A new note on the increasing sequences.

1. Introduction

Let [SIGMA][a.sub.n] be a given infinite series with partial sums ([s.sub.n]). We denote by [u.sub.n.sup.[alpha]] and [t.sub.n.sup.[alpha]] the n-th Cesaro means of order [alpha], with [alpha] > -1, of the sequence ([s.sub.n]) and ([na.sub.n]), respectively, i.e.,

[u.sub.n.sup.[alpha]] = [1/[A.sub.n.sup.[alpha]]][n.summation over (v=0)][A.sub.[n-v].sup.[[alpha]-1]][s.sub.v], (1)

[t.sub.n.sup.[alpha]] = [1/[A.sub.n.sup.[alpha]]][n.summation over (v=1)][A.sub.[n-v].sup.[[alpha]-1]]v[a.sub.v], (2)

where

[A.sub.n.sup.[alpha]] = O([n.sup.[alpha]]), [alpha] > -1, [A.sub.0.sup.[alpha]] = 1 and [A.sub.-n.sup.[alpha]] = 0 for n > 0. (3)

A series [SIGMA][a.sub.n] is said to be summable [[absolute value of C, [alpha]].sub.k], k [greater than or equal to] 1, if (see [7], [10])

[[infinity].summation over (n=1)][n.sup.[k-1]][[absolute value of [u.sub.n.sup.[alpha]] - [u.sub.[n-1].sup.[alpha]]].sup.k] = [[infinity].summation over (n=1)][[[absolute value of [t.sub.n.sup.[alpha]]].sup.k]/n] < [infinity]. (4)

and it is said to be summable [[absolute value of C, [alpha]; [delta]].sub.k], k [greater than or equal to] 1 and [delta] [greater than or equal to] 0, if (see [8])

[[infinity].summation over (n=1)][n.sup.[[delta]k-1]][[absolute value of [t.sub.n.sup.[alpha]]].sup.k] < [infinity]. (5)

Let ([p.sub.n]) be a sequence of positive numbers such that

[P.sub.n] = [n.summation over (v=0)][p.sub.v][right arrow] [infinity] as n [right arrow] [infinity], ([P.sub.-i] = [p.sub.-i] = 0, i [greater than or equal to] 1). (6)

The sequence-to-sequence transformation

[[sigma].sub.n] = [1/[P.sub.n]][n.summation over (v=0)][p.sub.v][s.sub.v] (7)

defines the sequence ([[sigma].sub.n]) of the Riesz mean or simply the ([bar.N], [p.sub.n]) mean of the sequence ([s.sub.n]), generated by the sequence of coefficients ([p.sub.n]) (see [9]). The series [SIGMA][a.sub.n] is said to be summable [[absolute value of [bar.N], [p.sub.n]].sub.k], k [greater than or equal to] 1, if (see [2], [3])

[[infinity].summation over (n=1)][([P.sub.n]/[p.sub.n]).sup.[k-1]][[absolute value of [DELTA][[sigma].sub.[n-1]]].sup.k] < [infinity], (8)

and it is said to be summable [[absolute value of [bar.N], [p.sub.n]; [delta]].sub.k], k [greater than or equal to] 1 and [delta] [greater than or equal to] 0, if (see [5])

[[infinity].summation over (n=1)][([P.sub.n]/[p.sub.n]).sup.[[delta]k+k-1]][[absolute value of[DELTA][[sigma].sub.[n-1]]].sup.k] < [infinity], (9)

where

[DELTA][[sigma].sub.[n-1]] = -[[p.sub.n]/[[P.sub.n][P.sub.[n-1]]]][n.summation over (v=1)][P.sub.[v-1]][a.sub.v], n [greater than or equal to] 1. (10)

In the special case [p.sub.n] = 1 for all values of n (resp. [delta] = 0) [[absolute value of [bar.N], [p.sub.n]; [delta]].sub.k] summability is the same as [[absolute value of C, 1; [delta]].sub.k] (resp. [[absolute value of [bar.N], [p.sub.n]].sub.k]) summability. Also if we take [delta] = 0 and k = 1, then we get [absolute value of [bar.N], [p.sub.n]] summability.

2. Known Results

Bor (4) has proved the following theorem for [[absolute value of [bar.N], [p.sub.n]].sub.k] summability factors.

