 1.

Lindenstrauss and Tzafriri [16] used the idea of Orlicz function to construct Orlicz sequence space

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The space [l.sub.M] with the norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

becomes a Banach space which is called an Orlicz sequence space. For M (t) = [t.sup.p] (1 [less than or equal to] p < [infinity]), the spaces [l.sub.M] coincide with the classical sequence space [l.sub.p].

If X is a sequence space, we give the following definitions:

(i) X' = the continuous dual of X.

(ii) [X.sup.[alpha]] = {a = ([a.sub.mn]) : [[SIGMA].sup.[infinity].sub.m,n=1] [absolute value of [a.sub.mn[x.sub.mn]] < [infinity], for each x [member of] X}.

(iii) [X.sup.[beta]] = {a = ([a.sub.mn]) : [[SIGMA].sup.[infinity].sub.m,n=1] [a.sub.mn]mn[x.sub.mn] is convegent, for each x [member of] X}.

(iv) [X.sup.[gamma]] = {a = ([a.sub.mn]) : [sup.sub.m,n] [greater than or equal to] 1 [[SIGMA].sup.M,N.sub.m,n=1] [a.sub.mn][x.sub.mn] < [infinity], for each x [member of] X}.

(v) let X be an FK-space [contains] [phi], then [X.sup.f] = {f([[??].sub.mn]) : f [member of] X'}.

(vi) [X.sup.[delta]] = {a = ([a.sub.mn]) : [sup.sub.m,n] [[absolute value of [a.sub.mn][x.sub.mn]]/sup.1/m+n] < [infinity], for each x [member of].

[X.sup.[alpha]], [X.sup.[beta]], [X.sup.[gamma]] and [X.sup.[delta]] are called [alpha]- (or Kothe-Toeplitz) dual of X, ([beta]- (or generalized-Kothe-Toeplitz) dual of X, 7-dual of X, [gamma]-dual of X respectively. It is clear that [x.sup.[alpha]] [subset] [X.sup.[beta]] and [X.sup.[alpha]] [subset] [X.sup.[gamma]], but [X.sup.[alpha]] [subset] [X.sup.[gamma]] does not hold, since the sequence of partial sums of a double convergent series need not be bounded.

The notion of difference sequence spaces (for single sequences) was introduced by Kizmaz [15] as follows:

Z ([DELTA]) = {x = ([x.sub.k]) [member of] w : ([DELTA][x.sub.k]) [member of] Z},

for Z = c, [c.sub.o] and [l.sub.[infinity]], where [DELTA][x.sub.k] = [x.sub.k] - [x.sub.k+1] for all k [member of] [??]. Here w, c, [c.sub.o] and [l.sub.[infinity]] denote the classes of all, convergent, null and bounded scalar valued single sequences respectively. The above spaces are Banach spaces normed by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Later on the notion was further investigated by many others. We now introduce the following difference double sequence spaces defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where Z = [[LAMBDA].sup.2], [[GAMMA].sup.2] and [chi square] respectively. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let r [member of] [??] be fixed, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now we introduced a generalized difference double operator as follows:

Let r, [mu] [member of] [??] be fixed, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all m, n [member of] [??].

The notion of a modulus function was introduced by Nakano [24]. We recall that a modulus f is a function from [0, [infinity]) [right arrow] [0,[infinity]), such that

(i) f (x) = 0 if and only if x = 0.

(ii) f (x + y) [less than or equal to] f (x) + f(y), for all x [greater than or equal to] 0, y [greater than or equal to] 0.

(iii) f is increasing.

(iv) f is continuous from the right at 0. Since [absolute value of f (x) - f (y)] [less than or equal to] f ([absolute value of x - y]), it follows from condition (iv) that f is continuous on [0, [infinity]).

It is immediate from (ii) and (iv) that f is continuous on [0, [infinity]). Also from condition (ii), we have f (nx) [less than or equal to] nf (x) for all n [member of] n and [n.sup.-1] f(x) [less than or equal to] f ([xn.sup.-1]) , for all n [member of] [??].

[section]2. Definitions and preliminaries

Let [w.sup.2] denote the set of all complex double sequences. A sequence x = ([x.sub.mn]) is said to be double analytic if [sup.sub.mn][absolute value of [x.sub.mn]]1/m+n] < [infinity]. The vector space of all prime sense double analytic sequences will be denoted by [[LAMBDA].sup.2]. A sequence x = ([x.sub.mn]) is called prime sense double entire sequence if [absolute value of [x.sub.mn].sup.1/m+n] [right arrow] 0 as m, n [right arrow] [infinity]. The double entire sequences will be denoted by [[GAMMA].sup.2].

The space [[LAMBDA].sup.2] and [[GAMMA].sup.2] is a metric space with the metric

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

for all x = {[x.sub.mn]} and y = {[y.sub.mn]} in [[GAMMA].sup.2].

Let [pi] = {[[pi.sub.mn]]} be a sequence of positive numbers. If X is a sequence space, we write [X.sub.[pi]] = {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where X = [[GAMMA].sup.2], [[LAMBDA].sup.2].

A sequence x = ([x.sub.mn]) is called prime sense double gai sequence if ((m + n)! [[absolute value of [x.sub.mn]]).sup.1/m+n] [] 0 as m, n [right arrow] [infinity]. The double gai sequences will be denoted by [chi square]. The space [chi square] is a metric space with the metric

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

for all x = {[x.sub.mn]} and y = {[y.sub.mn]} in [chi square].

