- Email: [email protected]
Some inequalities related to the concept of double statistical summability
Journal of the Egyptian Mathematical Society (2013) 21, 254–258
Egyptian Mathematical Society
Journal of the Egyptian Mathematical Society www.etms-eg.org www.elsevier.com/locate/joems
ORIGINAL ARTICLE
Some inequalities related to the concept of double statistical summability H. S ß evli a, S.A. Mohiuddine a b
b,*
_ Department of Mathematics, Istanbul Commerce University, U¨skudar, 34672 Istanbul, Turkey Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Received 5 February 2013; accepted 15 March 2013 Available online 23 April 2013
KEYWORDS Statistical convergence Cesa´ro summability Conservative matrices Double sequences
Abstract In this paper, we prove some inequalities related to the concept of C11(st2)-conservative matrices, C11(st2) lim sup and C11(st2) lim inf which are natural analogues of ðCbp ; st2 \ L1 Þmatrices, st2 lim sup and st2 lim inf, respectively. MSC:
40C05; 40J05
ª 2013 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.
1. Introduction First, we recall some notations and definitions which we will use throughout the paper. A double sequence x = (xjk) of real or complex numbers is said to be bounded if kxk1 ¼ supj;k jxjk j < 1: The space of bounded double sequences is denoted by L1 . A double sequence x = (xjk) is said to be convergent to the limit l in the Pringsheim sense (shortly, p-convergent to l) if for every > 0 there exists N 2 N such that Œxjk lŒ < when* Corresponding author. Tel.: +966 595116518. E-mail addresses: [email protected] (H. S ß evli·), mohiuddine@gmail. com (S.A. Mohiuddine). Peer review under responsibility of Egyptian Mathematical Society.
Production and hosting by Elsevier
ever j, k > N. In this case, l is called the p-limit of x. We denote by Cp , the space of p-convergent sequences [1]. Note that, in contrast to the case for single sequences, a convergent double sequence need not be bounded. By Cbp , we denote the space of double sequences which are bounded convergent. Let A = (amnjk:m, n, j, k = 0, 1, . . .) be a four-dimensional infinite matrix of real numbers. The double series 1 X 1 X amnjk xjk j¼0 k¼0
is called A transform of the double sequence x = (xjk) and denoted by Ax. We say that a sequence x is A-summable to the limit s if the A-transform of x exists for all m, n = 0, 1, . . . and convergent in the Pringsheim sense, i.e., p X q X p- lim amnjk xjk ¼ ymn p;q!1
j¼0 k¼0
and p- lim ymn ¼ s: m;n!1
1110-256X ª 2013 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. http://dx.doi.org/10.1016/j.joems.2013.03.008
Some inequalities related to the concept of double statistical summability
255
Let X and Y be any two sequence spaces. If x 2 X implies Ax 2 Y, then we say that the matrix A maps X into Y. We denote the class of all matrices A which map X into Y by (X, Y). It is known that A 2 ðCbp ; Cbp Þ, that is, A is conservative if and only if (see [2])
A double infinite Cesa´ro matrix (C, a, b) is a double infinite Hausdorff matrix with entries
p-limamnjk ¼ ajk for each j; k; m;n XX amnjk ¼ a; p-lim
ð1:1Þ
where
ð1:2Þ
Eam ¼
ð1:3Þ
Let rmn denotes the mn-term of the (C, 1, 1) transform of a double sequence x = (xjk), that is, m X n X 1 rmn ¼ rmn ðxjk Þ ¼ xjk : ð1:10Þ ðm þ 1Þðn þ 1Þ j¼0 k¼0
m;n
j
j
X p-lim jamnjk j ¼ aj0 for each j; m;n
p-lim m;n
ð1:4Þ
k
XX jamnjk j exists; j
ð1:5Þ
k
XX jamnjk j < 1: kAk ¼ sup m;n
j
ð1:6Þ
k
PP If A is conservative, the functional ðAÞ ¼ a j k ajk is called the characteristic of four-dimensional matrix A, [3]. If A is conservative and p-lim Ax = p-lim x for all x 2 Cbp , the matrix A is said to be RH-regular, and we denote A 2 ðCbp ; Cbp Þreg . Also, It is known that A 2 ðCbp ; Cbp Þreg if and only if conditions (1.5) and (1.6) hold, and also (1.1) with ajk = 0, (1.2) with a = 1, (1.3) and (1.4) with a0k = aj0 = 0 hold. If A is RH-regular then (A) = 1. For more details of double sequences and related concepts, we refer to [4–11] and references therein. The concept of statistical convergence of single sequences was first introduced by Fast [12]. Further, this concept was studied by Schoenberg [13], Fridy [14], Connor [15], C¸akan and Altay [16], Mohiuddine and Alghamdi [17] and many others. Let E # N N be a two-dimensional set of positive integers and let E(m, n) be the numbers of (j, k) in E such that j 6 m and k 6 n. Then, the two-dimensional analogue of natural density can be defined as follows [18]. The lower asymptotic density of the set E # N N is defined as Eðm; nÞ d2 ðKÞ ¼ lim inf : m;n mn
ð1:7Þ
In case, the sequence (E(m, n)/mn) has a limit in Pringsheim’s sense, then we say that E has a double natural density and is defined as Eðm; nÞ p-lim ¼ d2 ðEÞ: m;n mn
Statistical convergence for double sequences x = (xjk) of real numbers has been defined and studied by Mursaleen and Edely [18] and for double sequences of fuzzy real numbers by Savasß and Mursaleen [19]. Mohiuddine et al. [20] have recently introduced the concept of statistical convergence for double sequences in locally solid Riesz spaces. A real double sequence x = (xjk) is said to be statistically convergent [18] to L if for every e > 0, the set fðj; kÞ; j 6 m and k 6 n : jxjk Lj P g
ð1:9Þ
has double natural density zero. In this case, we write st2 limj,kxjk = L, and we denote the set of all statistically convergent double sequences by st2.
