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Log convexity and concavity of some double sequences.

[section] 1. Introduction

In literature, the well known means respectively called Arithmetic mean, Geometric mean and Harmonic mean are as follows;

For a,b > 0, then

A(a, b) = a + b/2, G(a, b) = [square root of ab] and H(a, b) = 2ab/a + b].

Several researchers introduced and studied some interesting results on double sequences in the form of above said means. Also, proved the convergence properties and obtained there limit values. As an application to estimate the best accurate value of [pi], the authors considered the following double sequences [4-8];

[a.sub.n+1] = H([a.sub.n], [b.sub.n]) and [b.sub.n+1] = G([a.sub.n+1], [b.sub.n]), (1)

where H and G stands for harmonic mean and geometric mean respectively.

In finding the roots of an equation one of the famous iteration method is called the Heron's method of extracting of square root is achieved by the following double sequences;

[a.sub.n+1] = H([a.sub.n], [b.sub.n]) and [b.sub.n+1] = A([a.sub.n], [b.sub.n]), (2)

where H and A stands for Harmonic mean and Arithmetic mean respectively.

Also, several researchers introduced and studied the double sequences for other applications as follows;

[a.sub.n+1] = A([a.sub.n], [b.sub.n]) and [b.sub.n+1] = A([a.sub.n], [b.sub.n]), (3)

where A and G stands for Arithmetic mean and Geometric mean respectively.

In [1, 3], contains many results on convergence and monotonicity. The double sequences were generalized as Archimedean double sequences and Gauss double sequences.

The sequence [c.sub.n] is said to be log-convex, if [c.sup.2.sub.n] < [c.sub.n+1][c.sub.n-1] and the sequence [c.sub.n] is said to be log-concave, if [c.sup.2.sub.n] [less than or equal to] [c.sub.n+1][c.sub.n-1].

In this paper, the logarithmic convexity and logarithmic concavity of the double sequences are presented (1)-(3).

[section] 2. Results

In this section, some results on log convexity and concavity of double sequences are proved.

Theorem 2.1. For n [greater than or equal to] 0, [a.sub.0] < [b.sub.0], then the sequences [a.sub.n+1] = H([a.sub.n], [b.sub.n]) and [b.sub.n+1] = G([a.sub.n+1], [b.sub.n]) are respectively Log-concave and Log-convex.

Proof. From definitions of harmonic mean and geometric mean, consider

[a.sub.n+1] = H([a.sub.n], [b.sub.n]) = 2[a.sub.n][b.sub.n]/[a.sub.n] + [b.sub.n] and [b.sub.n+1] = G([a.sub.n+1], [b.sub.n]) = [square root of [a.sub.n+1][b.sub.n]]. (4)

It is proved that for [a.sub.0] < [b.sub.0],

[a.sub.0] < [a.sub.1] < [a.sub.2] < ... < [a.sub.n] < [a.sub.n+1] < ... < [b.sub.n+i] < [b.sub.n] < ... < [b.sub.2] < [b.sub.i1] < [b.sub.0] (5)

from (4), [b.sup.2.sub.n] = [a.sub.n][b.sub.n-1] and [a.sub.n] < [b.sub.n+1] from (5), implies that [b.sup.2.sub.n] = [a.sub.n][b.sub.n-1] < [b.sub.n+1][b.sub.n-1], which is equivalently

[b.sup.2.sub.n] < [b.sub.n+1][b.sub.n-1], (6)

consider

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

form (5), [b.sub.1] < [b.sub.0], -[b.sub.1] > - [b.sub.0], - [a.sub.0][b.sub.1] > -[a.sub.1] [b.sub.0], this leads to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This proves that

[a.sup.2.sub.n] > [a.sub.n+1][a.sub.n-1]. (8)

Thus the (6) and (8) satisfies the conditions of log concave and log convex for sequence. That is the sequence an [a.sub.n] log concave and the sequence [b.sub.n] is log convex.

Theorem 2.2. For n [greater than or equal to] 0, [a.sub.0] < [b.sub.0], then the sequences [a.sub.n+1] = H([a.sub.n], [b.sub.n]) and [b.sub.n+1] = A([a.sub.n], [b.sub.n]) are respectively log-concave and log-convex.