Definition 2.1. A sequence t is called a double analytic growth sequence for a set [[LAMBDA].sup.2] of sequences if [x.sub.mn] = O ([t.sub.mn]) for all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[section]3. Main results

Theorem 3.1. If [[LAMBDA].sup.2] has a growth sequence, then [[LAMBDA].sup.2.sub.] has a growth sequence.

Proof. Let be a growth sequence for A2. Then \[x.sub.mn]\l I m+n < M \[t.sub.mn]\ for some M > 0.

Let x [member of] [[LAMBDA].sup.2.sub.[pi]]. Then [{[x.sub.mn/[[pi].sub.mn]}.sup.1/m+n] [member of] [[LAMBDA].sup.2]. We have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which means that [[absolute value of [x.sub.mn]].sup.1/m+n] [less than or equal to] M [[[t.sub.mn][[pi].sub.mn].sup.1/m+n]. Thus {[[pi].sub.mn][t.sub.mn]} is a growth sequence of [[LAMBDA].sup.2.sub.mn]. In other words, [[LAMBDA].sup.2.sub.[pi]] has the growth sequence [pi]t.

Theorem 3.2. Let [[LAMBDA].sup.2] be a BK-space. Then the rate space [[LAMBDA].sup.2.sub.[pi]] has a growth sequence.

Proof. Let x [member of] [[LAMBDA].sup.2.sub.[pi]]. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Put [P.sub.mn] (x) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then [P.sub.mn] is a continuous functional on [[GAMMA].sup.2.sub.[pi]]. Hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Also for every positive integer m, n, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence [x.sub.mn] = O ([P.sub.mn][[pi].sub.mn]). Thus {[P.sub.mn][[pi].sub.mn]} is a growth sequence for [[LAMBDA].sup.2.sub.[pi]].

Theorem 3.3. [([[GAMMA].sup.2.sub.[pi]]).sup.[alpha]] = [[LAMBDA].sup.2.sub.1/[pi]].

Proof. Let x [member of] [[LAMBDA].sup.2.sub.1/[pi]]. Then there exists M > 0 with [absolute value of [[pi].sub.mn][x.sub.mn]] [less than or equal to] [M.sup.m+n], [for all]m, n [greater than or equal to] 1.

Choose [epsilon] > 0 such that [epsilon]M < 1.

If y [member of] [[GAMMA].sup.2.[pi]], we have n [absolute value of [y.sub.mn]/[[pi].sub.mn]] [less than or equal to] [[epsilon].sup.m+n, [for all]m, n [greater than or equal to] [m.sub.0][n.sub.0] depending on [epsilon].

Therefore [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

On the other hand, let x [member of] [([[LAMBDA].sup.2.sub.[pi]]).sup.[alpha]]. Assume that x [not member of] [[LAMBDA].sup.2.sub.1/[pi]]. Then there exists an increasing sequence {[p.sub.mn][q.sub.mn]} of positive integers such that [absolute value of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] n > [for all]m, n > [m.sub.0][n.sub.0], Take y = {[y.sub.mn]} by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

Then {[y.sub.mn]} [member of] [[GAMMA].sup.2.sub.[pi]], but [SIGMA] [absolute value of[x.sub.mn][y.sub.mn]] = [infinity], a contradiction. This contradiction shows that

[([[GAMMA].sup.2.sub.[pi]]).sup.[alpha]] [subset] [[LAMBDA].sup.2.sub.1/[pi]]. (6)

From (4) and (6) it follows that [([[GAMMA].sup.2.sub.[pi]]).sup.[alpha]] [subset] = [[LAMBDA].sup.2.sub.1/[pi]].

Theorem 3.4. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. (i) First we show that [[eta].sup.2.sub.M[pi]] [subset] [[[LAMBDA].sup.2.sub.M[pi]]].sup.[beta]]. Let x [member of] [[eta].sup.2.sub.M[pi]] and y [member of] [[LAMBDA].sup.2.sub.M[pi]]. Then we can find a positive integer N such that ([[absolute value of [y.sub.mn]].sup.1/m+n)] < max (1, [sup.sub.m,n [greater than or equal to] 1] ([[absolute value of [y.sub.mn]].sup.1/m+n]) < N, for all m, n.

Hence we may write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since x [member of] [[eta].sup.2.sub.M[pi]], the series on the right side of the above inequality is convergent, whence x [member of] [[[LAMBDA].sup.2.sub.M[pi]].sup.[beta]]. Hence [[eta].sup.2.sub.M[pi]] [subset] [[[LAMBDA].sup.2.sub.M[pi]].sup.[beta]].

Now we show that [[[LAMBDA].sup.2.sub.M[pi]].sup.[beta]] [subset] [[eta].sup.2.sub.M[pi]].

For this, let x [member of] [[[LAMBDA].sup.2.sub.M[pi]].sup.[beta]], and suppose that x [not member of] [[[LAMBDA].sup.2.sub.M[pi]]. Then there exists a positive integer N > 1 such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If we define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] m, n = 1, 2, ... , then y [member of] [[[LAMBDA].sup.2.sub.M[pi]]. But, since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we get x [member of] [[[[LAMBDA].sup.2.sub.M[pi]].sup.[beta]], which contradicts to the assumption x [member of] [[[[LAMBDA].sup.2.sub.M[pi]].sup.[beta]]]. Therefore x [member of] [[eta].sup.2.sub.M[pi]]. Therefore [[[LAMBDA].sup.2.sub.M[pi]].sup.[beta]] = [[[eta].sup.2.sub.M[pi]].

