# On [[absolute value of A].sub.k] summability factors of infinite series/ Lopmatute ridade [[absolute value of A].sub.k]-summeeruvusteguritest.

Let A be a lower triangular matrix, {[s.sub.n]} any sequence. Then

[A.sub.n] := [n.summation over (v=0)] [a.sub.nv] [s.sub.v]:

A series [SIGMA][a.sub.n], with partial sums [s.sub.n], is said to be summable [[absolute value of A].sub.k],k [greater than or equal to] 1 if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

We may associate with A two lower triangular matrices [bar.N] and [??] as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

[[??].sub.nv] = [[bar.a].sub.nv] - [[bar.a].sub.n-1,v], n = 1,2,3,...

In our previous work on absolute summability [1,2] we have assumed that the triangular matrix A had row sums one. This condition rules out the consideration of factorable matrices that are not weighted mean matrices. A lower triangular matrix A is said to be factorable if the nonzero terms [a.sub.nk] can be written as [a.sub.n][b.sub.k] for 0 [less than or equal to] k [less than or equal to] n. If A is a factorable matrix with row sums one, then it is a weighted mean matrix.

We shall first establish a general theorem for triangular matrices, which also applies to factorable matrices which need not be weighted mean matrices, and then we shall specialize this result to triangular matrices with row sums one.

A series [SIGMA][a.sub.n] with partial sums [s.sub.n] is said to be bounded [[absolute value of A].sub.k],k [greater than or equal to] 1, if [[SIGMA].sup.m.sub.v=1][a.sub.mv][[absolute value of [s.sub.v]].sup.k] = O (1) m [right arrow] [infinity].

Theorem 1. Let A be a lower triangular matrix satisfying

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

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

(iii) n[absolute value of [a.sub.nn]] = O(1),

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

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

If [SIGMA][a.sub.n] is bounded [[absolute value of A].sub.k] and {[[lambda].sub.n]} is a bounded nonzero sequence satisfying

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

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

then the series [SIGMA][a.sub.n] ln is summable [[absolute value of A].sub.k],k > 1.

Proof. Let ([y.sub.n]) be the nth term of the A-transform of [[SIGMA].sup.n.sub.i=0] [[lambda].sub.i][a.sub.i]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

[y.sub.n] =

and, for n > 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Using Abel's transformation, we have, for n > 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Since Y1 is bounded, in order to prove our theorem, it is sufficient, by Minkowski's inequality, to show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Using Holder's inequality and (i), (iii), and (ii),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Using the boundedness of [SIGMA][a.sub.n] and {[[lambda].sub.n]}, (iv), (viii), and (vii),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Using (viii), Holder's inequality, (v), (iii), and (vi),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

as in the proof of [I.sub.1].

Finally, using (iii),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

as in the proof of [I.sub.1]. [there does not exist]

Theorem 2. Let A be a lower triangular matrix with nonnegative entries satisfying

(ix) [[bar.a].sub.n0] = 1,n = 0,1,2,...,

(x) [a.sub.n]-[1.sub.v] [greater than or equal to] [a.sub.nv] for n [greater than or equal to] v + 1,

and conditions (iii)-(v) of Theorem 1.

If [SIGMA][a.sub.n] is bounded [[absolute value of A].sub.k] and {[[lambda].sub.n]} is a bounded nonzero sequence satisfying conditions (vii) and (viii) of

Theorem 1; then the series [SIGMA][a.sub.n] ln is summable [[absolute value of A].sub.k],k > 1.

Proof. Upon examining the conditions of Theorem 1 it is clear that one needs to show that conditions (ix) and (x) imply that [[??].sub.n,v+1] [greater than or equal to] 0 and that conditions (i), (ii), and (vi) of Theorem 1 hold.

Using the definitions of [[??].sub.nv] and [[bar.a].sub.nv], and (ix) and (x),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and condition (i) of Theorem 1 is true.

Also,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and condition (ii) of Theorem 1 is true.

Finally,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and condition (vi) of Theorem 1 is satisfied. [there does not exist]

If one is dealing with absolute summability of order 1, then conditions (iii) and (iv) of Theorem 1 are not needed.

