Internat. J. Math. & Math. Sci.
VOL. 13 NO. 3 (1990) 501-506
501
CONVEXITY PRESERVING SUMMABILITY MATRICES
C.R. SELVARAJ
Pennsylvania State University
Shenango Valley Campus
147 Shenango Avenue
Sharon, PA 16146
(Received February 28, 1989 and in revised form May 27, 1989)
ABSTRACT.
The maln result of thls paper gives the necessary and sufficient conditions
for the Abel matrices to preserve the convexity of sequences.
Also, the higher orders
of the Cesro method are shown to be convexlty-preservlng matrices.
KEY WORDS AND PHRASES.
Convex sequences, convexlty-preservlng matrices, domain of a
matrix.
1980 AMS SUBJECT CLASSIFICATION CODE.
1.
40
INTRODUCTION.
A real sequence
(Xn)n= 0
is called convex if Its second order differences
A2xn Xn+2 2Xn+l + Xn
(1.1)
> 0
0,1,2,
for n
G.H.
Toader
[I]
proved
following two theorems characterizing the convex
the
sequences.
THEOREM I.I.
If the sequence (xk) is given by
k
x
k
then
A2Xk
(k- i + I)
(1.2)
bi,
bk+ 2.
THEOREM 1.2.
bk’S
[
i=O
k)
The sequence (x
is convex if and only if b
k
)
0 for k
)
2 where
are given by (1.2).
The aim of the next section is to prove the necessary and sufficient conditions
for the Abel matrix to preserve the convexity of sequences, using the above theorems.
502
2.
C.R. SELVARAJ
CONVEXITY PRESERVATION OF ABEL MATRIX.
For each sequence (t n) satisfying 0
(Frldy [2, p. 421]) is defined by
<
t
n
<
and llm
n
0
t
the Abel matrix A t
At In,k] tn(1 tn)k.
In [2], Frldy gives the characterization of the domain of the matrix A
A sequence x
t
It follows that all convex
I.
is in the domain of At if and only if
ll/k’llmlxk
k
sequences need not be in the domain of At
An obvious example is the convex sequence
k
k for k
given by x
we restrict ourselves to the convex
1,2,3,
So,
k
sequences that are in the domaln of the matrix.
x
THEOREM 2.1. The Abel matrix A t preserves the convexity of the sequences that are
in the domain of A if and only if for all n
0,1,2,...,
t
(a)
0 and
A2yn(1)
A2yn(k)
(b)
0, for k
2,3,...,
0,I,2,3,..., the sequence {Yn(k)}nO is defined by
where for each k
(1-t)k
n
Yn (k)
PROOF.
t
n
It is useful to note that
A2yn
A2yn (0).
(2.1)
First we prove a general result for a convex sequence in terms of the sequences
{Yn(k)}n.0.
If x is a convex sequence in the domain of
.. .
At,
then the nth term of the
transformed sequence is given by
(AtX)n
tn
tn
k=0
b
i=0
i
tn,k
(k- i + l) b I, using (1.2)
(22)
(I
k=i
t
)k(k
n
i
+ I).
The change in the order of summation is justified as follows:
k
k-0
n
i-0
(Atx) n
+
(Ibol
)k
b
O)
tn)k + (Ibll
(k + I)(I
k =0
(1-t)
(AtX)n
i 0
bl
k(1
tn )k,
(2.3)
for all n, the rlght-hand side
(0,I)
n
Therefore from (2.2),
and since x is in the domain of A and
t
of the equation (2.3) converges.
.
bl) k =0
n
t
n
-t
CONVEXITY PRESERVING SUMMABILITY MATRICES
503
0,1,2,...,
It follows that for n
A2(AtX)n i=OE b
(I
i[ -tn+2tn+2
)i
2(I
t-n+Itn+l)i -ttn)i ]
(I
+
.
Using the equation (2.|), we can write the above equation as follows:
A
2(Atx)
(b
n
0
+ b I)A2 (1) +
Yn
kA2yn(k).
(2.4)
b
kffi2
Now, assume that conditions (a) and (b) given in the theorem are true.
equation (2.4) together with Theorem 1.2 implies that
A2(AtX)n
Then the
0.
)
Hence, the sequence At x is convex.
Conversely, assume that the matrix At preserves convexity of the sequences in its
Then there exists a nonSuppose that the condition (a) fails to hold.
L
#
If
0.
O, consider the convex
L
negative integer N such that
do,in.
A2yN(1)
k)
that any sequence (x
>
It is easy to verify
in the domain of A t
may be written as in equation (1.2) by taking
{-I, -2, -3,...}, which
sequence u
is
k-I
b
x
0
and
0
b
(k
E
x
k
k
i
+
if0
l)bl,
for k
)
(2.5)
I.
Thus for the above sequence u,
b
and b
k
0
0,
for k
)
I.
Substituting these values of b’s into the equation (2.4) we get
A2yN(I)
A2(AtU)N
-L
<
O,
Atu
which contradicts that the transformed sequence
is convex.
<
0, we repeat a similar argument considering the convex sequence
Hence, if the matrix At preserves
to get a contradiction.
is true.
(a)
convexity of sequences, then the condition
If L
v
the
{1,2,3,...}
Next, suppose that the condition (a)
non-negative integer
J
is true, but not
such that the sequence
{yn(J)}n= 0
(b).
Then there exists a
is not convex.
That is, for
some N,
A2yN(J)
<
(2.6)
0.
Now, construct a convex sequence x as follows:
-(I
xj
-(I
tN+2) j
xj_
tN+2
tN+ 1)j
tN+l
tS)J
-(I
xj_ 2
tN
and choose the other terms of the sequence (x k) in such a way that
all k #
J
2.
By construction,
A2xj_2
bj >
0,
by (2.6).