Eam Ebn mþa m
;
¼
Cða þ m þ 1Þ ma : Cðm þ 1ÞCða þ 1Þ Cða þ 1Þ
Savasß et al. [21] studied the concept of the absolute Cesa´ro summability of double series. Recently, Moricz [22] defined the concept of statistical (C, 1,1) summability as follows. A real double sequence x = (xjk) is said to be statistically summable (C,1, 1) to some number L if the sequence r = rmn is statistically convergent to L, i.e. st2 lim rmn = L. We denote by C11(st2) the set of all double sequences which are statistically summable (C,1, 1). We claim that if a double sequence (xjk) is bounded, then st2 lim xjk = L implies st2 lim rmn = L. For any real bounded double sequence x = (xjk), the concepts l(x) = p lim inf x and L(x) = p lim sup x have been introduced in [2]. In this paper, we prove some inequalities related to the concepts of C11(st2)-conservative matrices, C11(st2) lim sup and C11(st2) lim inf which are natural analogues of ðCbp ; st2 \ L1 Þ-matrices, st2 lim sup and st2 lim inf, respectively. Such type of inequalities are also considered by C¸akan and Altay [3] and C¸akan et al [23]. The one-dimensional version of such type inequalities appears in [24] and [25]. 2. The main results Theorem 2.1. Let A = (amnjk) be conservative and x 2 L1 . Then XX k þ ðAÞ k ðAÞ p-lim sup ðamnjk ajk Þxjk 6 uðxÞ þ wðxÞ 2 2 m;n j k
for any k P j ðAÞ j, if and only if XX p-lim sup jamnjk ajk j 6 k; m;n
p-lim m;n
ð1:8Þ
b1 Ea1 mj Enk
k
X p-lim jamnjk j ¼ a0k for each k; m;n
hmnjk ¼
X
j
ð2:1Þ
ð2:2Þ
k
jamnjk ajk j ¼ 0
ð2:3Þ
ðj;kÞ2E
for every E#N N with d2(E) = 0, where u(x) = C11(st2) lim sup x and w(x) = C11(st2) lim inf x. Theorem 2.2. Let A be C11(st2)-conservative. Then, for some constant k P ŒjŒ and for all x 2 L1 , XX kþj kj C11 ðst2 Þ lim sup ðamnjk ajk Þxjk 6 LðxÞ lðxÞ 2 2 m;n j k
ð2:4Þ
if and only if C11 ðst2 Þ lim sup m;n
XX jamnjk ajk j 6 k: j
ð2:5Þ
k
Theorem 2.3. Let A be C11(st2)-conservative. Then, for some constant k P ŒjŒ and for all x 2 L1 ,
256
H. S ß evli, S.A. Mohiuddine
XX kþj kj C11 ðst2 Þ lim sup ðamnjk ajk Þ 6 uðxÞ þ wðxÞ 2 2 m;n j k
if and only if (2.5) holds and X C11 ðst2 Þ lim ðamnjk ajk Þ ¼ 0 m;n
ð2:6Þ
ð2:7Þ
ðj;kÞ2E
XX kK C11 ðst2 Þ lim sup ðamnjk ajk Þ 6 ; 2 n j k
ð3:13Þ
where for any g 2 R; gþ ¼ maxf0; gg and g = max{g,0}.
3. Lemmas
Lemma 3.4 [3]. Let A be conservative. Then for some constant k P j ðAÞ j and for all x 2 L1 , one has
We shall need the following lemmas for the proof of our Theorems. The following lemma is a double sequence version of Corollary 3 of Kolk [26] and Lemma 2.1 of Mursaleen et al [25]. Lemma 3.1. A 2 ðCbp ; st2 \ L1 Þ if and only if
XX k þ ðAÞ k ðAÞ p lim sup ðamnjk ajk Þxjk 6 LðxÞ lðxÞ 2 2 m;n j k
if and only if XX p lim sup jamnjk ajk j 6 k m;n
XX sup jamnjk j < 1; j
ð3:12Þ
and
for every E # N N with d2(E) = 0.
m;n
XX kþK C11 ðst2 Þ lim sup ðamnjk ajk Þþ 6 2 m;n j k
ð3:1Þ
j
ð3:14Þ
ð3:15Þ
k
holds.
k
C11 ðst2 Þ limamnjk ¼ ajk for each j; k; m;n X C11 ðst2 Þ lim jamnjk j ¼ a0k for each k; m;n
ð3:2Þ
j
C11 ðst2 Þ lim
X jamnjk j ¼ aj0 for each j;
ð3:4Þ
C11 ðst2 Þ lim
k X X amnjk ¼ a:
ð3:5Þ
m;n
m;n
j
4. Proof of theorems
ð3:3Þ Proof of Theorem 2.1. Necessity. Let L(x) = p lim sup x and l(x) = p lim inf x. Since u(x) 6 L(x) and w(x) 6 l(x) for all x 2 L1 , we have p lim sup
k
m;n
We call such matrices as C11(st2)-conservative matrices, and in this case K¼a
XX ajk j
ð3:6Þ
k
is defined which is known as the C11(st2) characteristic of A. This number is analogous to the number st2 ðAÞ defined by Cosßkun and C¸akan [24] for single sequences. The following lemma which is C11(st2)-analogue of a result of Patterson [2]. Lemma 3.2. Let A = (amnjk) be a four-dimensional matrix and iAi < 1. If conditions
j
m;n
ð3:7Þ
and the necessity of (2.2) follows from Lemma 3.4. Denote by B = (bmnjk) the doubly infinite matrix with entries bmnjk ¼
amnjk ajk for ðj; kÞ 2 E; 0 for ðj; kÞ R E:
Since A is conservative, the matrix B satisfies the conditions of Lemma 3.2. Hence there exists a y 2 L1 such that iyi 6 1 and p lim sup
j
m;n
ð3:9Þ
k
are satisfied then there exists y 2 L1 such that iyi 6 1 and C11 ðst2 Þ lim sup
XX XX amnjk yjk ¼ C11 ðst2 Þ lim sup jamnjk j: j
j
k
Lemma 3.3. Let A be C11(st2)-conservative and k > 0. Then
m;n
if and only if
XX jamnjk ajk j 6 k j
k
ð3:11Þ
j
m;n
k
j
ð4:1Þ
k
1 for ðj; kÞ 2 E; 0 for ðj; kÞ R E:
So that, C11(st2) lim y = u(y) = w(y) = 0; and by (2.1) and (4.1) we have p lim sup
The following lemma is derived by replacing st with C1(st) from Lemma 2.3 of Cosßkun and C¸akan [24].
