Özarslan and Kartal Journal of Inequalities and Applications (2017) 2017:179 DOI 10.1186/s13660-017-1455-3
RESEARCH
Open Access
A generalization of a theorem of Bor ˘ Hikmet Seyhan Özarslan* and Bagdagül Kartal *
Correspondence: [email protected] Department of Mathematics, Erciyes University, Kayseri, 38039, Turkey
Abstract In this paper, a general theorem concerning absolute matrix summability is established by applying the concepts of almost increasing and δ -quasi-monotone sequences. MSC: 26D15; 40D15; 40F05; 40G99 Keywords: matrix transformations; almost increasing sequences; quasi-monotone sequences; Hölder inequality; Minkowski inequality
1 Introduction A positive sequence (yn ) is said to be almost increasing if there is a positive increasing sequence (un ) and two positive constants K and M such that Kun ≤ yn ≤ Mun (see []). A sequence (cn ) is said to be δ-quasi-monotone, if cn → , cn > ultimately and cn ≥ –δn , where cn = cn – cn+ and δ = (δn ) is a sequence of positive numbers (see []). Let an be a given infinite series with partial sums (sn ). Let T = (tnv ) be a normal matrix, i.e., a lower triangular matrix of nonzero diagonal entries. At that time T describes the sequence-tosequence transformation, mapping the sequence s = (sn ) to Ts = (Tn (s)), where
Tn (s) =
n
tnv sv ,
n = , , . . .
()
v=
Let (ϕn ) be any sequence of positive real numbers. The series ϕ – |T, pn |k , k ≥ , if (see []) ∞
¯ n (s)k < ∞, ϕnk– T
an is said to be summable
()
n=
where ¯ n (s) = Tn (s) – Tn– (s). T If we take ϕn = Ppnn , then ϕ – |T, pn |k summability reduces to |T, pn |k summability (see []). If we set ϕn = n for all n, ϕ – |T, pn |k summability is the same as |T|k summability (see []). ¯ pn |k summability (see []). Also, if we take ϕn = Ppnn and tnv = Ppnv , then we get |N, © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Özarslan and Kartal Journal of Inequalities and Applications (2017) 2017:179
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2 Known result ¯ pn |k summability facIn [, ], Bor has established the following theorem dealing with |N, tors of infinite series. Theorem . Let (Yn ) be an almost increasing sequence such that |Yn | = O(Yn /n) and λn → as n → ∞. Assume that there is a sequence of numbers (Bn ) such that it is δ-quasi monotone with nYn δn < ∞, Bn Yn is convergent and |λn | ≤ |Bn | for all n. If m |λn | = O() as m → ∞, n n=
()
m |zn |k = O(Ym ) as m → ∞, n n=
()
and m pn n=
Pn
|zn |k = O(Ym ) as m → ∞,
()
where (zn ) is the nth (C, ) mean of the sequence (nan ), then the series ¯ pn |k , k ≥ . |N,
an λn is summable
3 Main result The purpose of this paper is to generalize Theorem . to the ϕ – |T, pn |k summability. Before giving main theorem, let us introduce some well-known notations. Let T = (tnv ) be a normal matrix. Lower semimatrices T¯ = (t¯nv ) and Tˆ = (tˆnv ) are defined as follows: t¯nv =
n
tni ,
n, v = , , . . .
()
i=v
and tˆ = t¯ = t ,
tˆnv = t¯nv – t¯n–,v ,
n = , , . . .
()
Here, T¯ and Tˆ are the well-known matrices of series-to-sequence and series-to-series transformations, respectively. Then we write
Tn (s) =
n
tnv sv =
v=
n
t¯nv av
()
v=
and ¯ n (s) = T
n
tˆnv av .
()
v=
By taking the definition of general absolute matrix summability, we established the following theorem.