A2xk
bk+2
0 for
C. R. SELVARAJ
504
bk’S
Using these values of
N,
together with our assumption (a) in equation (2.4) for n
we get
A2(Atx)N bjA2yN(j)
which again contradicts that
Atx
<
0,
is a convex sequence.
This completes the proof.
We give below an example of an Abel matrix A t which preserves the convexity of
sequences.
The Abel matrix A t where the sequence (t
EXAMPLE 2.1.
is given by
+ 2’ for n=O,l,2,...
’
n
n)
We shall show that this matrix A t satisfies the
conditions (a) and (b) asserted in Theorem 2.1. Since
preserves the convexity of sequences.
t
Yn
(1)
t
a2yn (1)
n + I,
n
O.
In order to see that the condition (b) is satisfied, for each k
A2yn(k)
k
(n + 3)
k-1
(n + 4)
2,3,...,we have
k
(n + 1)
+
k-1
k-1
(n + 2)
(n + 3)
2(n + 2)
k
It suffices to prove that f(n + I) > f(n) where
k
(n + 2)
k-I
(n + 3)
f(n)
k
(n + I)
k-I
(n + 2)
By differentiation it can be easily proved that f(n) is an increasing function.
3.
PRESERVING CONVEXITY BY CESARO.
In this section we use Theorems I.I and 1.2 to prove that the higher orders of
Cesro means preserve
introduce
In order to prove this result, we
Let K represent the set of all real
the convexity of sequences.
some notations and terminologies.
convex sequences.
DEFINITION 3.1.
For a summabillty matrix A we define the set K(A) as the set of
all sequences such that the transformed sequences are convex, i.e.,
K(A)
DEFINITION 3.2.
K(A)
K}.
The matrix A is K-included by the matrix B provided that
K(B).
DEFINITION 3.3.
K(A)
{x: Ax
c
The matrix B is K-stronger than the matrix A provided that
K(B).
Before we prove the main result we state the following preliminary theorems which
are easy to prove.
CONVEXITY PRESERVING SUMMABILITY MATRICES
505
Let A be a row-finite matrLx and T be a trLangle (lower triangular
-I preserves the convexLty
matrix with non-zero dagonal entres). Then the matrix AT
THEOREM 3.1.
of sequences if and only if T is K-included by A.
THEOREM 3.2. Let A be a row-finlte matrix and T be a triangle. Then the matrix
AT -! is K-stronger than the identity matrix if and only if A is K-stronger than T.
For j e]q, the jth order of the Cesro matrix is given by Hardy [3, pp. 96-98] as
(n-k+j+l
if k
n+j
j
Cj [n,k]
if k
0,
>
n.
In [I], G.H. Toader proved that the Cesro matrix C
is K-stronger than the identity
matrix.
THEOREM
For
3.3.
each
e
IN, the
matrix
-I
Cj+ICj
preserves
the
convexity
sequences.
PROOF.
If
Cj+I
C
is represented by
it is easy to see that
Aj
(j+k)
---------,
if k
O,
if k>n.
n+j+!
j+1
Aj [n,kl
Let x be a convex sequence.
n,
Then
n
E
(Ajx) n
Aj[n,k]xk.
k=0
A simple calculation shows that
(J
+ n
j
)Xn (n
Writing
+ j +
J +
n +
(hX)n- (J + ) (Aj X)n-l"
(3.1)
n
(Ajx) n
[
i--O
(3.2)
(n- i + l)e i
we can see that the equation (3.1) can be simplified to
n
+
Xn =J
Therefore for each n
82 xn
In
+ (J + l)(n- i + I)]
i=O
e
i.
0,I,2,...,
[(n-3+J )en+2
nen+
which implies that
(n- 3 +
J)en+ 2
nen+
+ (J +
l)A2Xn"
(3.3)
of
506
C. R. SELVARAJ
Since x is a convex sequence, it follows from Theorem 1.2 that
A2xn bn+2
0 for n
0, I,
Using this fact in the equation (3.3), we obtain
en+ 2
0 for n
(3.4)
0,I,...
Hence, the equation (3.2) together with (3.4) implies that [cf. Theorem 1.2] the
transformed sequence
REMARK 3.1.
Combining the theorems 3.1 and 3.3 we see that
K(CI)
Since the
This completes the proof.
is convex.
Ajx
e:
c_
K(C2)
c_
K(Cj) K(Cj+ I)
preserves the convexity of sequences [I], it follows that
Cesro matrix C
for each j elNthe jth order of the Cesro matrix
Cj
{0,I,0,0,...,} which
considering the sequence x
preserves the convexity.
Also, by
is not convex, it is easy to prove
that
A2(AjX)n > 0.
Cj+iC
Thus the matrix
is K-stronger than the identity matrix.
Therefore
K(C
K(C I)
21
K(Cj)
K(Cj+ I)
Now we state this result in the following theorem.
THEOREM 3.4.
the
sequences.
For j ]N, the Jth order Cesro matrix Cj preserves the convexity of
Also, Cj+ is K-stronger than Cj, which is K-stronger than the
identity matrix.
REMARK 3.2.
The H6"ider’s summabillty matrix of order
Hj
(Cl)J
Therefore the matrix
Hi+
is
Rj
for
J
J
is given by
1,2
preserves the convexity of the sequences and also the matrix
K-stronger than the matrix
REFERENCES
I.
TOADER, G.H., A Hierarchy of Convexity for Sequences, L’anal. Num. et la Th. de
l’approx. 12 (1983), 187-192.
2
A
Abel Transformations
FRIDY, J ..,
3.
MARDY, G.H., Divergent Series, Oxford University Press, London, 1949.
into
I
Canad. Math. Bull. 25 (1982), 421-427.