C11 ðst2 Þ lim sup
yjk ¼
ð3:10Þ
k
XX XX jbmnjk j ¼ p lim sup bmnjk yjk :
Now, let y = (yjk) be defined by
ð3:8Þ
X C11 ðst2 Þ lim jamnjk j ¼ 0 for each j;
k
k þ ðAÞ k ðAÞ 6 LðxÞ lðxÞ; 2 2
m;n
C11 ðst2 Þ limamnjk ¼ 0 for each j; k; m;n X C11 ðst2 Þ lim jamnjk j ¼ 0 for each k;
XX ðamnjk ajk Þxjk
m;n
X
jamnjk ajk j 6
ðj;kÞ2E
k þ ðAÞ k ðAÞ uðyÞ þ 2 2
wðxÞ ¼ 0 and we get (2.3). Sufficiency. Let x 2 L1 . Write E1 ¼ fðj; kÞ : rjk > uðxÞ þ g N N and E2 ¼ fðj; kÞ : rjk < wðxÞ g N N. Then we have d2(E1) = d2(E2) = 0; and hence d2(E) = 0 for E = E1 \ E2. We can write
Some inequalities related to the concept of double statistical summability XX X X þ ðamnjk ajk Þxjk ¼ ðamnjk ajk Þxjk þ ðamnjk ajk Þ xjk j
k
ðj;kÞ2E
X
m;n
ðj;kÞRE
ðamnjk ajk Þ xjk :
Hence m;n
X
XX j
ðamnjk ajk Þxjk 6 p lim sup m;n
k
which gives (2.4), since e was arbitrary.
m;n
ðamnjk ajk Þþ rjk þ p lim sup m;n
ðj;kÞRE
X
Proof of Theorem 2.3. Necessity. Let (2.6) hold. Since u(x) 6 L(x) and w(x) 6 l(x), (2.5) follows from Theorem 2.2. Now let us show the necessity of (2.7). For any E # N N with d2(E) = 0, let us define a matrix B = (bmnjk) by the following
! ðamnjk ajk Þ rjk
ðj;kÞRE
¼ I1 ðxÞ þ I2 ðxÞ þ I3 ðxÞ; say:
From condition (2.3), we have I1(x) = 0. Let e > 0, then there is a set E as defined above such that for (j,k) R E, wðxÞ < rp < uðxÞ þ ;
uðxÞ < rj;k < wðxÞ þ :
ð4:2Þ
Therefore from conditions (2.2) and (4.2) we get kþ ðuðxÞ þ Þ 2 k ðwðxÞ þ Þ: I3 ðxÞ 6 2 I2 ðxÞ 6
m;n
j
bmnjk ¼
amnjk ajk 0
kþ k uðxÞ þ wðxÞ þ k; 2 2 kþ k 6 uðxÞ þ wðxÞ; 2 2
C11 ðst2 Þ lim sup m;n
6
since e was arbitrary. This completes the proof of the theorem. Proof of Theorem 2.2. Necessity. If we define the matrix B = (bmnjk) by bmnjk = amnjk ajk for all m, n, j, k, then, since A is C11(st2)-conservative, the matrix B satisfies the hypothesis of Lemma 3.2. Hence for a y 2 L1 such that iyi 6 1 we have XX C11 ðst2 Þ lim sup jbmnjk j ¼ C11 ðst2 Þ k
j
2.1,
þ
j>j0 k>k0
j6j0 k6k0
ðamnjk ajk Þþ xjk
XX
XX
ðamnjk ajk Þxjk ¼
k
X
ðamnjk ajk Þxjk þ
ðj;kÞ2E
C11 ðst2 Þ lim sup m;n
X
X
þ
ðamnjk ajk Þ xjk
ðj;kÞRE
ðamnjk ajk Þ xjk :
XX kþj kj amnjk ajk 6 uðxÞ þ wðxÞ þ k: 2 2 j k
But e was arbitrary, so (2.6) holds and this completes the proof of theorem. References some
XX XX ðamnjk ajk Þxjk ¼ ðamnjk ajk Þxk XX
Sufficiency. Let (2.5) and (2.7) hold and x 2 L1 . As in Theorem 2.1, we can write
Using Lemma 3.3 and since A is C11(st2)-conservative, we have
k
for
kþj kj uðyÞ þ wðyÞ ¼ 0; 2 2
ðj;kÞRE
since kyk 6 1:
Sufficiency. As in Theorem j0 ; k0 2 N ðj > j0 ; k > k0 Þ, we can write
jamnjk ajk jxjk
ðj;kÞ2E
Using (2.4), we get XX kþj kj C11 ðst2 Þ lim sup jbmnjk j 6 LðyÞ lðyÞ 2 2 m;n j k kþj kj kyk þ 6 2 2 6 k;
X
and hence we get (2.7).