(ii) and (iii) can be shown in a similar way of (i). Therefore we omit it.

Theorem 3.5. Let M be an Orlicz function or modulus function which satisfies the [[DELTA].sub.2]-condition and if [[GAMMA].sup.2.sub.M[pi]] is a growth sequence then [[GAMMA].sup.2.sub.[pi]] [subset] [[GAMMA].sup.2.sub.M[pi]].

Proof. Let

x [member of] [[GAMMA].sup.2.sub.[pi]]. (7)

Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for sufficiently large m, n and every [epsilon] > 0. But then by taking [rho] [greater than or equal to] 1/2,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (because M is non-decreasing)

[less than or equal to] (M (2[epsilon])).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [less than or equal to] KM ([epsilon]) (by the [[DELTA].sub.2]- condition, for some k > 0)

[less than or equal to] [member of] (by defining M ([epsilon]) < [epsilon]/K).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Hence

x [member of] [[GAMMA].sup.2.sub.M[pi]]. (9)

From (7) and (9) we get [[GAMMA].sup.2.sub.[pi]] [subset] [[GAMMA].sup.2.sub.M[pi]]. This completes the proof.

Theorem 3.6. If [[GAMMA].sup.2.sub.M[pi]] is a growth sequence then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. (i) First we show that [[eta].sup.2.sub.M[pi]] [subset] [[[GAMMA].sup.2.sub.M[pi]].sup.[beta]]. We know that [[GAMMA].sup.2.sub.M[pi]] [subset] [[LAMBDA].sup.2.sub.M[pi]], [[[LAMBDA].sup.2.sub.M[pi]].sup.[beta]] [subset] [[[GAMMA].sub.M[pi]].sup.[beta]]. But [[[LAMBDA].sup.2.sub.M[pi]].sup.[beta]] = [[eta].sup.2.sub.M[pi]], by Theorem 3.4, therefore

[[eta].sup.2.sub.M[pi]] [subset] [[GAMMA].sup.2.sub.M[pi]]. (10)

(ii) Now we show that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let y = {[y.sub.mn]} be an arbitrary point in ([[GAMMA].sup.2.sub.M[pi]]).sup.[beta]]. If y is not in [[LAMBDA].sup.2.sub.[pi]], then for each natural number q, we can find an index [m.sub.q][n.sub.q] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] > q, (1, 2, 3, ...).

Define x = {[x.sub.mn]} by ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for (m,n) = ([m.sub.q,nq) for some q [member of] [??], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = 0 otherwise.

Then x is in [[GAMMA].sup.2.sub.M[pi]], but for infinitely mn,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Consider the sequence z = {[z.sub.mn]}, where (M(M([z.sub.mn][[pi].sub.11][t.sub.11]/[rho])) = (M([x.sub.11][[pi].sub.11][t.sub.11]/[rho])) with s = [SIGMA] (M((m+n)![x.sub.mn]/[rho])), and (M([z.sub.mn]/[[pi].sub.mn][t.sub.mn]/[rho])) = (M([z.sub.mn]/[[pi].sub.mn][t.sub.mn]/[rho] (m,n = 1, 2, 3, ...). Then z is a point of [[GAMMA].sup.2.sub.M[pi]]. Also [SIGMA] ((M([z.sub.mn][y.sub.mn]/[rho])) = 0. Hence z is in [[GAMMA].sup.2.sub.M[pi]]. But, by the equation (11), [SIGMA] (M([z.sub.mn][y.sub.mn]/[rho])) does not converge [??] [SIGMA][x.sub.mn][y.sub.mn] diverges.

Thus the sequence y would not be in [([[GAMMA].sup.2.sub.M[pi]]).sup.3]. This contradiction proves that

[([[GAMMA].sup.2.sub.M[pi]]).sup.3] [subset] [[LAMBDA].sup.2.sub.[pi]]. (12)

If we now choose M = id, where id is the identity and [y.sub.1n]/[[pi].sub.1n][t.sub.1n] = [x.sub.1n]/[[pi].sub.1n][t.sub.1n] = 1 and [y.sub.mn]/[[pi].sub.][t.sub.mn] = [x.sub.mn]/[[pi].sub.mn][t.sub.mn] = 0 (m > 1) for all n, then obviously x [member of] [[GAMMA].sup.2.sub.M[pi]] and y [member of] [[LAMBDA].sup.2.sub.[pi]], but [[SIGMA].sup.[infinity].sub.m,n=1] [x.sub.mn][y.sub.mn] = [infinity], hence

y [member of] [([[GAMMA].sup.2.sub.M[pi]]).sup.[beta]]. (13)

From (12) and (13) we are granted

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)

Hence (10) and (14) we are granted [[eta].sup.2.sub.M[pi]] [subset] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This completes the proof.

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[24] H. Nakano, Concave modulars, J. Math. Soc. Japan, 5(1953), 29-49.

[25] W. Orlicz, Uber Raume ([L.sup.M]), Bull. Int. Acad. Polon. Sci. A, 1936, 93-107.

[26] S. D. Parashar and B. Choudhary, Sequence spaces defined by Orlicz functions, Indian J. Pure Appl. Math., 25(1994), No. 4, 419-428.

[27] W. H. Ruckle, FK spaces in which the sequence of coordinate vectors is bounded, Canad. J. Math., 25(1973), 973-978.

[28] N. Subramanian, R. Nallswamy and N. Saivaraju, Characterization of entire sequences via double Orlicz space, Internaional Journal of Mathematics and Mathemaical Sciences, 2007, Article ID 59681, 10 pages.