Özarslan and Kartal Journal of Inequalities and Applications (2017) 2017:179
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Theorem . Let T = (tnv ) be a positive normal matrix such that t¯n = ,
n = , , . . . ,
()
tn–,v ≥ tnv , for n ≥ v + , pn , tnn = O Pn
() ()
and ( ϕPn npn ) be a non-increasing sequence. If all conditions of Theorem . with conditions () and () are replaced by m
ϕnk–
n=
pn Pn
k–
|zn |k = O(Ym ) as m → ∞ n
()
and m
ϕnk–
n=
then the series
pn Pn
k |zn |k = O(Ym ) as m → ∞,
()
an λn is ϕ – |T, pn |k summable, k ≥ .
We need the following lemmas for the proof of Theorem .. Lemma . ([]) Let (Yn ) be an almost increasing sequence and λn → as n → ∞. If (Bn ) is δ-quasi-monotone with Bn Yn is convergent and |λn | ≤ |Bn | for all n, then we have |λn |Yn = O() as n → ∞.
()
Lemma . ([]) Let (Yn ) be an almost increasing sequence such that n|Yn | = O(Yn ). If nYn δn < ∞, and Bn Yn is convergent, then
(Bn ) is δ-quasi monotone with
nBn Yn = O() as n → ∞, ∞
()
nYn |Bn | < ∞.
()
n=
4 Proof of Theorem 3.1 Let (In ) indicate the T-transform of the series an λn . Then we obtain ¯ n= I
n
tˆnv av λv =
v=
by means of () and ().
n tˆnv λv v=
v
vav
()
Özarslan and Kartal Journal of Inequalities and Applications (2017) 2017:179
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Using Abel’s formula for (), we obtain
¯ n= I
n–
v
v=
=
n– v=
+
v n tˆnv λv tˆnn λn rar + rar v n r= r=
v+ v+ v (tˆnv )λv zv + tˆn,v+ λv zv v v v=
n–
n–
tˆn,v+ λv+
v=
zv n + + tnn λn zn v n
= In, + In, + In, + In, . For the proof of Theorem ., it suffices to prove that ∞
ϕnk– |In,r |k < ∞
n=
for r = , , , . By Hölder’s inequality, we have m+
ϕnk– |In, |k
n=
= O()
m+
ϕnk–
k n– v (tˆnv )|λv ||zv |
n=
v=
n=
v=
k– n– n– m+ k k k– = O() ϕn . v (tˆnv ) |λv | |zv | v (tˆnv ) v=
By () and (), we have v (tˆnv ) = tˆnv – tˆn,v+ = t¯nv – t¯n–,v – t¯n,v+ + t¯n–,v+ = tnv – tn–,v .
()
Thus using (), () and () n– n– v (tˆnv ) = (tn–,v – tnv ) ≤ tnn . v=
v=
Hence, we get m+ n=
ϕnk– |In, |k
= O()
m+ n=
k– ϕnk– tnn
n– k k v (tˆnv )|λv | |zv | v=
Özarslan and Kartal Journal of Inequalities and Applications (2017) 2017:179
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by using () m+
ϕnk– |In, |k
n=
n– m+ k k ϕn pn k– v (tˆnv )|λv | |zv | = O() Pn n= v= = O()
m v=
= O()
m+ ϕn pn k– v (tˆnv ) |λv | |zv | P n n=v+ k
k
m ϕv pv k– v=
Pv
|λv |k |zv |k
m+ v (tˆnv ). n=v+
Now, using () and (), we obtain m+ m+ v (tˆnv ) = (tn–,v – tnv ) ≤ tvv . n=v+
n=v+
Thus, by using Abel’s formula, we obtain m+
ϕnk– |In, |k
= O()
n=
m ϕv pv k– v=
= O()
m
Pv ϕvk–
v=
= O()
m–
|λv |
v=
= O()
m–
pv Pv
|λv |k– |λv ||zv |k tvv
k
v
|λv ||zv |k ϕrk–
r=
pr Pr
k |zr |k + O()|λm |
v=
|λv |Yv + O()|λm |Ym
v=
= O()
m–
|Bv |Yv + O()|λm |Ym
v=
= O() as m → ∞, in view of () and (). Again, using Hölder’s inequality, we have m+
ϕnk– |In, |k
= O()
n=
m+
ϕnk–
n=
= O()
m+ n=
×
n– v=
n–
k |tˆn,v+ ||λv ||zv |
v=
ϕnk–
n– v=
|tˆn,v+ ||Bv ||zv |k k–
|tˆn,v+ ||Bv |
.
