On \N, pn| summability of some numerical series

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ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXVIII (1989) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXVIII (1989)

Hü s e y in Bor (Kayseri)

On \N, pn| summability of some numerical series

Abstract. In this paper a theorem on |N , p„\ summability factors, which generalizes a theorem of Bhatt [1], has been proved.

1. Let be a given infinite series with the sequence of partial sums (s„). Let (p„) be a sequence of positive numbers such that

П

(1.1) Pn = X P, 00 as и->оо {P-k =

P-k

= 0, /с ^ 1).

v = 0

The sequence-to-sequence transformation 1 "

(1-2) tn = — Y,PvSv

* n v = 0

defines the sequence of the {N, p„) mean of the sequence (s„), generated by the sequence of coefficients (pn). The series is said to be summable \N, p„\ if (see [2])

OO

(i.3) X lfi.-fi.-il < °°-

n = 1

It is known that, in the special case when pn = 1/(и+1), the summability IN, pnI is equivalent to the summability jR, log n, 1|.

A sequence (A„) is said to be convex (see [4]) if A1 /„ ^ 0 for n — 1, 2, ...

2. Bhatt [1] proved the following theorem.

Th e o r e m

A. I f (Àn) is a convex sequence such that the series IS convergent and the sequence (s„) is hounded, then the series Х ап^п1°Яи is summable \R ,\ogn, 1[.

3. The purpose of this paper is to extend Theorem A for |N, pn\

summability in the form of the following theorem.

Th e o r e m.

I f (Xn) is non-negative, non-increasing, is convergent, and

(s„) is bounded, then the series ^ апРпЛ„ is summable |iV, pj.

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172 Hiiseyin Bor

It should be noted that the conditions on the sequence (A„) in our theorem, which is somewhat more general than Theorem A.

4. We need the following lemma for the proof of our theorem.

Le m m a.

I f the sequence

(A„)

satisfies the conditions of the theorem, then (4.1) Y P„AÀ„ = 0(1) as m -> oo.

n— 0

Proof. First we observe that if AA„ ^ 0 and < со, then

(4.2) P„An = 0 ( 1) as и -* со.

Indeed, since (A„) is non-increasing, we have that

Рт Ят = Am

Y

Pn = 0(1) X = 0 (1 ) as m ^ o o .

n = 0 n=0

m

Applying Abel transformation to the sum Y p„A„, we have n= 0

m m — 1 m

Y PnK — Y РпАК + Рщ^т = S P / i ^ „ - P mZUm + P„A /1=0

m

= Y Pm^m+ 1

n= 0 n= 0

n= 0 Hence

Y ? пл к = ЛпЛп+i- E P/A.

n= 0 /1=0

Since A„^An+1, we obtain

X P„zU„ ^ PmAm + X риЛ„ = 0(1)+ 0(1) = 0(1) as m-+ со

n = 0 n = 0

by virtue of the hypothesis and (4.2).

5. P r o o f o f th e th e o re m . Let (Tn) be the sequence of (N, pn) mean of the series J]a „P nA„. Then, by definition, we have

Tn = - w Y P v Y ar P r k = ~ X ( P n - P v - l ) a v Pv k -

•* n V = 0 r = 0 •*/! t>= 0

Tn-Tn- 1 =

^{Pn *}¹

P p ^

1 n 1 и- l v=Y P v P V- l a v k , 0 n ^ l .

Then

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Summability o f some numerical series

Using Abel transformation, we get

Tn — Tn- x —

^Pn ^£ ^ ( P v P v - l K ) s v + P „ S „ 2 . n

Pn Pn- 1 17= 0

P” I Ч

р

. ^ - Н

г

— "z V p . s A Pn Pn- 1 17= 0 PnPn-

^{1 D = 0}

Pn n - 1

P n P n- 1 1)=0

— T n, l + T „ ' 2 + T „ ,3 + T n A , s a y -

To prove the theorem it is sufficient to show that

£ Pf ) Pv + 1 ^17 + 1 ^17 T" P n

£ |T J < o o for r = 1,2, 3,4.

Firstly, using the fact that

Z - j r j — o W P J

71=1 7 + 1 - * 7 1 *71— 1

(see [3]), we have

in m n 71—1

Z K i l ^ Z -Б -Б Г - Z Л А 1 Р . A

71= 1 71= 1 * 7 1 * 71- 1 17= 0

771 — 1 771 7 7 1 - 1

= X P „ k |P „ A X _ Ï Î _ = 0(1) J P „ |s„ |P „ A

1 7 = 0 71= 17 + 1 * 7 1 * 7 1 — 1 V — i )

Since s„ — 0(1), by hypothesis, we have that

771 771 — 1

Z |Г„д| = 0(1) £ PvAX„ = 0 (l) as m со,

71= 1 17= 0

by the lemma.

Again

771 771 _ 71— 1

Z K

2

K z H r — Z Л , к

1

р А

71= 1 71= 1 * 7 1 * 71- 1 17= 0

771— 1 771 771— 1

= £ P .k J p A I — = 0(1) £ p A K I

1 7 = 0 71=1 7 + 1 * 7 1* 71— 1 1 7 = 0

771- 1

= 0(1) £ pvkv = 0(1) as m-+co,

17= 0

by virtue of the hypothesis.

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174 Hliseyin Bor

Similarly we have that

m m — 1

Z \TnJ = 0(1) X Pv+

1 ^ + 1

= 0 (1 ) as m->oo.

n= 1 i>= 0

Finally, we have that

m m m

X l7^ = Z P»AIS».I = 0(1) Z Pnh = 0(1) as m -> x ,

n= 1 n= 1 n= 0

by virtue of the hypothesis.

Therefore, we get

m

Z |T J = 0(1) as m -> x for r = 1, 2, 3, 4.

n= 1

This completes the proof of the theorem. In particular, if we take

p„

= 1 for all values of n in our theorem, then we get the following corollary.

Co r o l l a r y.

I f the sequence

(/„ )

is non-negative, non-increasing, Z

К

convergent and (s„) = 0(1), then the series Z nanA„ is summahle |C, 1|.

References

[1] S. N. B h a tt, An aspect o f local property o f\R , lo g n, 1| summability of the Fourier Series, Proc. Nat. Inst. Sci. Ind. 26 (1960), 69-73.

[2] E. C. D a n ie l, On absolute summability factors o f infinite series, Japan Acad. Journal 40 (1964), 65-69.

[3] N. S in g h , On |N , p„\ summability factors o f infinite series, Indian J. Math. 10 (1968), 19-24.

[4] A. Z y g m u n d , Trigonometric series, Cambridge 1979.

DEPARTMENT OF MATHEMATICS, ERCIYES UNIVERSITY, KAYSERI, TURKEY