[29] N. Subramanian and U. K. Misra, The semi normed space defined by a double gai sequence of modulus function, Fasciculi Math., 46(2010).

[30] N. Subramanian and U. K. Misra, Characterization of gai sequences via double Orlicz space, Southeast Asian Bulletin of Mathematics, 35(2011), 687-697.

[31] N. Subramanian, B. C. Tripathy and C. Murugesan, The double sequence space of [[GAMMA].sup.2] Fasciculi Math., 40(2008), 91-103.

[32] N. Subramanian, B. C. Tripathy and C. Murugesan, The Cesaro of double entire sequences, International Mathematical Forum, 4(2009), No. 2, 49-59.

[33] N. Subramanian and U. K. Misra, The Generalized double of gai sequence spaces, Fasciculi Math., 43(2010).

[34] N. Subramanian and U. K. Misra, Tensorial transformations of double gai sequence spaces, International Journal of Computational and Mathematical Sciences, 3(2009), 186-188.

[35] B. C. Tripathy, On statistically convergent double sequences, Tamkang J. Math., 34(2003), No. 3, 231-237.

[36] B. C. Tripathy, Y. Altin and M. Et, Generalized difference sequence spaces on seminormed spaces defined by Orlicz functions, Maathematica Slovaca, 58(2008), No. 3, 315-324.

[37] B. C. Tripathy, On generalized difference paranormed statistically convergent sequences, Indian Journal of Pure and Applied Mathematics, 35(2004), No. 5, 655-663.

[38] B. C. Tripathy and A. Baruah, Norlund and Riesz mean of sequences of fuzzy real numbers, Applied Mathematics Letters, 23(2010), 651-655.

[39] B. C. Tripathy and S. Borgogain, The sequence space m [(M, [phi], [[DELTA].sup.n.sub.m], p).sup.F], Mathematical Modelling and Analysis, 13(2008), No. 4, 577-586.

[40] B. C. Tripathy and P. Chandra, On some generalized difference paranormed sequence spaces associated with multiplier sequences defined by modulus function, Anal. Theory Appl., 27(2011), No. 1, 21-27.

[41] B. C. Tripathy and A. J. Dutta, On fuzzy real-valued double sequence spaces [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], Mathematical and Computer Modelling, 46(2007), No. 9-10, 1294-1299.

[42] B. C. Tripathy and A. J. Dutta, Bounded variation double sequence space of fuzzy real numbers, Computers and Mathematics with Applications, 59(2010), No. 2, 1031-1037.

[43] B. C. Tripathy and B. Sarma, Statistically convergent difference double sequence spaces, Acta Mathematica Sinica, 24(2008), No. 5, 737-742.

[44] B. C. Tripathy and B. Sarma, Sequence spaces of fuzzy real numbers defined by Orlicz functions, Mathematica Slovaca, 58(2008), No. 5, 621-628.

[45] B. C. Tripathy and B. Sarma, Vector valued double sequence spaces defined by Orlicz function, Mathematica Slovaca, 59(2009), No. 6, 767-776.

[46] B. C. Tripathy and B. Sarma, Double sequence spaces of fuzzy numbers defined by Orlicz function, Acta Mathematica Scientia, 31(2011), 134-140.

[47] B. C. Tripathy and S. Mahanta, On a class of generalized lacunary difference sequence spaces defined by Orlicz function, Acta Mathematica Applicata Sinica (Eng. Ser.), 20(2004), No. 2, 231-238.

[48] B. C. Tripathy and M. Sen, Characterization of some matrix classes involving paranormed sequence spaces, Tamkang Journal of Mathematics, 37(2006), No. 2, 155-162.

[49] A. Turkmenoglu, Matrix transformation between some classes of double sequences, Jour. Inst. of math. and Comp. Sci. (Math. Seri.), 12(1999), No. 1, 23-31.

[50] M. Zeltser, Investigation of Double Sequence Spaces by Soft and Hard Analitical Methods, Dissertationes Mathematicae Universitatis Tartuensis 25, Tartu University Press, Univ. of Tartu, Faculty of Mathematics and Computer Science, Tartu, 2001.

N. Subramanian ([dagger]) and C. Murugesan ([double dagger])

([dagger]) Department of Mathematics, Sastra University, Thanjavur, 613401, India

([double dagger]) Department of Mathematics, Sathyabama University, Chennai, 600119, India

E-mail: nsmaths@yahoo.com muruga23@sify.com

Throughout w, x and A denote the classes of all, gai and analytic scalar valued single sequences, respectively. We write [w.sup.2] for the set of all complex sequences ([x.sub.mn]), where m, n [member of] [??], the set of positive integers. Then, [w.sup.2] is a linear space under the Coordinatewise addition and scalar multiplication.

Some initial work on double sequence spaces were found in Bromwich [4]. Later on, they were investigated by Hardy [13], Moricz [19], Moricz and Rhoades [20], Basarir and Solankan [3], Tripathy [37], Tripathy and Dutta ([41],[42]), Tripathy and Sarma ([43],[44],[45],[46]), Tripathy and Sen [48], Turkmenoglu [49], and many others.