m
ϕvk–
pv Pv
k |zv |k
Özarslan and Kartal Journal of Inequalities and Applications (2017) 2017:179
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By means of (), () and (), we have tˆn,v+ = t¯n,v+ – t¯n–,v+ =
n
n–
tni –
i=v+
tn–,i
i=v+ n–
= tnn +
(tni – tn–,i )
i=v+
≤ tnn . In this way, we have m+
ϕnk– |In, |k
= O()
n=
m+
k– ϕnk– tnn
n–
n=
= O()
m
Pn |Bv ||zv |k
|tˆn,v+ ||Bv ||zv |
m+ ϕn pn k– Pn
n=v+
m ϕv pv k– v=
k
v=
v=
= O()
|tˆn,v+ ||Bv ||zv |
v=
n– m+ ϕn pn k– n=
= O()
k
Pv
m+
|Bv ||zv |k
|tˆn,v+ |
|tˆn,v+ |.
n=v+
By (), (), () and (), we obtain |tˆn,v+ | =
v
(tn–,i – tni ).
i=
Thus, using () and (), we have m+
m+ v
|tˆn,v+ | =
n=v+
(tn–,i – tni ) ≤ ,
n=v+ i=
then we get m+
ϕnk– |In, |k
= O()
n=
m v=
ϕvk–
pv Pv
k–
v|Bv | |zv |k v
k– m– v
k k– pr |zr | = O() v|Bv | ϕr P r r v= r= k– m pv k |zv | + O()m|Bm | ϕvk– P v v v= = O()
m–
v|Bv |Yv + O()
v=
= O() as m → ∞, in view of (), () and ().
m– v=
|Bv |Yv + O()m|Bm |Ym
Özarslan and Kartal Journal of Inequalities and Applications (2017) 2017:179
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Also, we have m+
ϕnk– |In, |k
≤
n=
m+
ϕnk–
n=
≤
m+
n– v=
ϕnk–
n=
n– v=
|zv | |tˆn,v+ ||λv+ | v
k
|zv |k |tˆn,v+ ||λv+ | v
n– v=
|λv+ | |tˆn,v+ | v
k–
n– k– m+ |λv+ | |zv |k k– k– ≤ ϕn tnn |tˆn,v+ ||λv+ | v v n= v= v= k– m+ n– |zv |k k– pn = O() ϕn |tˆn,v+ ||λv+ | Pn v n= v= m+ n– |zv |k ϕn pn k– |tˆn,v+ ||λv+ | = O() Pn v n= v= = O()
n–
m
|λv+ |
v=
= O()
m ϕv pv k– Pv
v=
= O()
m
ϕvk–
v=
= O()
m+ |zv | k ϕn pn k– |tˆn,v+ | v n=v+ Pn
m–
pv Pv
|λv+ |
|λv+ |
k– |λv+ |
v
v= m
k–
pr Pr
k–
ϕvk–
v=
= O()
|zv | k v
ϕrk–
r=
+ O()|λm+ | m–
m+ |zv | k |tˆn,v+ | v n=v+
pv Pv
k |zr | r
k |zv | v
|Bv+ |Yv+ + O()|λm+ |Ym+
v=
= O() as m → ∞, in view of (), (), () and (). Finally, as in In, , we have m
ϕnk– |In, |k = O()
n=
m
k ϕnk– tnn |λn |k |zn |k
n=
= O()
m
ϕnk–
n=
= O()
m n=
ϕnk–
pn Pn pn Pn
k |λn |k– |λn ||zn |k k |λn ||zn |k = O() as m → ∞,
in view of (), () and (). Finally, the proof of Theorem . is completed.