Theorem A. Let ([X.sub.n]) be a positive non-decreasing sequence and suppose that there exists sequences ([[beta].sub.n]) and ([[lambda].sub.n]) such that

[absolute value of [DELTA][[lambda].sub.n]][less than or equal to][[beta].sub.n], (11)

[[beta].sub.n] [right arrow] 0 as n [right arrow] [infinity], (12)

[[infinity].summation over (n=1)]n[absolute value of [DELTA][[beta].sub.n]][X.sub.n] < [infinity], (13)

[absolute value of [[lambda].sub.n]][X.sub.n] = O(1). (14)

If

[n.summation over (v=1)][[[absolute value of [s.sub.v]].sup.k]/v] = O([X.sub.n]) as n [right arrow] [infinity], (15)

and ([p.sub.n]) is a sequence such that

[P.sub.n] = O(n[p.sub.n]), (16)

[P.sub.n][DELTA][p.sub.n] = O([p.sub.n][p.sub.[n+1]]), (17)

then the series [[summation].sub.[n=1].sup.[infinity]][a.sub.n][[[P.sub.n][[lambda].sub.n]]/[np.sub.n]] is summable [[absolute value of [bar.N], [p.sub.n]].sub.k], k [greater than or equal to] 1.

If we take k = 1 in Theorem A, then we get a theorem due to Mishra and Srivastava (12) concerning the [absolute value of [bar.N], [p.sub.n]] summability factors.

Recently Bor (6) generalized Theorem A for [[absolute value of [bar.N], [p.sub.n]; [delta]].sub.k] summability in the following form.

Theorem B. Let ([X.sub.n]) be a positive non-decreasing sequence and the sequences ([[beta].sub.n]) and ([[lambda].sub.n]) are such that conditions (11)-(17) of Theorem A are satisfied with the condition (15) replaced by:

[n.summation over (v=1)][([P.sub.v]/[p.sub.v]).sup.[delta]k][[absolute value of [S.sub.v]].sup.k]/v] = O([X.sub.n]) as n [right arrow] [infinity], (18)

and

[[m+1].summation over (n=v+1)][([P.sub.n]/[p.sub.n]).sup.[[delta]k-1]][1/[P.sub.[n-1]]] = O([([P.sub.n]/[p.sub.n]).sup.[delta]k][1/[P.sub.v]]) as m [right arrow] [infinity], (19)

then the series [[summation].sub.[n=1].sup.[infinity]][a.sub.n][[[P.sub.n][[lambda].sub.n]]/[np.sub.n]] is summable [[absolute value of [bar.N], [p.sub.n]; [delta]].sub.k], k [greater than or equal to] 1 and 0 [less than or equal to] [delta] < 1/k.

Remark. It should be noted that if we take [delta] = 0, then we get Theorem A. In this case condition (18) reduces to condition (15) and condition (19) reduces to

[[m+1].summation over (n=v+1)][[p.sub.n]/[[P.sub.n][P.sub.[n-1]]]] = [[m+1].summation over (n=v+1)]([1/[P.sub.[n-1]]] - [1/[P.sub.n]]) = O(1/[P.sub.v]) as m [right arrow] [infinity], (20)

which always holds. Also it may be noticed that, under the conditions on the sequence ([[lambda].sub.n]) we have that ([[lambda].sub.n]) is bounded and [DELTA][[lambda].sub.n] = O(1/n) (see [4]).

3. Main Result

The aim of this paper is to prove Theorem B under weaker conditions. For this we need the concept of almost increasing sequence. A positive sequence ([b.sub.n]) is said to be almost increasing if there exists a positive increasing sequence ([c.sub.n]) and two positive constants A and B such that A[c.sub.n] [less than or equal to] [b.sub.n] [less than or equal to] B[c.sub.n] (see [1]). Obviously every increasing sequence is almost increasing. However, the converse need not be true as can be seen by taking the example, say [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Now, we shall prove the following theorem.

Theorem. Let ([X.sub.n]) be an almost increasing sequence. If the conditions (11)-(14) and (16)-(19) are satisfied, then the series [[summation].sub.[n=1].sup.[infinity]][a.sub.n][[[P.sub.n][[lambda].sub.n]]/[np.sub.n]] is summable [[absolute value of [bar.N], [p.sub.n]; [delta]].sub.k], k [greater than or equal to] 1 and 0 [less than or equal to] [delta] < 1/k.

We require the following lemma for the proof of the theorem.