Let us define the following sets of double sequences:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where t = ([t.sub.mn]) is the sequence of positive reals for all m, n [member of] n and P-[lim.sub.m,n [right arrow][infinity]] denotes the limit in the Pringsheim's sense. In the case [t.sub.mn] = 1 for all m, n [member of] [??]; [M.sub.u] (t), [C.sub.p](t), [C.sub.0p](t), [L.sub.u] (t), [C.sub.bp](t) and [C.sub.0bp](t) reduce to the sets [M.sub.u], [C.sub.p], [C.sub.0p], [L.sub.u], [C.sub.bp] and [C.sub.0bp], respectively. Now, we may summarize the knowledge given in some document related to the double sequence spaces. Gokhan and Colak [11,12] have proved that [M.sub.u] (t) and Cp (t), [C.sub.bp] (t) are complete paranormed spaces of double sequences and obtained the [alpha]-, [beta], [gamma]-duals of the spaces [M.sub.u] (t) and [C.sub.bp] (t). Quite recently, in her PhD thesis, Zelter [50] has essentially studied both the theory of topological double sequence spaces and the theory of summability of double sequences. Mursaleen and Edely [21] and Tripathy [35] have recently introduced the statistical convergence and Cauchy for double sequences independently and given the relation between statistical convergent and strongly Cesaro summable double sequences. Later, Mursaleen [22] and Mursaleen and Edely [23] have defined the almost strong regularity of matrices for double sequences and applied these matrices to establish a core theorem and introduced the M-core for double sequences and determined those four dimensional matrices transforming every bounded double sequences x = ([x.sub.jk]) into one whose core is a subset of the M-core of x. More recently, Altay and Basar [1] have defined the spaces BS, BS (t), [CS.sub.p], [CS.sub.bp], [CS.sub.r] and BV of double sequences consisting of all double series whose sequence of partial sums are in the spaces [M.sub.u], [M.sub.u](t), [C.sub.p], [C.sub.bp], [C.sub.r] and [L.sub.u], respectively, and have also examined some properties of those sequence spaces and determined the a-duals of the spaces BS, BV, [CS.sub.bp] and the [beta] ([??})-duals of the spaces [CS.sub.bp] and [CS.sub.r] of double series. Quite recently Basar and Sever [5] have introduced the Banach space [L.sub.q] of double sequences corresponding to the well-known space [l.sub.q] of single sequences and have examined some properties of the space [L.sub.q]. Quite recently Subramanian and Misra ([29],[30]>[33]) have studied the space [s.sup.2.sub.M] (p, q, u) and the generalized gai of double sequences and have proved some inclusion relations.

We need the following inequality in the sequel of the paper. For a, b > 0 and 0 < p < 1, we have

[(a + b).sup.p] [less than or equal to] [a.sup.p] + [b.sup.p]. (1)

The double series [[SIGMA].sup.[infinity].sub.m,n=1] [x.sub.mn] is called convergent if and only if the double sequence ([s.sub.mn]) is convergent, where [s.sub.mn] = [[SIGMA].sup.m,n.sub.i,j=1] [x.sub.ij] (m, n [member of] [??]).

A sequence x = ([x.sub.mn]) is said to be double analytic if [sup.sub.mn] [[absolute value of [x.sub.mn]].sup.1/m+n] < [infinity]. The vector space of all double analytic sequences will be denoted by [[LAMBDA].sup.2]. A sequence x = ([x.sub.mn]) is called double entire sequence if [[absolute value of [x.sub.mn]].sup.1/m+n] [right arrow] 0 as m,n [right arrow] [infinity]. The double entire sequences will be denoted by [[GAMMA].sup.2]. A sequence x = ([x.sub.mn]) is called double gai sequence if ((m + n)! [[absolute value of [x.sub.mn]]).sup.1/m+n] [right arrow] 0 as m,n [right arrow] [infinity]. The double gai sequences will be denoted by [chi square]. Let [phi] denote the set of all finite sequences.

Consider a double sequence x = ([x.sub.ij]). The [(m, n).sup.th] section [x.sup.[m,n]] of the sequence is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all m, n [member of] [??]; where [[??].sub.ij] denotes the double sequence whose only non-zero term is 1/(i+j)! in the [(i,j).sup.th] place for each i, j [member of] [??].

An FK-space (or a metric space) X is said to have AK property if ([[??].sub.mn]) is a Schauder basis for X or equivalently [x.sup.[m,n]] [right arrow] x.

An FDK-space is a double sequence space endowed with a complete metrizable; locally convex topology under which the coordinate mappings x = ([x.sub.k]) [right arrow] ([x.sub.mn]) (m, n [member of] [??]) are also continuous.

Orlicz [25] used the idea of Orlicz function to construct the space ([L.sup.M]). Lindenstrauss and Tzafriri [16] investigated Orlicz sequence spaces in more detail, and they proved that every Orlicz sequence space [l.sub.M] contains a subspace isomorphic to [l.sub.p] (1 [less than or equal to] p < [infinity]). subsequently, different classes of sequence spaces were defined by Parashar and Choudhary [26], Mursaleen et al. [23], Tripathy et al. [36], Rao and Subramanian [6], and many others. The Orlicz sequence spaces are the special cases of Orlicz spaces studied in [25].

Recalling [14] and [25], an Orlicz function is a function M : [0, [infinity]) [right arrow] [0, [infinity]) which is continuous, non-decreasing, and convex with M (0) = 0, M (x) > 0, for x > 0 and M (x) [right arrow] [infinity] as x [right arrow] [infinity]. If convexity of Orlicz function M is replaced by subadditivity of M, then this function is called modulus function, defined by Nakano [24] and further discussed by Ruckle [27] and Maddox [18], Tripathy and Chandra [40] and many others.