Özarslan and Kartal Journal of Inequalities and Applications (2017) 2017:179
5 Corollary If we take ϕn = Ppnn and tnv = Ppnv in Theorem ., then we get Theorem .. In this case, conditions () and () reduce to conditions () and (), respectively. Also, the condition ‘( ϕPn npn ) is a non-increasing sequence’ and the conditions ()-() are clearly satisfied. 6 Conclusions In this study, we have generalized a well-known theorem dealing with an absolute summability method to a ϕ – |T, pn |k summability method of an infinite series by using almost increasing sequences and δ-quasi-monotone sequences. Acknowledgements This work was supported by Research Fund of the Erciyes University, Project Number: FBA-2014-3846. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the manuscript and read and approved the final manuscript.
Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 3 May 2017 Accepted: 18 July 2017 References 1. Bari, NK, Steˇckin, SB: Best approximations and differential properties of two conjugate functions. Tr. Mosk. Mat. Obˆs. 5, 483-522 (1956) 2. Boas, RP: Quasi-positive sequences and trigonometric series. Proc. Lond. Math. Soc. 14A, 38-46 (1965) 3. Özarslan, HS, Keten, A: A new application of almost increasing sequences. An. S¸ tiin¸t. Univ. ‘Al.I. Cuza’ Ia¸si, Mat. 61, 153-160 (2015) 4. Sulaiman, WT: Inclusion theorems for absolute matrix summability methods of an infinite series. IV. Indian J. Pure Appl. Math. 34, 1547-1557 (2003) 5. Tanovi˘c-Miller, N: On strong summability. Glas. Mat. Ser. III 14, 87-97 (1979) 6. Bor, H: On two summability methods. Math. Proc. Camb. Philos. Soc. 97, 147-149 (1985) 7. Bor, H: An application of almost increasing and δ -quasi-monotone sequences. JIPAM. J. Inequal. Pure Appl. Math. 1(2), Article ID 18 (2000) 8. Bor, H: Corrigendum on the paper ‘An application of almost increasing and δ -quasi-monotone sequences’ published in JIPAM, Vol.1, No.2. (2000), Article 18. JIPAM. J. Inequal. Pure Appl. Math. 3(1), Article ID 16 (2002)
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RESEARCH
Open Access
A generalization of a theorem of Bor ˘ Hikmet Seyhan Özarslan* and Bagdagül Kartal *
Correspondence: [email protected] Department of Mathematics, Erciyes University, Kayseri, 38039, Turkey
Abstract In this paper, a general theorem concerning absolute matrix summability is established by applying the concepts of almost increasing and δ -quasi-monotone sequences. MSC: 26D15; 40D15; 40F05; 40G99 Keywords: matrix transformations; almost increasing sequences; quasi-monotone sequences; Hölder inequality; Minkowski inequality
1 Introduction A positive sequence (yn ) is said to be almost increasing if there is a positive increasing sequence (un ) and two positive constants K and M such that Kun ≤ yn ≤ Mun (see []). A sequence (cn ) is said to be δ-quasi-monotone, if cn → , cn > ultimately and cn ≥ –δn , where cn = cn – cn+ and δ = (δn ) is a sequence of positive numbers (see []). Let an be a given infinite series with partial sums (sn ). Let T = (tnv ) be a normal matrix, i.e., a lower triangular matrix of nonzero diagonal entries. At that time T describes the sequence-tosequence transformation, mapping the sequence s = (sn ) to Ts = (Tn (s)), where
Tn (s) =
n
tnv sv ,
n = , , . . .