j
XX bmnjk yjk : lim sup m;n
k
j;k
Using the fact that C11(st2) lim y = u(y) = w(y) = 0 and (2.6), we get
<
j
m;n
j;k
Let us define the sequence y = (yjk) by 1 for ðj; kÞ 2 E; yjk ¼ 0 for ðj; kÞ R E:
k
m;n
for ðj; kÞ 2 E; for ðj; kÞ R E:
Then, it is clear that B satisfies the conditions of Lemma 3.2 and hence there exists a y 2 L1 such that iyi 6 1 and X X C11 ðst2 Þ lim sup bmnjk yjk ¼ C11 ðst2 Þ lim sup jbmnjk j: m;n
Hence we get XX p lim sup ðamnjk ajk Þxk
j
k
jamnjk ajk jjxjk j þ p lim sup
ðj;kÞ2E
X
j
kþj kj ðlðxÞ Þ 6 ðLðxÞ þ Þ 2 2 kþj kj LðxÞ lðxÞ þ k; ¼ 2 2
ðj;kÞRE
p lim sup
257
XX C11 ðst2 Þ lim sup ðamnjk ajk Þxjk
ðamnjk ajk Þ xjk :
j>j0 k>k0
Since for any e > 0, l(x) e < xjk < L(x) + e; and A is C11(st2)-conservative, we get by Lemma 3.2 that
[1] A. Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53 (1900) 289–321. [2] R.F. Patterson, Double sequence core theorems, Int. J. Math. Math. Sci. 22 (4) (1999) 785–793. [3] C. C¸akan, B. Altay, A class of conservative four dimensional matrices, J. Ineq. Appl. (2006). Article ID 14721, pages 1. [4] M. Mursaleen, S.A. Mohiuddine, Some matrix transformations of convex and paranormed sequence spaces into the spaces of invariant means, J. Function Spaces Appl. 2012 (2012). Article ID 612671, 10 pages.
258 [5] M. Mursaleen, S.A. Mohiuddine, Double r-multiplicative matrices, J. Math. Anal. Appl. 327 (2007) 991–996. [6] M. Mursaleen, S.A. Mohiuddine, Regularly r-conservative and r-coercive four dimensional matrices, Comput. Math. Appl. 56 (2008) 1580–1586. [7] M. Mursaleen, S.A. Mohiuddine, On r-conservative and boundedly r-conservative four-dimensional matrices, Comput. Math. Appl. 59 (2010) 880–885. [8] M. Mursaleen, S.A. Mohiuddine, Some new double sequence spaces of invariant means, Glasnik Matematicki 45 (65) (2010) 139–153. [9] M. Mursaleen, S.A. Mohiuddine, Almost bounded variation of double sequences and some four dimensional summability matrices, Publ. Math. Debrecen 75/3–4 (2009) 495–508. [10] M. Mursaleen, S.A. Mohiuddine, Invariant mean and some core theorems for double sequences, Taiwanese J. Math. 14 (1) (2010) 21–33. [11] M. Mursaleen, S.A. Mohiuddine, Some inequalities on sublinear functionals related to the invariant mean for double sequences, Math. Ineq. Appl. 13 (1) (2010) 157–163. [12] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951) 241–244. [13] I.J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959) 361–375. [14] J.A. Fridy, On statistical convergence, Analysis 5 (1985) 301– 313. [15] J.S. Connor, The statistical and strong p-Cesa`ro convergence of sequences, Analysis 8 (1988) 47–63.
H. S ß evli, S.A. Mohiuddine [16] C. C¸akan, B. Altay, Statistically boundedness and statistical core of double sequences, J. Math. Anal. Appl. 317 (2006) 690– 697. [17] S.A. Mohiuddine, M.A. Alghamdi, Statistical summability through a lacunary sequence in locally solid Riesz spaces, J. Inequal. Appl. (2012), 2012:225. [18] M. Mursaleen, Osama H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl. 288 (2003) 223–231. [19] E. Savasß, M. Mursaleen, On statistically convergent double sequences of fuzzy numbers, Inform. Sci. 162 (2004) 183–192. [20] S.A. Mohiuddine, A. Alotaibi, M. Mursaleen, Statistical convergence of double sequences in locally solid Riesz spaces, Abstr. Appl. Analy. 2012 (2012) 9. Article ID 719729. [21] E. Savasß, H. S ß evli, B.E. Rhoades, On the Cesa´ro summability of double series, J. Ineq. Appl. (2008) 4. Article ID 257318. [22] F. Moricz, Tauberian theorems for double sequences that are statistically summable (C, 1, 1), J. Math. Anal. Appl. 286 (2003) 340–350. [23] C. C¸akan, B. Altay, H. Cosßkun, Double lacunary and lacunary statistical convergence of double sequences. Stud. Sci. Math. Hungarica 47 (1) (2010) 35–45. [24] H. Cosßkun, C. C¸akan, A class of statistical and r-conservative matrices, Czech. Math. J. 55 (103) (2005) 791–801. [25] M. Mursaleen, A. Alotaibi, S.A. Mohiuddine, Some inequalities on statistical summability (C, 1), J. Math. Ineq. 2 (2) (2008) 239– 245. [26] E. Kolk, Matrix maps into the space of statistically convergent bounded sequences, Proc. Estonian Acad. Sci. Phys. Math. 45 (1996) 187–192.