Proof. Consider

[a.sup.2.sub.n] - [a.sub.n-1][a.sub.n+1] = [(2[a.sub.n-1][b.sub.n-1]/[a.sub.n-1] + [b.sub.n-1]).sup.2] = -[a.sub.n-1] (2[a.sub.n][b.sub.n]/[a.sub.n] + [b.sub.n])

on rearranging the above expression leads to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

this proves that

[a.sup.2.sub.n] > [a.sub.n+1][a.sub.n-1], (9)

again consider

[b.sup.2.sub.n] - [b.sub.n-1][b.sub.n+1] = [([a.sub.n+1] + [b.sub.n-1]/2).sup.2] - [b.sub.n-1]([a.sub.n] + [b.sub.n]/2)

on rearranging the above expression leads to

([a.sup.2.sub.n-1] + 3[a.sub.n-1][b.sub.n-1])/4([a.sub.n-1] + [b.sub.n-1])) ([a.sub.n-1] - [b.sub.n-1]) < 0,

this proves that

[b.sup.2.sub.n] < [b.sub.n+1][b.sub.n-1]. (10)

Thus the (9) and (10) satisfies the conditions of log concave and log convex. That is the sequence [a.sub.n] is log concave and the sequence [b.sub.n] is log convex.

Theorem 2.3. For n [greater than or equal to] 0, [a.sub.0] < [b.sub.0], then the sequences [a.sub.n+1] = A([a.sub.n], [b.sub.n]) and [b.sub.n+1] = G([a.sub.n], [b.sub.n]) are respectively log concave and log convex.

Proof. Consider

[a.sup.2.sub.n] - [a.sub.n-1][a.sub.n+1] = [([a.sub.n-1] + [b.sub.n-1]/2).sup.2] - [a.sub.n-1] ([a.sub.n] + [b.sub.n]/2)

on substituting and using the fact A(a, b) > G(a, b), from (5) the above expression leads to

1/2 ([b.sub.n-1]([a.sub.n-1] + [b.sub.n-1]/2 - [a.sub.n-1][square root of [a.sub.n-1][b.sub.n-1]] < 0,

this proves that

[a.sup.2.sub.n] > [a.sub.n+1][a.sub.n-1], (11)

again consider

[b.sup.2.sub.n] - [b.sub.n-1][b.sub.n+1] = [a.sub.n-1][b.sub.n-1] - [b.sub.n-1] [square root of [a.sub.n][b.sub.n]]

on substituting and using the fact A(a, b) > [a.sub.n-1], G(a, b) > [a.sub.n-1], the above expression leads to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

this proves that

[b.sup.2.sub.n] < [b.sub.n+1][b.sub.n-1]. (12)

Thus the (11) and (12) satisfies the conditions of log concave and log convex. That is the sequence [a.sub.n] is log concave and the sequence [b.sub.n] is log convex.

References

[1] P. S. Bullen, Handbook of means and their inequalities, Kluwer Acad. Publ., Dordrecht, 2003.

[2] G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, 2nd edition, Cambridge University Press, Cambridge, 1959.

[3] G. Toader and S. Toader, Greek means and Arithmetic mean and Geometric mean, RGMIA Monograph, 2005.

[4] G. M. Phillips, Two Millennia of Mathematics: From Archimedes to Gauss, CMS Books in Mathematics 6, Springer-Verlag, New York, 2000.

[5] G. M. Phillips, Archimedes the numerical analyst, Amer. Math. Monthly, 88(1981), No. 3, 165-169.

[6] T. Nowicki, On the arithmetic and harmonic means, Dynamical Systems, World Sci. Publishing, Singapore, 1998, 287-290.

[7] Iulia Costin and G. Toader, Gaussian double sequences, 5th Joint Conference on Mathematics and Computer Science, Debrecen, Hungary 2004.

[8] Iulia Costin and G. Toader, Archimedean double sequences, preprint, 2004.

K. M. Nagaraja ([dagger]) and P. Siva. Kota. Reddy ([double dagger])

([dagger]) Department of Mathematics, Sri Krishna Institute of Technology, Bangalore, 560090, India

([double dagger]) Department of Mathematics, Acharya Institute of Technology, Bangalore, 560090, India

E-mail: kmn_2406@yahoo.co.in pskreddy@acharya.ac.in
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Author:Nagaraja, K.M.; Reddy, P. Siva. Kota
Publication:Scientia Magna
Date:Jun 1, 2011
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