()
v=
Let (ϕn ) be any sequence of positive real numbers. The series ϕ – |T, pn |k , k ≥ , if (see []) ∞
¯ n (s)k < ∞, ϕnk– T
an is said to be summable
()
n=
where ¯ n (s) = Tn (s) – Tn– (s). T If we take ϕn = Ppnn , then ϕ – |T, pn |k summability reduces to |T, pn |k summability (see []). If we set ϕn = n for all n, ϕ – |T, pn |k summability is the same as |T|k summability (see []). ¯ pn |k summability (see []). Also, if we take ϕn = Ppnn and tnv = Ppnv , then we get |N, © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Özarslan and Kartal Journal of Inequalities and Applications (2017) 2017:179
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2 Known result ¯ pn |k summability facIn [, ], Bor has established the following theorem dealing with |N, tors of infinite series. Theorem . Let (Yn ) be an almost increasing sequence such that |Yn | = O(Yn /n) and λn → as n → ∞. Assume that there is a sequence of numbers (Bn ) such that it is δ-quasi monotone with nYn δn < ∞, Bn Yn is convergent and |λn | ≤ |Bn | for all n. If m |λn | = O() as m → ∞, n n=
()
m |zn |k = O(Ym ) as m → ∞, n n=
()
and m pn n=
Pn
|zn |k = O(Ym ) as m → ∞,
()
where (zn ) is the nth (C, ) mean of the sequence (nan ), then the series ¯ pn |k , k ≥ . |N,
an λn is summable
3 Main result The purpose of this paper is to generalize Theorem . to the ϕ – |T, pn |k summability. Before giving main theorem, let us introduce some well-known notations. Let T = (tnv ) be a normal matrix. Lower semimatrices T¯ = (t¯nv ) and Tˆ = (tˆnv ) are defined as follows: t¯nv =
n
tni ,
n, v = , , . . .
()
i=v
and tˆ = t¯ = t ,
tˆnv = t¯nv – t¯n–,v ,
n = , , . . .
()
Here, T¯ and Tˆ are the well-known matrices of series-to-sequence and series-to-series transformations, respectively. Then we write
Tn (s) =
n
tnv sv =
v=
n
t¯nv av
()
v=
and ¯ n (s) = T
n
tˆnv av .
()
v=
By taking the definition of general absolute matrix summability, we established the following theorem.
Özarslan and Kartal Journal of Inequalities and Applications (2017) 2017:179
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Theorem . Let T = (tnv ) be a positive normal matrix such that t¯n = ,
n = , , . . . ,
()
tn–,v ≥ tnv , for n ≥ v + , pn , tnn = O Pn
() ()
and ( ϕPn npn ) be a non-increasing sequence. If all conditions of Theorem . with conditions () and () are replaced by m
ϕnk–
n=
pn Pn
k–
|zn |k = O(Ym ) as m → ∞ n
()
and m
ϕnk–
n=
then the series
pn Pn
k |zn |k = O(Ym ) as m → ∞,
()
an λn is ϕ – |T, pn |k summable, k ≥ .
We need the following lemmas for the proof of Theorem .. Lemma . ([]) Let (Yn ) be an almost increasing sequence and λn → as n → ∞. If (Bn ) is δ-quasi-monotone with Bn Yn is convergent and |λn | ≤ |Bn | for all n, then we have |λn |Yn = O() as n → ∞.
()
Lemma . ([]) Let (Yn ) be an almost increasing sequence such that n|Yn | = O(Yn ). If nYn δn < ∞, and Bn Yn is convergent, then
(Bn ) is δ-quasi monotone with
nBn Yn = O() as n → ∞, ∞
()
nYn |Bn | < ∞.
()
n=
4 Proof of Theorem 3.1 Let (In ) indicate the T-transform of the series an λn . Then we obtain ¯ n= I
n
tˆnv av λv =
v=
by means of () and ().
n tˆnv λv v=
v
vav
()
Özarslan and Kartal Journal of Inequalities and Applications (2017) 2017:179
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Using Abel’s formula for (), we obtain
¯ n= I
n–
v
v=
=
n– v=
+
v n tˆnv λv tˆnn λn rar + rar v n r= r=
v+ v+ v (tˆnv )λv zv + tˆn,v+ λv zv v v v=
n–
n–
tˆn,v+ λv+
v=
zv n + + tnn λn zn v n
= In, + In, + In, + In, . For the proof of Theorem ., it suffices to prove that ∞
ϕnk– |In,r |k < ∞
n=
for r = , , , . By Hölder’s inequality, we have m+
ϕnk– |In, |k
n=
= O()
m+
ϕnk–
k n– v (tˆnv )|λv ||zv |
n=
v=
n=
v=
k– n– n– m+ k k k– = O() ϕn . v (tˆnv ) |λv | |zv | v (tˆnv ) v=
By () and (), we have v (tˆnv ) = tˆnv – tˆn,v+ = t¯nv – t¯n–,v – t¯n,v+ + t¯n–,v+ = tnv – tn–,v .