Egyptian Mathematical Society
Journal of the Egyptian Mathematical Society www.etms-eg.org www.elsevier.com/locate/joems
ORIGINAL ARTICLE
Some inequalities related to the concept of double statistical summability H. S ß evli a, S.A. Mohiuddine a b
b,*
_ Department of Mathematics, Istanbul Commerce University, U¨skudar, 34672 Istanbul, Turkey Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
Received 5 February 2013; accepted 15 March 2013 Available online 23 April 2013
KEYWORDS Statistical convergence Cesa´ro summability Conservative matrices Double sequences
Abstract In this paper, we prove some inequalities related to the concept of C11(st2)-conservative matrices, C11(st2) lim sup and C11(st2) lim inf which are natural analogues of ðCbp ; st2 \ L1 Þmatrices, st2 lim sup and st2 lim inf, respectively. MSC:
40C05; 40J05
ª 2013 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.
1. Introduction First, we recall some notations and definitions which we will use throughout the paper. A double sequence x = (xjk) of real or complex numbers is said to be bounded if kxk1 ¼ supj;k jxjk j < 1: The space of bounded double sequences is denoted by L1 . A double sequence x = (xjk) is said to be convergent to the limit l in the Pringsheim sense (shortly, p-convergent to l) if for every > 0 there exists N 2 N such that Œxjk lŒ < when* Corresponding author. Tel.: +966 595116518. E-mail addresses: [email protected] (H. S ß evli·), mohiuddine@gmail. com (S.A. Mohiuddine). Peer review under responsibility of Egyptian Mathematical Society.
Production and hosting by Elsevier
ever j, k > N. In this case, l is called the p-limit of x. We denote by Cp , the space of p-convergent sequences [1]. Note that, in contrast to the case for single sequences, a convergent double sequence need not be bounded. By Cbp , we denote the space of double sequences which are bounded convergent. Let A = (amnjk:m, n, j, k = 0, 1, . . .) be a four-dimensional infinite matrix of real numbers. The double series 1 X 1 X amnjk xjk j¼0 k¼0
is called A transform of the double sequence x = (xjk) and denoted by Ax. We say that a sequence x is A-summable to the limit s if the A-transform of x exists for all m, n = 0, 1, . . . and convergent in the Pringsheim sense, i.e., p X q X p- lim amnjk xjk ¼ ymn p;q!1
j¼0 k¼0
and p- lim ymn ¼ s: m;n!1
1110-256X ª 2013 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society. http://dx.doi.org/10.1016/j.joems.2013.03.008
Some inequalities related to the concept of double statistical summability
255
Let X and Y be any two sequence spaces. If x 2 X implies Ax 2 Y, then we say that the matrix A maps X into Y. We denote the class of all matrices A which map X into Y by (X, Y). It is known that A 2 ðCbp ; Cbp Þ, that is, A is conservative if and only if (see [2])
A double infinite Cesa´ro matrix (C, a, b) is a double infinite Hausdorff matrix with entries
p-limamnjk ¼ ajk for each j; k; m;n XX amnjk ¼ a; p-lim
ð1:1Þ
where
ð1:2Þ
Eam ¼
ð1:3Þ
Let rmn denotes the mn-term of the (C, 1, 1) transform of a double sequence x = (xjk), that is, m X n X 1 rmn ¼ rmn ðxjk Þ ¼ xjk : ð1:10Þ ðm þ 1Þðn þ 1Þ j¼0 k¼0
m;n
j
j
X p-lim jamnjk j ¼ aj0 for each j; m;n
p-lim m;n
ð1:4Þ
k
XX jamnjk j exists; j
ð1:5Þ
k
XX jamnjk j < 1: kAk ¼ sup m;n
j
ð1:6Þ
k
PP If A is conservative, the functional ðAÞ ¼ a j k ajk is called the characteristic of four-dimensional matrix A, [3]. If A is conservative and p-lim Ax = p-lim x for all x 2 Cbp , the matrix A is said to be RH-regular, and we denote A 2 ðCbp ; Cbp Þreg . Also, It is known that A 2 ðCbp ; Cbp Þreg if and only if conditions (1.5) and (1.6) hold, and also (1.1) with ajk = 0, (1.2) with a = 1, (1.3) and (1.4) with a0k = aj0 = 0 hold. If A is RH-regular then (A) = 1. For more details of double sequences and related concepts, we refer to [4–11] and references therein. The concept of statistical convergence of single sequences was first introduced by Fast [12]. Further, this concept was studied by Schoenberg [13], Fridy [14], Connor [15], C¸akan and Altay [16], Mohiuddine and Alghamdi [17] and many others. Let E # N N be a two-dimensional set of positive integers and let E(m, n) be the numbers of (j, k) in E such that j 6 m and k 6 n. Then, the two-dimensional analogue of natural density can be defined as follows [18]. The lower asymptotic density of the set E # N N is defined as Eðm; nÞ d2 ðKÞ ¼ lim inf : m;n mn
ð1:7Þ
In case, the sequence (E(m, n)/mn) has a limit in Pringsheim’s sense, then we say that E has a double natural density and is defined as Eðm; nÞ p-lim ¼ d2 ðEÞ: m;n mn
Statistical convergence for double sequences x = (xjk) of real numbers has been defined and studied by Mursaleen and Edely [18] and for double sequences of fuzzy real numbers by Savasß and Mursaleen [19]. Mohiuddine et al. [20] have recently introduced the concept of statistical convergence for double sequences in locally solid Riesz spaces. A real double sequence x = (xjk) is said to be statistically convergent [18] to L if for every e > 0, the set fðj; kÞ; j 6 m and k 6 n : jxjk Lj P g
ð1:9Þ
has double natural density zero. In this case, we write st2 limj,kxjk = L, and we denote the set of all statistically convergent double sequences by st2.
