CONVEXITY PRESERVING SUMMABILITY MATRICES

The main result of this paper gives the necessary and sufficient conditions for the Abel matrices to preserve the convexity of sequences. Also, the higher orders of the Cesaro method are shown to be convexity-preserving matrices.

A real sequence (Xn)n= 0 is called convex if Its second order differences A2x n Xn+2 2Xn+l + Xn > 0  The sequence (x k) is convex if and only if b k ) 0 for k ) 2 where bk'S 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.

CONVEXITY PRESERVATION OF ABEL MATRIX.
For each sequence (t n) satisfying 0 < t n < and llm t 0 the Abel matrix A t (Frldy [2, p. 421]) is defined by n At In,k] tn(1 tn)k.
In [2], Frldy gives the characterization of the domain of the matrix A t A sequence x A t if and only if llmlxk ll/k'-I.It follows that all convex is in the domain of k sequences need not be in the domain of A t An obvious example is the convex sequence x given by x k k k for k 1,2,3, So, we restrict ourselves to the convex sequences that are in the domaln of the matrix.THEOREM  A2yn(k) 0, for k 2,3,..., where for each k 0,I,2,3,..., the sequence {Yn(k)}nO is defined by (

.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 The change in the order of summation is justified as follows: (2.3) and since x is in the domain of A t and -t (0,I) for all n, the rlght-hand side n of the equation (2.3)Using the equation (2.|), we can write the above equation as follows: A 2(Atx) n (b 0 + b I)A 2 Yn (1) + .b kA2yn(k).kffi2 (2.4) Now, assume that conditions (a) and (b) given in the theorem are true.
Hence, the sequence A x is convex.
t Conversely, assume that the matrix A t preserves convexity of the sequences in its do,in.
Suppose that the condition (a) fails to hold.
Then there exists a non- negative integer N such that A2yN(1) L # 0. If L > O, consider the convex sequence u {-I, -2, -3,...}, which is in the domain of A t It is easy to verify that any sequence (x k) may be written as in equation (1.2) by taking  Substituting these values of b's into the equation (2.4) we get A2(AtU)N A2yN(I) -L < O, which contradicts that the transformed sequence Atu is convex.
Hence, if the matrix A t preserves the convexity of sequences, then the condition (a) is true.
Next, suppose that the condition (a) is true, but not (b).Then there exists a non-negative integer J such that the sequence {yn(J)}n= 0 is not convex.That is, for some N, A2yN(J) < 0.
(2.6) Now, construct a convex sequence x as follows: tN tN+2 tN+l and choose the other terms of the sequence (x k) in such a way that A2xk bk+2 0 for all k # J 2. By construction, A2xj_2 bj > 0, by (2.6).
Using these values of bk'S together with our assumption (a) in equation (2.4) for n N, we get A2(Atx)N bjA2yN(j) < 0, which again contradicts that Atx 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.
EXAMPLE 2.1.The Abel matrix A t where the sequence (t n) is given by n ' + 2' for n=O,l,2,... preserves the convexity of sequences.We shall show that this matrix A t satisfies the In order to see that the condition (b) is satisfied, for each k 2,3,...,we have A2yn(k It suffices to prove that f(n + I) > f(n) where By differentiation it can be easily proved that f(n) is an increasing function.
In this section we use Theorems I.I and 1.2 to prove that the higher orders of Cesro means preserve the convexity of sequences.In order to prove this result, we introduce some notations and terminologies.
Let K represent the set of all real 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) {x: Ax K}.
DEFINITION 3.2.The matrix A is K-included by the matrix B provided that K(A) K(B).
The matrix B is K-stronger than the matrix A provided that K(A) c K(B).
Before we prove the main result we state the following preliminary theorems which are easy to prove.
. Toader [I] proved the following two theorems characterizing the convex sequences.THEOREM I.I.If the sequence (x k) is given by )bl, for k ) I.
conditions (a) and (b) asserted in Theorem 2.1.
2.1.The Abel matrix A t preserves the convexity of the sequences that are in the domain of A t if and only if for all n converges.Therefore from (2.2),