()
Thus using (), () and () n– n– v (tˆnv ) = (tn–,v – tnv ) ≤ tnn . v=
v=
Hence, we get m+ n=
ϕnk– |In, |k
= O()
m+ n=
k– ϕnk– tnn
n– k k v (tˆnv )|λv | |zv | v=
Özarslan and Kartal Journal of Inequalities and Applications (2017) 2017:179
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by using () m+
ϕnk– |In, |k
n=
n– m+ k k ϕn pn k– v (tˆnv )|λv | |zv | = O() Pn n= v= = O()
m v=
= O()
m+ ϕn pn k– v (tˆnv ) |λv | |zv | P n n=v+ k
k
m ϕv pv k– v=
Pv
|λv |k |zv |k
m+ v (tˆnv ). n=v+
Now, using () and (), we obtain m+ m+ v (tˆnv ) = (tn–,v – tnv ) ≤ tvv . n=v+
n=v+
Thus, by using Abel’s formula, we obtain m+
ϕnk– |In, |k
= O()
n=
m ϕv pv k– v=
= O()
m
Pv ϕvk–
v=
= O()
m–
|λv |
v=
= O()
m–
pv Pv
|λv |k– |λv ||zv |k tvv
k
v
|λv ||zv |k ϕrk–
r=
pr Pr
k |zr |k + O()|λm |
v=
|λv |Yv + O()|λm |Ym
v=
= O()
m–
|Bv |Yv + O()|λm |Ym
v=
= O() as m → ∞, in view of () and (). Again, using Hölder’s inequality, we have m+
ϕnk– |In, |k
= O()
n=
m+
ϕnk–
n=
= O()
m+ n=
×
n– v=
n–
k |tˆn,v+ ||λv ||zv |
v=
ϕnk–
n– v=
|tˆn,v+ ||Bv ||zv |k k–
|tˆn,v+ ||Bv |
.
m
ϕvk–
pv Pv
k |zv |k
Özarslan and Kartal Journal of Inequalities and Applications (2017) 2017:179
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By means of (), () and (), we have tˆn,v+ = t¯n,v+ – t¯n–,v+ =
n
n–
tni –
i=v+
tn–,i
i=v+ n–
= tnn +
(tni – tn–,i )
i=v+
≤ tnn . In this way, we have m+
ϕnk– |In, |k
= O()
n=
m+
k– ϕnk– tnn
n–
n=
= O()
m
Pn |Bv ||zv |k
|tˆn,v+ ||Bv ||zv |
m+ ϕn pn k– Pn
n=v+
m ϕv pv k– v=
k
v=
v=
= O()
|tˆn,v+ ||Bv ||zv |
v=
n– m+ ϕn pn k– n=
= O()
k
Pv
m+
|Bv ||zv |k
|tˆn,v+ |
|tˆn,v+ |.
n=v+
By (), (), () and (), we obtain |tˆn,v+ | =
v
(tn–,i – tni ).
i=
Thus, using () and (), we have m+
m+ v
|tˆn,v+ | =
n=v+
(tn–,i – tni ) ≤ ,
n=v+ i=
then we get m+
ϕnk– |In, |k
= O()
n=
m v=
ϕvk–
pv Pv
k–
v|Bv | |zv |k v
k– m– v
k k– pr |zr | = O() v|Bv | ϕr P r r v= r= k– m pv k |zv | + O()m|Bm | ϕvk– P v v v= = O()
m–
v|Bv |Yv + O()
v=
= O() as m → ∞, in view of (), () and ().