Eam Ebn mþa m
;
¼
Cða þ m þ 1Þ ma : Cðm þ 1ÞCða þ 1Þ Cða þ 1Þ
Savasß et al. [21] studied the concept of the absolute Cesa´ro summability of double series. Recently, Moricz [22] defined the concept of statistical (C, 1,1) summability as follows. A real double sequence x = (xjk) is said to be statistically summable (C,1, 1) to some number L if the sequence r = rmn is statistically convergent to L, i.e. st2 lim rmn = L. We denote by C11(st2) the set of all double sequences which are statistically summable (C,1, 1). We claim that if a double sequence (xjk) is bounded, then st2 lim xjk = L implies st2 lim rmn = L. For any real bounded double sequence x = (xjk), the concepts l(x) = p lim inf x and L(x) = p lim sup x have been introduced in [2]. In this paper, we prove some inequalities related to the concepts of C11(st2)-conservative matrices, C11(st2) lim sup and C11(st2) lim inf which are natural analogues of ðCbp ; st2 \ L1 Þ-matrices, st2 lim sup and st2 lim inf, respectively. Such type of inequalities are also considered by C¸akan and Altay [3] and C¸akan et al [23]. The one-dimensional version of such type inequalities appears in [24] and [25]. 2. The main results Theorem 2.1. Let A = (amnjk) be conservative and x 2 L1 . Then XX k þ ðAÞ k ðAÞ p-lim sup ðamnjk ajk Þxjk 6 uðxÞ þ wðxÞ 2 2 m;n j k
for any k P j ðAÞ j, if and only if XX p-lim sup jamnjk ajk j 6 k; m;n
p-lim m;n
ð1:8Þ
b1 Ea1 mj Enk
k
X p-lim jamnjk j ¼ a0k for each k; m;n
hmnjk ¼
X
j
ð2:1Þ
ð2:2Þ
k
jamnjk ajk j ¼ 0
ð2:3Þ
ðj;kÞ2E
for every E#N N with d2(E) = 0, where u(x) = C11(st2) lim sup x and w(x) = C11(st2) lim inf x. Theorem 2.2. Let A be C11(st2)-conservative. Then, for some constant k P ŒjŒ and for all x 2 L1 , XX kþj kj C11 ðst2 Þ lim sup ðamnjk ajk Þxjk 6 LðxÞ lðxÞ 2 2 m;n j k
ð2:4Þ
if and only if C11 ðst2 Þ lim sup m;n
XX jamnjk ajk j 6 k: j
ð2:5Þ
k
Theorem 2.3. Let A be C11(st2)-conservative. Then, for some constant k P ŒjŒ and for all x 2 L1 ,
256
H. S ß evli, S.A. Mohiuddine
XX kþj kj C11 ðst2 Þ lim sup ðamnjk ajk Þ 6 uðxÞ þ wðxÞ 2 2 m;n j k
if and only if (2.5) holds and X C11 ðst2 Þ lim ðamnjk ajk Þ ¼ 0 m;n
ð2:6Þ
ð2:7Þ
ðj;kÞ2E
XX kK C11 ðst2 Þ lim sup ðamnjk ajk Þ 6 ; 2 n j k
ð3:13Þ
where for any g 2 R; gþ ¼ maxf0; gg and g = max{g,0}.
3. Lemmas
Lemma 3.4 [3]. Let A be conservative. Then for some constant k P j ðAÞ j and for all x 2 L1 , one has
We shall need the following lemmas for the proof of our Theorems. The following lemma is a double sequence version of Corollary 3 of Kolk [26] and Lemma 2.1 of Mursaleen et al [25]. Lemma 3.1. A 2 ðCbp ; st2 \ L1 Þ if and only if
XX k þ ðAÞ k ðAÞ p lim sup ðamnjk ajk Þxjk 6 LðxÞ lðxÞ 2 2 m;n j k
if and only if XX p lim sup jamnjk ajk j 6 k m;n
XX sup jamnjk j < 1; j
ð3:12Þ
and
for every E # N N with d2(E) = 0.
m;n
XX kþK C11 ðst2 Þ lim sup ðamnjk ajk Þþ 6 2 m;n j k
ð3:1Þ
j
ð3:14Þ
ð3:15Þ
k
holds.
k
C11 ðst2 Þ limamnjk ¼ ajk for each j; k; m;n X C11 ðst2 Þ lim jamnjk j ¼ a0k for each k; m;n
ð3:2Þ
j
C11 ðst2 Þ lim
X jamnjk j ¼ aj0 for each j;
ð3:4Þ
C11 ðst2 Þ lim
k X X amnjk ¼ a:
ð3:5Þ
m;n
m;n
j
4. Proof of theorems
ð3:3Þ Proof of Theorem 2.1. Necessity. Let L(x) = p lim sup x and l(x) = p lim inf x. Since u(x) 6 L(x) and w(x) 6 l(x) for all x 2 L1 , we have p lim sup
k
m;n
We call such matrices as C11(st2)-conservative matrices, and in this case K¼a
XX ajk j
ð3:6Þ
k
is defined which is known as the C11(st2) characteristic of A. This number is analogous to the number st2 ðAÞ defined by Cosßkun and C¸akan [24] for single sequences. The following lemma which is C11(st2)-analogue of a result of Patterson [2]. Lemma 3.2. Let A = (amnjk) be a four-dimensional matrix and iAi < 1. If conditions
j
m;n
ð3:7Þ
and the necessity of (2.2) follows from Lemma 3.4. Denote by B = (bmnjk) the doubly infinite matrix with entries bmnjk ¼
amnjk ajk for ðj; kÞ 2 E; 0 for ðj; kÞ R E:
Since A is conservative, the matrix B satisfies the conditions of Lemma 3.2. Hence there exists a y 2 L1 such that iyi 6 1 and p lim sup
j
m;n
ð3:9Þ
k
are satisfied then there exists y 2 L1 such that iyi 6 1 and C11 ðst2 Þ lim sup
XX XX amnjk yjk ¼ C11 ðst2 Þ lim sup jamnjk j: j
j
k
Lemma 3.3. Let A be C11(st2)-conservative and k > 0. Then
m;n
if and only if
XX jamnjk ajk j 6 k j
k
ð3:11Þ
j
m;n
k
j
ð4:1Þ
k
1 for ðj; kÞ 2 E; 0 for ðj; kÞ R E:
So that, C11(st2) lim y = u(y) = w(y) = 0; and by (2.1) and (4.1) we have p lim sup
The following lemma is derived by replacing st with C1(st) from Lemma 2.3 of Cosßkun and C¸akan [24].