Theorem 3. Let A be a lower triangular matrix satisfying conditions (ii), (iv), and (vi) of Theorem 1. If [SIGMA][a.sub.n] is bounded [absolute value of A] and {[[lambda].sub.n]} is a bounded nonzero sequence satisfying conditions (vii) and (viii) of Theorem 1 (with k = 1); then the series [SIGMA][a.sub.n] [[lambda].sub.n] is summable [absolute value of A].

Proof. This can be proved by using the techniques similar to that of Theorem 1. So we omit it.

Theorem 4. Let A be a lower triangular matrix with nonnegative entries satisfying conditions (ix) and (x) of Theorem 2 and condition (iv) of Theorem 1. If [SIGMA][a.sub.n] is bounded [absolute value of A] and {[[lambda].sub.n]} is a bounded nonzero sequence satisfying conditions (vii) and (viii) of Theorem 1; then the the series [SIGMA][a.sub.n] [[lambda].sub.n] is summable [absolute value of A].

Proof. As in the proof of Theorem 2, conditions (ix) and (x) of Theorem 2 imply conditions (i) and (ii) of

Theorem 1. [there does not exist]

A weighted mean matrix is a lower triangular matrix with entries [a.sub.nk] = [p.sub.k]/[P.sub.n]; where {[p.sub.k]} is a nonnegative sequence with [p.sub.0] > 0 and [P.sub.n] := [[SIGMA].sup.n.sub.k=0] [p.sub.k]. A weighted mean matrix is denoted by ([bar.N]; [p.sub.n]):

Corollary 1. Let {[p.sub.n]} be a positive sequence such that [P.sub.n] := [[SIGMA].sup.n.sub.k=0] [p.sub.k] [right arrow] [infinity], and satisfies

(xi) [np.sub.n] = O([P.sub.n]).

If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is bounded [absolute value of [bar.N],[p.sub.n]].sub.k] and {[[lambda].sub.n]} is a bounded nonzero sequence satisfying

(xii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]; and

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

then the series [SIGMA][a.sub.n] [[lambda].sub.n] is summable [absolute value of [bar.N],[p.sub.n]].sub.k],k [greater than or equal to] 1.

Proof. Conditions (i), (iv), and (v) of Theorem 1 are automatically satisfied for any weighted mean method. Conditions (iii), (vii), and (viii) of Theorem 1 become, respectively, conditions (xi), (xii), and (xiii) of Corollary 1. [there does not exist]

Corollary 2. If [SIGMA][a.sub.n] is bounded [absolute value of [bar.N],p] and {[[lambda].sub.n]} is a bounded nonzero sequence satisfying

(a) [m.summation over (n=1)] [p.sub.n]/[P.sub.n] [absolute value of [[lambda].sub.n]] = O(1); and

(b) [P.sub.n]/[p.sub.n] [absolute value of [DELTA][[lambda].sub.n] = O([absolute value of [[lambda].sub.n]]);

then [SIGMA][a.sub.n] [[lambda].sub.n] is summable [absolute value of [bar.N],p].

Proof. A weighted mean matrix automatically satisfies conditions (i)-(iii) of Theorem 1. Conditions (vii) and (viii) of Theorem 1 reduce to conditions (a) and (b) of Corollary 2, respectively.

Corollary 2 is a result of [3]. [there does not exist]

doi: 10.3176/proc.2010.3.02

ACKNOWLEDGEMENTS

The second author acknowledges support from the Scientific and Technical Research Council of Turkey. The authors thank the referees for their comments and suggestions.

Received 18 February 2009, revised 1 September 2009, accepted 8 September 2009

REFERENCES

[1.] Rhoades, B. E. Inclusion theorems for absolute matrix summability methods. J. Math. Anal. Appl., 1999, 238, 82-90.

[2.] Rhoades, B. E. and Savas, E. Some necessary conditions for absolute matrix summability factors. Indian J. Pure Appl. Math., 2002, 33(7), 1003-1009.

[3.] Singh, N. On [absolute value of [bar.N], [p.sub.n]] summability factors of infinite series. Indian J. Math., 1968, 10, 19-24.

B. E. Rhoades (a) and Ekrem Savas (b)*

(a) Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, U.S.A.; rhoades@indiana.edu

(b) Department of Mathematics, Istanbul Commerce University, Uskudar, Istanbul, Turkey

* Corresponding author, ekremsavas@yahoo.com