An Orlicz function M is said to satisfy the [[DELTA].sub.2]-condition for all values of u if there exists a constant K > 0 such that M (2u) [less than or equal to] KM (u) (u [greater than or equal to] 0). The [[DELTA].sub.2]-condition is equivalent to M (Lu) [less than or equal to] KLM (u), for all values of u and for L > 1.

Lindenstrauss and Tzafriri [16] used the idea of Orlicz function to construct Orlicz sequence space

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The space [l.sub.M] with the norm

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

becomes a Banach space which is called an Orlicz sequence space. For M (t) = [t.sup.p] (1 [less than or equal to] p < [infinity]), the spaces [l.sub.M] coincide with the classical sequence space [l.sub.p].

If X is a sequence space, we give the following definitions:

(i) X' = the continuous dual of X.

(ii) [X.sup.[alpha]] = {a = ([a.sub.mn]) : [[SIGMA].sup.[infinity].sub.m,n=1] [absolute value of [a.sub.mn[x.sub.mn]] < [infinity], for each x [member of] X}.

(iii) [X.sup.[beta]] = {a = ([a.sub.mn]) : [[SIGMA].sup.[infinity].sub.m,n=1] [a.sub.mn]mn[x.sub.mn] is convegent, for each x [member of] X}.

(iv) [X.sup.[gamma]] = {a = ([a.sub.mn]) : [sup.sub.m,n] [greater than or equal to] 1 [[SIGMA].sup.M,N.sub.m,n=1] [a.sub.mn][x.sub.mn] < [infinity], for each x [member of] X}.

(v) let X be an FK-space [contains] [phi], then [X.sup.f] = {f([[??].sub.mn]) : f [member of] X'}.

(vi) [X.sup.[delta]] = {a = ([a.sub.mn]) : [sup.sub.m,n] [[absolute value of [a.sub.mn][x.sub.mn]]/sup.1/m+n] < [infinity], for each x [member of].

[X.sup.[alpha]], [X.sup.[beta]], [X.sup.[gamma]] and [X.sup.[delta]] are called [alpha]- (or Kothe-Toeplitz) dual of X, ([beta]- (or generalized-Kothe-Toeplitz) dual of X, 7-dual of X, [gamma]-dual of X respectively. It is clear that [x.sup.[alpha]] [subset] [X.sup.[beta]] and [X.sup.[alpha]] [subset] [X.sup.[gamma]], but [X.sup.[alpha]] [subset] [X.sup.[gamma]] does not hold, since the sequence of partial sums of a double convergent series need not be bounded.

The notion of difference sequence spaces (for single sequences) was introduced by Kizmaz [15] as follows:

Z ([DELTA]) = {x = ([x.sub.k]) [member of] w : ([DELTA][x.sub.k]) [member of] Z},

for Z = c, [c.sub.o] and [l.sub.[infinity]], where [DELTA][x.sub.k] = [x.sub.k] - [x.sub.k+1] for all k [member of] [??]. Here w, c, [c.sub.o] and [l.sub.[infinity]] denote the classes of all, convergent, null and bounded scalar valued single sequences respectively. The above spaces are Banach spaces normed by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Later on the notion was further investigated by many others. We now introduce the following difference double sequence spaces defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where Z = [[LAMBDA].sup.2], [[GAMMA].sup.2] and [chi square] respectively. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let r [member of] [??] be fixed, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now we introduced a generalized difference double operator as follows:

Let r, [mu] [member of] [??] be fixed, then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all m, n [member of] [??].

The notion of a modulus function was introduced by Nakano [24]. We recall that a modulus f is a function from [0, [infinity]) [right arrow] [0,[infinity]), such that

(i) f (x) = 0 if and only if x = 0.

(ii) f (x + y) [less than or equal to] f (x) + f(y), for all x [greater than or equal to] 0, y [greater than or equal to] 0.

(iii) f is increasing.

(iv) f is continuous from the right at 0. Since [absolute value of f (x) - f (y)] [less than or equal to] f ([absolute value of x - y]), it follows from condition (iv) that f is continuous on [0, [infinity]).

It is immediate from (ii) and (iv) that f is continuous on [0, [infinity]). Also from condition (ii), we have f (nx) [less than or equal to] nf (x) for all n [member of] n and [n.sup.-1] f(x) [less than or equal to] f ([xn.sup.-1]) , for all n [member of] [??].

[section]2. Definitions and preliminaries

Let [w.sup.2] denote the set of all complex double sequences. A sequence x = ([x.sub.mn]) is said to be double analytic if [sup.sub.mn][absolute value of [x.sub.mn]]1/m+n] < [infinity]. The vector space of all prime sense double analytic sequences will be denoted by [[LAMBDA].sup.2]. A sequence x = ([x.sub.mn]) is called prime sense double entire sequence if [absolute value of [x.sub.mn].sup.1/m+n] [right arrow] 0 as m, n [right arrow] [infinity]. The double entire sequences will be denoted by [[GAMMA].sup.2].

The space [[LAMBDA].sup.2] and [[GAMMA].sup.2] is a metric space with the metric

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

for all x = {[x.sub.mn]} and y = {[y.sub.mn]} in [[GAMMA].sup.2].

Let [pi] = {[[pi.sub.mn]]} be a sequence of positive numbers. If X is a sequence space, we write [X.sub.[pi]] = {[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where X = [[GAMMA].sup.2], [[LAMBDA].sup.2].

A sequence x = ([x.sub.mn]) is called prime sense double gai sequence if ((m + n)! [[absolute value of [x.sub.mn]]).sup.1/m+n] [] 0 as m, n [right arrow] [infinity]. The double gai sequences will be denoted by [chi square]. The space [chi square] is a metric space with the metric

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

for all x = {[x.sub.mn]} and y = {[y.sub.mn]} in [chi square].