C11 ðst2 Þ lim sup
yjk ¼
ð3:10Þ
k
XX XX jbmnjk j ¼ p lim sup bmnjk yjk :
Now, let y = (yjk) be defined by
ð3:8Þ
X C11 ðst2 Þ lim jamnjk j ¼ 0 for each j;
k
k þ ðAÞ k ðAÞ 6 LðxÞ lðxÞ; 2 2
m;n
C11 ðst2 Þ limamnjk ¼ 0 for each j; k; m;n X C11 ðst2 Þ lim jamnjk j ¼ 0 for each k;
XX ðamnjk ajk Þxjk
m;n
X
jamnjk ajk j 6
ðj;kÞ2E
k þ ðAÞ k ðAÞ uðyÞ þ 2 2
wðxÞ ¼ 0 and we get (2.3). Sufficiency. Let x 2 L1 . Write E1 ¼ fðj; kÞ : rjk > uðxÞ þ g N N and E2 ¼ fðj; kÞ : rjk < wðxÞ g N N. Then we have d2(E1) = d2(E2) = 0; and hence d2(E) = 0 for E = E1 \ E2. We can write
Some inequalities related to the concept of double statistical summability XX X X þ ðamnjk ajk Þxjk ¼ ðamnjk ajk Þxjk þ ðamnjk ajk Þ xjk j
k
ðj;kÞ2E
X
m;n
ðj;kÞRE
ðamnjk ajk Þ xjk :
Hence m;n
X
XX j
ðamnjk ajk Þxjk 6 p lim sup m;n
k
which gives (2.4), since e was arbitrary.
m;n
ðamnjk ajk Þþ rjk þ p lim sup m;n
ðj;kÞRE
X
Proof of Theorem 2.3. Necessity. Let (2.6) hold. Since u(x) 6 L(x) and w(x) 6 l(x), (2.5) follows from Theorem 2.2. Now let us show the necessity of (2.7). For any E # N N with d2(E) = 0, let us define a matrix B = (bmnjk) by the following
! ðamnjk ajk Þ rjk
ðj;kÞRE
¼ I1 ðxÞ þ I2 ðxÞ þ I3 ðxÞ; say:
From condition (2.3), we have I1(x) = 0. Let e > 0, then there is a set E as defined above such that for (j,k) R E, wðxÞ < rp < uðxÞ þ ;
uðxÞ < rj;k < wðxÞ þ :
ð4:2Þ
Therefore from conditions (2.2) and (4.2) we get kþ ðuðxÞ þ Þ 2 k ðwðxÞ þ Þ: I3 ðxÞ 6 2 I2 ðxÞ 6
m;n
j
bmnjk ¼
amnjk ajk 0
kþ k uðxÞ þ wðxÞ þ k; 2 2 kþ k 6 uðxÞ þ wðxÞ; 2 2
C11 ðst2 Þ lim sup m;n
6
since e was arbitrary. This completes the proof of the theorem. Proof of Theorem 2.2. Necessity. If we define the matrix B = (bmnjk) by bmnjk = amnjk ajk for all m, n, j, k, then, since A is C11(st2)-conservative, the matrix B satisfies the hypothesis of Lemma 3.2. Hence for a y 2 L1 such that iyi 6 1 we have XX C11 ðst2 Þ lim sup jbmnjk j ¼ C11 ðst2 Þ k
j
2.1,
þ
j>j0 k>k0
j6j0 k6k0
ðamnjk ajk Þþ xjk
XX
XX
ðamnjk ajk Þxjk ¼
k
X
ðamnjk ajk Þxjk þ
ðj;kÞ2E
C11 ðst2 Þ lim sup m;n
X
X
þ
ðamnjk ajk Þ xjk
ðj;kÞRE
ðamnjk ajk Þ xjk :
XX kþj kj amnjk ajk 6 uðxÞ þ wðxÞ þ k: 2 2 j k
But e was arbitrary, so (2.6) holds and this completes the proof of theorem. References some
XX XX ðamnjk ajk Þxjk ¼ ðamnjk ajk Þxk XX
Sufficiency. Let (2.5) and (2.7) hold and x 2 L1 . As in Theorem 2.1, we can write
Using Lemma 3.3 and since A is C11(st2)-conservative, we have
k
for
kþj kj uðyÞ þ wðyÞ ¼ 0; 2 2
ðj;kÞRE
since kyk 6 1:
Sufficiency. As in Theorem j0 ; k0 2 N ðj > j0 ; k > k0 Þ, we can write
jamnjk ajk jxjk
ðj;kÞ2E
Using (2.4), we get XX kþj kj C11 ðst2 Þ lim sup jbmnjk j 6 LðyÞ lðyÞ 2 2 m;n j k kþj kj kyk þ 6 2 2 6 k;
X
and hence we get (2.7).
