NEW CHEBYSHEV TYPE INEQUALITIES FOR SEQUENCES OF REAL NUMBERS

. Some new inequalities of Chebyshev type for sequences of real numbers are pointed out. MSC 2000. Primary 26D15; Secondary 26D10.

The inequality (2) was mentioned by Hardy, Littlewood and Polya in their book [3] in 1934 in the more general setting of synchronous sequences, i.e., if a, b are synchronous (asynchronous), this means that holds true.A relaxation of the synchronicity condition was provided by M. Biernacki in 1951, [4] who showed that for a nonnegative p, if a, b are monotonic in mean in the same sense, i.e. for P k := k i=1 p i , Victoria University, Melbourne, Australia, e-mail: sever@matilda.vu.edu.au,url: http://rgmia.vu.edu.au/SSDragomirWeb.html.then (2) holds true for the " ≥ " sign.If they are monotonic in mean in the opposite sense, then (2) holds true for the " ≤ " sign.
For general real weights p, Mitrinović and Pečarić has shown in [9] that the inequality (2) holds true if (5) 0 and a, b are monotonic in the same (opposite) sense.
The following identity is well known in the literature as Sonin's identity (see [5] and [6, p. 246 for any real number b.
Another well known identity in terms of double sums is the following one known in the literature as Korkine's identity (see [7] and [6, p. 242 The purpose of this work is to point out other identities of interest in obtaining Chebyshev's type inequalities.Some natural applications for studying the positivity of the Chebyshev's functional are mentioned.

THE IDENTITIES
The first result is embodied in the following

tuples of real numbers. If we define
then we have the identity where Proof.We use the following well known summation by parts formula (9) q−1 l=p where d l , v l are real numbers, l = p, ..., q (q > p; p, q are natural numbers) .
If we choose in ( 9), p = 1, q = n, which produce the first identity in (8) .
The second and the third identity are obvious and we omit the details.
Before we prove the second result, we need the following lemma providing an identity that is interesting in itself as well.
Lemma 2. Let p = (p 1 , ..., p n ) and a = (a 1 , ..., a n ) be n-tuples of real numbers.Then we have the equality (10) det We have Using the summation by parts formula, we have Using ( 12) and (13) we have and the identity is proved.
We are able now to state and prove the second identity for the Chebyshev functional The proof is obvious by Theorem 1 and Lemma 2.
Remark 1.The identity (14) was stated without a proof in paper [9].It also may be found in [6, p. 281], again without a proof.

SOME NEW INEQUALITIES
We may point out the following result concerning the positivity of the Chebyshev functional  If b is monotonic nonincreasing and either (i) or (ii) or (iii) from above holds, then the reverse inequality in (15) holds true.
The proof of the theorem follows from the identities incorporated in Theorem 1 and we omit the details.
Using the second theorem, we may state the following result as well

p i a i b i − n i=1 p i a i • n i=1 p i b i ,
1. INTRODUCTION For p = (p 1 , ..., p n ) , a = (a 1 , ..., a n ) and b = (b 1 , ..., b n ) n-tuples of real numbers, consider the Chebyshev functional (1) T n (p; a, b) := P n n i=1

Theorem 3 .P
With the assumptions of Theorem 1, we have the equality (14) T n (p; a, b) = min{i,j} Pmax{i,j} • ∆a j • ∆b j .