Definition 2.1. A sequence t is called a double analytic growth sequence for a set [[LAMBDA].sup.2] of sequences if [x.sub.mn] = O ([t.sub.mn]) for all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

[section]3. Main results

Theorem 3.1. If [[LAMBDA].sup.2] has a growth sequence, then [[LAMBDA].sup.2.sub.] has a growth sequence.

Proof. Let be a growth sequence for A2. Then \[x.sub.mn]\l I m+n < M \[t.sub.mn]\ for some M > 0.

Let x [member of] [[LAMBDA].sup.2.sub.[pi]]. Then [{[x.sub.mn/[[pi].sub.mn]}.sup.1/m+n] [member of] [[LAMBDA].sup.2]. We have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which means that [[absolute value of [x.sub.mn]].sup.1/m+n] [less than or equal to] M [[[t.sub.mn][[pi].sub.mn].sup.1/m+n]. Thus {[[pi].sub.mn][t.sub.mn]} is a growth sequence of [[LAMBDA].sup.2.sub.mn]. In other words, [[LAMBDA].sup.2.sub.[pi]] has the growth sequence [pi]t.

Theorem 3.2. Let [[LAMBDA].sup.2] be a BK-space. Then the rate space [[LAMBDA].sup.2.sub.[pi]] has a growth sequence.

Proof. Let x [member of] [[LAMBDA].sup.2.sub.[pi]]. Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Put [P.sub.mn] (x) = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then [P.sub.mn] is a continuous functional on [[GAMMA].sup.2.sub.[pi]]. Hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Also for every positive integer m, n, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Hence [x.sub.mn] = O ([P.sub.mn][[pi].sub.mn]). Thus {[P.sub.mn][[pi].sub.mn]} is a growth sequence for [[LAMBDA].sup.2.sub.[pi]].

Theorem 3.3. [([[GAMMA].sup.2.sub.[pi]]).sup.[alpha]] = [[LAMBDA].sup.2.sub.1/[pi]].

Proof. Let x [member of] [[LAMBDA].sup.2.sub.1/[pi]]. Then there exists M > 0 with [absolute value of [[pi].sub.mn][x.sub.mn]] [less than or equal to] [M.sup.m+n], [for all]m, n [greater than or equal to] 1.

Choose [epsilon] > 0 such that [epsilon]M < 1.

If y [member of] [[GAMMA].sup.2.[pi]], we have n [absolute value of [y.sub.mn]/[[pi].sub.mn]] [less than or equal to] [[epsilon].sup.m+n, [for all]m, n [greater than or equal to] [m.sub.0][n.sub.0] depending on [epsilon].

Therefore [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], Hence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

On the other hand, let x [member of] [([[LAMBDA].sup.2.sub.[pi]]).sup.[alpha]]. Assume that x [not member of] [[LAMBDA].sup.2.sub.1/[pi]]. Then there exists an increasing sequence {[p.sub.mn][q.sub.mn]} of positive integers such that [absolute value of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] n > [for all]m, n > [m.sub.0][n.sub.0], Take y = {[y.sub.mn]} by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (5)

Then {[y.sub.mn]} [member of] [[GAMMA].sup.2.sub.[pi]], but [SIGMA] [absolute value of[x.sub.mn][y.sub.mn]] = [infinity], a contradiction. This contradiction shows that

[([[GAMMA].sup.2.sub.[pi]]).sup.[alpha]] [subset] [[LAMBDA].sup.2.sub.1/[pi]]. (6)

From (4) and (6) it follows that [([[GAMMA].sup.2.sub.[pi]]).sup.[alpha]] [subset] = [[LAMBDA].sup.2.sub.1/[pi]].

Theorem 3.4. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. (i) First we show that [[eta].sup.2.sub.M[pi]] [subset] [[[LAMBDA].sup.2.sub.M[pi]]].sup.[beta]]. Let x [member of] [[eta].sup.2.sub.M[pi]] and y [member of] [[LAMBDA].sup.2.sub.M[pi]]. Then we can find a positive integer N such that ([[absolute value of [y.sub.mn]].sup.1/m+n)] < max (1, [sup.sub.m,n [greater than or equal to] 1] ([[absolute value of [y.sub.mn]].sup.1/m+n]) < N, for all m, n.

Hence we may write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since x [member of] [[eta].sup.2.sub.M[pi]], the series on the right side of the above inequality is convergent, whence x [member of] [[[LAMBDA].sup.2.sub.M[pi]].sup.[beta]]. Hence [[eta].sup.2.sub.M[pi]] [subset] [[[LAMBDA].sup.2.sub.M[pi]].sup.[beta]].

Now we show that [[[LAMBDA].sup.2.sub.M[pi]].sup.[beta]] [subset] [[eta].sup.2.sub.M[pi]].

For this, let x [member of] [[[LAMBDA].sup.2.sub.M[pi]].sup.[beta]], and suppose that x [not member of] [[[LAMBDA].sup.2.sub.M[pi]]. Then there exists a positive integer N > 1 such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If we define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] m, n = 1, 2, ... , then y [member of] [[[LAMBDA].sup.2.sub.M[pi]]. But, since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we get x [member of] [[[[LAMBDA].sup.2.sub.M[pi]].sup.[beta]], which contradicts to the assumption x [member of] [[[[LAMBDA].sup.2.sub.M[pi]].sup.[beta]]]. Therefore x [member of] [[eta].sup.2.sub.M[pi]]. Therefore [[[LAMBDA].sup.2.sub.M[pi]].sup.[beta]] = [[[eta].sup.2.sub.M[pi]].