j
XX bmnjk yjk : lim sup m;n
k
j;k
Using the fact that C11(st2) lim y = u(y) = w(y) = 0 and (2.6), we get
<
j
m;n
j;k
Let us define the sequence y = (yjk) by 1 for ðj; kÞ 2 E; yjk ¼ 0 for ðj; kÞ R E:
k
m;n
for ðj; kÞ 2 E; for ðj; kÞ R E:
Then, it is clear that B satisfies the conditions of Lemma 3.2 and hence there exists a y 2 L1 such that iyi 6 1 and X X C11 ðst2 Þ lim sup bmnjk yjk ¼ C11 ðst2 Þ lim sup jbmnjk j: m;n
Hence we get XX p lim sup ðamnjk ajk Þxk
j
k
jamnjk ajk jjxjk j þ p lim sup
ðj;kÞ2E
X
j
kþj kj ðlðxÞ Þ 6 ðLðxÞ þ Þ 2 2 kþj kj LðxÞ lðxÞ þ k; ¼ 2 2
ðj;kÞRE
p lim sup
257
XX C11 ðst2 Þ lim sup ðamnjk ajk Þxjk
ðamnjk ajk Þ xjk :
j>j0 k>k0
Since for any e > 0, l(x) e < xjk < L(x) + e; and A is C11(st2)-conservative, we get by Lemma 3.2 that
[1] A. Pringsheim, Zur theorie der zweifach unendlichen Zahlenfolgen, Math. Ann. 53 (1900) 289–321. [2] R.F. Patterson, Double sequence core theorems, Int. J. Math. Math. Sci. 22 (4) (1999) 785–793. [3] C. C¸akan, B. Altay, A class of conservative four dimensional matrices, J. Ineq. Appl. (2006). Article ID 14721, pages 1. [4] M. Mursaleen, S.A. Mohiuddine, Some matrix transformations of convex and paranormed sequence spaces into the spaces of invariant means, J. Function Spaces Appl. 2012 (2012). Article ID 612671, 10 pages.
258 [5] M. Mursaleen, S.A. Mohiuddine, Double r-multiplicative matrices, J. Math. Anal. Appl. 327 (2007) 991–996. [6] M. Mursaleen, S.A. Mohiuddine, Regularly r-conservative and r-coercive four dimensional matrices, Comput. Math. Appl. 56 (2008) 1580–1586. [7] M. Mursaleen, S.A. Mohiuddine, On r-conservative and boundedly r-conservative four-dimensional matrices, Comput. Math. Appl. 59 (2010) 880–885. [8] M. Mursaleen, S.A. Mohiuddine, Some new double sequence spaces of invariant means, Glasnik Matematicki 45 (65) (2010) 139–153. [9] M. Mursaleen, S.A. Mohiuddine, Almost bounded variation of double sequences and some four dimensional summability matrices, Publ. Math. Debrecen 75/3–4 (2009) 495–508. [10] M. Mursaleen, S.A. Mohiuddine, Invariant mean and some core theorems for double sequences, Taiwanese J. Math. 14 (1) (2010) 21–33. [11] M. Mursaleen, S.A. Mohiuddine, Some inequalities on sublinear functionals related to the invariant mean for double sequences, Math. Ineq. Appl. 13 (1) (2010) 157–163. [12] H. Fast, Sur la convergence statistique, Colloq. Math. 2 (1951) 241–244. [13] I.J. Schoenberg, The integrability of certain functions and related summability methods, Amer. Math. Monthly 66 (1959) 361–375. [14] J.A. Fridy, On statistical convergence, Analysis 5 (1985) 301– 313. [15] J.S. Connor, The statistical and strong p-Cesa`ro convergence of sequences, Analysis 8 (1988) 47–63.
H. S ß evli, S.A. Mohiuddine [16] C. C¸akan, B. Altay, Statistically boundedness and statistical core of double sequences, J. Math. Anal. Appl. 317 (2006) 690– 697. [17] S.A. Mohiuddine, M.A. Alghamdi, Statistical summability through a lacunary sequence in locally solid Riesz spaces, J. Inequal. Appl. (2012), 2012:225. [18] M. Mursaleen, Osama H. Edely, Statistical convergence of double sequences, J. Math. Anal. Appl. 288 (2003) 223–231. [19] E. Savasß, M. Mursaleen, On statistically convergent double sequences of fuzzy numbers, Inform. Sci. 162 (2004) 183–192. [20] S.A. Mohiuddine, A. Alotaibi, M. Mursaleen, Statistical convergence of double sequences in locally solid Riesz spaces, Abstr. Appl. Analy. 2012 (2012) 9. Article ID 719729. [21] E. Savasß, H. S ß evli, B.E. Rhoades, On the Cesa´ro summability of double series, J. Ineq. Appl. (2008) 4. Article ID 257318. [22] F. Moricz, Tauberian theorems for double sequences that are statistically summable (C, 1, 1), J. Math. Anal. Appl. 286 (2003) 340–350. [23] C. C¸akan, B. Altay, H. Cosßkun, Double lacunary and lacunary statistical convergence of double sequences. Stud. Sci. Math. Hungarica 47 (1) (2010) 35–45. [24] H. Cosßkun, C. C¸akan, A class of statistical and r-conservative matrices, Czech. Math. J. 55 (103) (2005) 791–801. [25] M. Mursaleen, A. Alotaibi, S.A. Mohiuddine, Some inequalities on statistical summability (C, 1), J. Math. Ineq. 2 (2) (2008) 239– 245. [26] E. Kolk, Matrix maps into the space of statistically convergent bounded sequences, Proc. Estonian Acad. Sci. Phys. Math. 45 (1996) 187–192.