Lemma 1 ([11]). If ([X.sub.n]) be an almost increasing sequence, then under the conditions (12)-(13) we have that

n[X.sub.n][[beta].sub.n] = O(1), (21)

[[infinity].summation over (n=1)][[beta].sub.n][X.sub.n] < [infinity]. (22)

Lemma 2 ([12]). If the conditions (16) and (17) are satisfied, then

[DELTA]([P.sub.n]/[np.sub.n]) = O(1/n) (23)

4. Proof of the Theorem

Let ([T.sub.n]) be the sequence of ([bar.N], [p.sub.n]) mean of the series [[summation].sub.[n=1].sup.[infinity]][[[a.sub.n][P.sub.n][[lambda].sub.n]]/[np.sub.n]]. Then, by definition, we have

[T.sub.n] = [1/[P.sub.n]][n.summation over (v=1)][p.sub.v][v.summation over (r=1)][[[a.sub.r][P.sub.r][[lambda].sub.r]]/[rp.sub.r]] = [1/[P.sub.n]][n.summation over (v=1)]([P.sub.n] - [P.sub.[v-1]])[[[a.sub.v][P.sub.v][[lambda].sub.v]]/[vp.sub.v]].

Then

[T.sub.n] - [T.sub.[n-1]] = [[p.sub.n]/[[P.sub.n][P.sub.[n-1]]]][n.summation over (v=1)][[[P.sub.[v-1]][P.sub.v][a.sub.v][[lambda].sub.v]]/[vp.sub.v]], n [greater than or equal to] 1, ([P.sub.-1] = 0).

Using Abel's transformation, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since

[[absolute value of [T.sub.[n,1]] + [T.sub.[n,2]] + [T.sub.[n,3]] + [T.sub.[n,4]]].sup.k] [less than or equal to] [4.sup.k]([[absolute value of [T.sub.[n,1]]].sup.k] + [[absolute value of [T.sub.[n,2]]].sup.k] + [[absolute value of [T.sub.[n,3]]].sup.k] + [[absolute value of [T.sub.[n,4]]].sup.k])

to complete the proof of the Theorem, it is sufficient to show that

[[infinity].summation over (n=1)][([P.sub.n]/[p.sub.n]).sup.[[delta]k+k-1]][[absolute value of [T.sub.[n,r]]].sup.k] < [infinity], for r = 1,2,3,4. (24)

Firstly, by Abel transformation, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by the hypotheses of the Theorem and Lemma 1.

Now, using the fact that [P.sub.[v+1]] = O((v+1)[p.sub.[v+1]]) by (16), and applying Holder's

inequality we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

as m [right arrow] [infinity], by the hypotheses of the Theorem and Lemma 1. Again, as in [T.sub.[n,1]], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

by the hypotheses of the Theorem, Lemma 1 and Lemma 2. Finally, using Holder's inequality, as in [T.sub.[n,3]], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore we get that

[m.summation over (n=1)][([P.sub.n]/[p.sub.n]).sup.[[delta]k+k-1]][[absolute value of [T.sub.[n,r]]].sup.k] = O(1), for r = 1, 2, 3, 4.

This completes the proof of the Theorem.

Finally if we take [p.sub.n] = 1 for all values of n in the Theorem, then we get a new result concerning the [[absolute value of C, 1; [delta]].sub.k] summability factors.

References

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(7) T. M. Flett, On an extension of absolute summability and some theorems of Littlewood and Paley, Proc. London Math. Soc., 7(1957), 113-141.

(8) T. M. Flett, Some more theorems concerning the absolute summability of Fourier series and power series, Proc. London Math. Soc., 8(1958), 357-387.

(9) G. H. Hardy, Divergent Series, Oxford Univ. Press., Oxford, (1949).

(10) E. Kogbetliantz, Sur les series absolument sommables par la methode des moyennes arithmetiques, Bull. Sci. Math., 49(1925), 234-256.

(11) S. M.Mazhar, A note on absolute summability factors, Bull. Inst. Math. Acad. Sinica, 25(1997), 233-242.

(12) K. N. Mishra and R. S. L. Srivastava, On [absolute value of [bar.N], [p.sub.n]] summability factors of in_nite series, Indian J. Pure Appl. Math., 15(1984), 651-656.

Huseyin Bor [dagger]

Department of Mathematics, Erciyes University, 38039 Kayseri, Turkey

Received February 19, 2008, Accepted January 13, 2009.

* 2000 AMS Subject Classification. 40D15, 40F05, 40G99.

[dagger] E-mail: [e-mail], [e-mail]