(ii) and (iii) can be shown in a similar way of (i). Therefore we omit it.

Theorem 3.5. Let M be an Orlicz function or modulus function which satisfies the [[DELTA].sub.2]-condition and if [[GAMMA].sup.2.sub.M[pi]] is a growth sequence then [[GAMMA].sup.2.sub.[pi]] [subset] [[GAMMA].sup.2.sub.M[pi]].

Proof. Let

x [member of] [[GAMMA].sup.2.sub.[pi]]. (7)

Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for sufficiently large m, n and every [epsilon] > 0. But then by taking [rho] [greater than or equal to] 1/2,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (because M is non-decreasing)

[less than or equal to] (M (2[epsilon])).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [less than or equal to] KM ([epsilon]) (by the [[DELTA].sub.2]- condition, for some k > 0)

[less than or equal to] [member of] (by defining M ([epsilon]) < [epsilon]/K).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Hence

x [member of] [[GAMMA].sup.2.sub.M[pi]]. (9)

From (7) and (9) we get [[GAMMA].sup.2.sub.[pi]] [subset] [[GAMMA].sup.2.sub.M[pi]]. This completes the proof.

Theorem 3.6. If [[GAMMA].sup.2.sub.M[pi]] is a growth sequence then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. (i) First we show that [[eta].sup.2.sub.M[pi]] [subset] [[[GAMMA].sup.2.sub.M[pi]].sup.[beta]]. We know that [[GAMMA].sup.2.sub.M[pi]] [subset] [[LAMBDA].sup.2.sub.M[pi]], [[[LAMBDA].sup.2.sub.M[pi]].sup.[beta]] [subset] [[[GAMMA].sub.M[pi]].sup.[beta]]. But [[[LAMBDA].sup.2.sub.M[pi]].sup.[beta]] = [[eta].sup.2.sub.M[pi]], by Theorem 3.4, therefore

[[eta].sup.2.sub.M[pi]] [subset] [[GAMMA].sup.2.sub.M[pi]]. (10)

(ii) Now we show that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let y = {[y.sub.mn]} be an arbitrary point in ([[GAMMA].sup.2.sub.M[pi]]).sup.[beta]]. If y is not in [[LAMBDA].sup.2.sub.[pi]], then for each natural number q, we can find an index [m.sub.q][n.sub.q] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] > q, (1, 2, 3, ...).

Define x = {[x.sub.mn]} by ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for (m,n) = ([m.sub.q,nq) for some q [member of] [??], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = 0 otherwise.

Then x is in [[GAMMA].sup.2.sub.M[pi]], but for infinitely mn,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Consider the sequence z = {[z.sub.mn]}, where (M(M([z.sub.mn][[pi].sub.11][t.sub.11]/[rho])) = (M([x.sub.11][[pi].sub.11][t.sub.11]/[rho])) with s = [SIGMA] (M((m+n)![x.sub.mn]/[rho])), and (M([z.sub.mn]/[[pi].sub.mn][t.sub.mn]/[rho])) = (M([z.sub.mn]/[[pi].sub.mn][t.sub.mn]/[rho] (m,n = 1, 2, 3, ...). Then z is a point of [[GAMMA].sup.2.sub.M[pi]]. Also [SIGMA] ((M([z.sub.mn][y.sub.mn]/[rho])) = 0. Hence z is in [[GAMMA].sup.2.sub.M[pi]]. But, by the equation (11), [SIGMA] (M([z.sub.mn][y.sub.mn]/[rho])) does not converge [??] [SIGMA][x.sub.mn][y.sub.mn] diverges.

Thus the sequence y would not be in [([[GAMMA].sup.2.sub.M[pi]]).sup.3]. This contradiction proves that

[([[GAMMA].sup.2.sub.M[pi]]).sup.3] [subset] [[LAMBDA].sup.2.sub.[pi]]. (12)

If we now choose M = id, where id is the identity and [y.sub.1n]/[[pi].sub.1n][t.sub.1n] = [x.sub.1n]/[[pi].sub.1n][t.sub.1n] = 1 and [y.sub.mn]/[[pi].sub.][t.sub.mn] = [x.sub.mn]/[[pi].sub.mn][t.sub.mn] = 0 (m > 1) for all n, then obviously x [member of] [[GAMMA].sup.2.sub.M[pi]] and y [member of] [[LAMBDA].sup.2.sub.[pi]], but [[SIGMA].sup.[infinity].sub.m,n=1] [x.sub.mn][y.sub.mn] = [infinity], hence

y [member of] [([[GAMMA].sup.2.sub.M[pi]]).sup.[beta]]. (13)

From (12) and (13) we are granted

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)

Hence (10) and (14) we are granted [[eta].sup.2.sub.M[pi]] [subset] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This completes the proof.

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N. Subramanian ([dagger]) and C. Murugesan ([double dagger])

([dagger]) Department of Mathematics, Sastra University, Thanjavur, 613401, India

([double dagger]) Department of Mathematics, Sathyabama University, Chennai, 600119, India

E-mail: nsmaths@yahoo.com muruga23@sify.com

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Author: | Subramanian, N.; Murugesan, C. |
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Publication: | Scientia Magna |

Article Type: | Report |

Date: | Jun 1, 2012 |

Words: | 4905 |

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