m– v=
|Bv |Yv + O()m|Bm |Ym
Özarslan and Kartal Journal of Inequalities and Applications (2017) 2017:179
Page 7 of 8
Also, we have m+
ϕnk– |In, |k
≤
n=
m+
ϕnk–
n=
≤
m+
n– v=
ϕnk–
n=
n– v=
|zv | |tˆn,v+ ||λv+ | v
k
|zv |k |tˆn,v+ ||λv+ | v
n– v=
|λv+ | |tˆn,v+ | v
k–
n– k– m+ |λv+ | |zv |k k– k– ≤ ϕn tnn |tˆn,v+ ||λv+ | v v n= v= v= k– m+ n– |zv |k k– pn = O() ϕn |tˆn,v+ ||λv+ | Pn v n= v= m+ n– |zv |k ϕn pn k– |tˆn,v+ ||λv+ | = O() Pn v n= v= = O()
n–
m
|λv+ |
v=
= O()
m ϕv pv k– Pv
v=
= O()
m
ϕvk–
v=
= O()
m+ |zv | k ϕn pn k– |tˆn,v+ | v n=v+ Pn
m–
pv Pv
|λv+ |
|λv+ |
k– |λv+ |
v
v= m
k–
pr Pr
k–
ϕvk–
v=
= O()
|zv | k v
ϕrk–
r=
+ O()|λm+ | m–
m+ |zv | k |tˆn,v+ | v n=v+
pv Pv
k |zr | r
k |zv | v
|Bv+ |Yv+ + O()|λm+ |Ym+
v=
= O() as m → ∞, in view of (), (), () and (). Finally, as in In, , we have m
ϕnk– |In, |k = O()
n=
m
k ϕnk– tnn |λn |k |zn |k
n=
= O()
m
ϕnk–
n=
= O()
m n=
ϕnk–
pn Pn pn Pn
k |λn |k– |λn ||zn |k k |λn ||zn |k = O() as m → ∞,
in view of (), () and (). Finally, the proof of Theorem . is completed.
Özarslan and Kartal Journal of Inequalities and Applications (2017) 2017:179
5 Corollary If we take ϕn = Ppnn and tnv = Ppnv in Theorem ., then we get Theorem .. In this case, conditions () and () reduce to conditions () and (), respectively. Also, the condition ‘( ϕPn npn ) is a non-increasing sequence’ and the conditions ()-() are clearly satisfied. 6 Conclusions In this study, we have generalized a well-known theorem dealing with an absolute summability method to a ϕ – |T, pn |k summability method of an infinite series by using almost increasing sequences and δ-quasi-monotone sequences. Acknowledgements This work was supported by Research Fund of the Erciyes University, Project Number: FBA-2014-3846. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the manuscript and read and approved the final manuscript.
Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 3 May 2017 Accepted: 18 July 2017 References 1. Bari, NK, Steˇckin, SB: Best approximations and differential properties of two conjugate functions. Tr. Mosk. Mat. Obˆs. 5, 483-522 (1956) 2. Boas, RP: Quasi-positive sequences and trigonometric series. Proc. Lond. Math. Soc. 14A, 38-46 (1965) 3. Özarslan, HS, Keten, A: A new application of almost increasing sequences. An. S¸ tiin¸t. Univ. ‘Al.I. Cuza’ Ia¸si, Mat. 61, 153-160 (2015) 4. Sulaiman, WT: Inclusion theorems for absolute matrix summability methods of an infinite series. IV. Indian J. Pure Appl. Math. 34, 1547-1557 (2003) 5. Tanovi˘c-Miller, N: On strong summability. Glas. Mat. Ser. III 14, 87-97 (1979) 6. Bor, H: On two summability methods. Math. Proc. Camb. Philos. Soc. 97, 147-149 (1985) 7. Bor, H: An application of almost increasing and δ -quasi-monotone sequences. JIPAM. J. Inequal. Pure Appl. Math. 1(2), Article ID 18 (2000) 8. Bor, H: Corrigendum on the paper ‘An application of almost increasing and δ -quasi-monotone sequences’ published in JIPAM, Vol.1, No.2. (2000), Article 18. JIPAM. J. Inequal. Pure Appl. Math. 3(1), Article ID 16 (2002)
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