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Relation between series and important means.

Abstract

In this paper, we obtain the relation between Logarithmic mean, Identric mean with series and pth - Logarithmic mean, Generalized power type Heron mean with Convolution product of sequences

2000 Mathematics Subject Classification: 26D15. 26D10.

Key words: Logarithmic mean, Identric mean, Z-transform and convolution.

1. Introduction

For positive numbers a and b, Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

Let a, b > 0, k is a natural number and r is a real number, then the generalized power type Heron mean [H.sub.r](a,b,k) can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

Let a, b > 0, p [not equal to] 0,-1, then pth - Logarithmic mean [L.sub.p] (a,b) can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

are respectively called the logarithmic mean, Heron mean, pth - Logarithmic mean and generalized power type Heron mean(see [2, 3, 5, 6, 7]).

Definition 1. [1] Let f(n), g(n) are two sequences defined for the discrete values of n = 0,1,2,..... Then convolution product of f(n) and g(n) is denoted by f(n) * g(n) and defined as

f (n) * g(n) = [n.summation over (m=0)] f(m).g(n - m). (1.5)

2. Main Results

Lemma 1. Let [a.sup.n], [b.sup.n] be two sequences then [a.sup.n] * [b.sup.n] = [a.sup.n+1] - [b.sup.n+1]/a - b.

Proof:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Theorem 1. Let a, b > 0, p [not equal to] 0, -1 (is an integer) and a [not equal to] b, the corresponding sequences are {[a.sup.p]},{[b.sup.p]} then [L.sub.p](a,b) = [(([a.sup.p] * [b.sup.p])/p + 1).sup.1/p].

Proof: From (1. 4), we have

[L.sub.p] = [L.sub.p](a,b) = [([a.sup.p+1] - [b.sup.p+1]/(p + 1)(a - b)).sup.1/p] for a [not equal to] b (2.1)

and by lemma.1,

[a.sup.p] * [b.sup.p] = [a.sup.p+1] - [b.sup.p+1]/a - b. (2.2)

Substitute (2.2) in (2.1), we obtain, [L.sub.p](a,b) = [(([a.sup.p] * [b.sup.p])/p + 1).sup.1/p]

Theorem 2. Let a, b > 0, r [not equal to] 0, k [not equal to] -1 (is an integer) and b a [not equal to], the corresponding sequences are {[A.sup.k]}, {[B.sup.k]} (where A = [a.sup.r/k], B = [b.sup.r/k]),

then [H.sub.r](a,b;k) = [(([A.sup.k] * [B.sup.k])/k + 1).sup.1/r].

Proof: From (1.3) we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

Let us denote A = [a.sup.r/k] and B = [b.sup.r/k]

= ([A.sup.(k+1)] - [B.sup.(k+1)]/(k+1)(A - B))1/r for A [not equal to] B, r [not equal to] o

and by lemma 1, for A = [a.sup.r/k] and B = [b.sup.r/k]. Where r/k is an integer

[A.sup.k] * [B.sup.k] = [A.sup.k+1] - [B.sup.k+1]/A - B. (2.4)

Substitute (2.4) in (2.3), we obtain, [H.sub.r](a,b;k) = [([A.sup.k] * [B.sup.k])/k+1).sup.1/r]

Theorem 3. Logarithmic mean is monotonically increasing for a > [e.sup.n+1] > b [greater than or equal to] 1.

Proof: Let f (n) = [[infinity].summation over (n=1)] [(ln a).sup.n] - [(ln b).sup.n]/n! (ln a - ln b), provided a, b.

Consider

(n+1)th term of f (n) - nth term of f(n) = [(ln a).sup.n+1] - [(ln b).sup.n+1]/(n+1)! (ln a - ln b) - [(ln a).sup.n] - [(ln b).sup.n]/n! (ln a - ln b)

Denote ln a = A, ln b = B.

(n+1)th term f(n) - nth term f(n) = [(A).sup.n+1] - [(B).sup.n+1]/(n + 1)! (A - B) - [(A).sup.n] - [(B).sup.n]/n! (A - B)

= 1/(n+1)!(A - B) [[A.sup.n](A - (n+1))+[B.sup.n]((n+1) - B)]

Since a > b [greater than or equal to] 1,

We have ln a > ln b [greater than or equal to] 0, [A.sup.n] >0, [B.sup.n] [greater than or equal to]0. A - (n+1)>0,

[??] ln a>(n+1)

[??] a > [e.sup.n+1]

Similarly (n+1) - B >0,

[??] (n+1) > ln b

[??] [e.sup.n+1] >b

That is (n+1)th term of f(n) - nth term of f(n) is positive provided a > [e.sup.n+1] > b[greater than or equal to] 1.

Hence f (n) is increasing monotonically for a > [e.sup.n+1] > b > 1.

From (1.1) we have, L [a, b] = a - b/ln a - ln b when a [not equal to] b

Let a = [e.sup.x], b = [e.sup.Y] (see [4])

when [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since f (n) is increasing.

Hence Logarithmic mean is monotonically increasing for a > [e.sup.n+1] > b [greater than or equal to] 1

Theorem 4. The function g (n) = [[infinity].summation over (n=1)] [(ln a).sup.n] - [(ln b).sup.n]/a - b is increasing monotonically for a > e > b [greater than or equal to] 1.

Proof: Let g (n) = [[infinity].summation over (n=1)] [(ln a).sup.n] - [(ln b).sup.n]/a - b, provided a [not equal to] b [not equal to] 1.

Consider

(n+1)th term of g (n) - nth term of g(n) = [(ln a).sup.n+1] - [(ln b).sup.n+1]/a - b - [(ln a).sup.n] - [(ln b).sup.n]/ a - b

Let us denote ln a = A, ln b = B.

(n+1)th term g(n) - nth term g(n) = [(A).sup.n+1] - [(B).sup.n+1]/a - b - [(A).sup.n] - [(B).sup.n]/a - b

= [[A.sup.n](A - 1)+ [B.sup.n](1 - B)/a - b]

Since a > b [greater than or equal to] 1,

We have ln a > ln b [greater than or equal to] 0, [A.sup.n] > 0, [B.sup.n] [greater than or equal to] 0. (A - 1)>0,

[??] ln a > 1

[??] a > e

Similarly 1 - B >0,

[??] 1 > ln b

[??] e > b

That is (n+1)th term of g(n) - nth term of g(n) is positive provided a > e > b [greater than or equal to] 1. Hence g(n) is increasing monotonically for a > e > b [greater than or equal to] 1.

Remark: 1. From (1.2), we have

I (a, b) = e (a(ln a-1)-b(ln b-1)/a - b) when a [not equal to] b

(n+1)th term of g(n) - nth term of g(n) = [(ln a).sup.n](ln a - 1) + [(ln b).sup.n](1 - ln b)/a - b

If [(ln a).sup.n] = a, [(ln b).sup.n] = b

ie [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

then (n+1)th term of g(n) - nth term of g(n) = (a(ln a-1)-b(ln b-1)/a - b) = ln I [(a, b)] that is exp {(n+1)th term of g(n) - nth term of g(n)} is Identric mean.

Remark: 2. From (1.1), we have

L (a,b) = a - b/ln a - ln b for a [not equal to] b.

and the first term of the series f(n) = ln a - ln b/a - b for a [not equal to] b.

we observe that reciprocal of the first term of g(n) is Logarithmic mean.

Remark: 3. let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is positive for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

i.e., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

References

[1] B.S. Grewal, A text book on Engg. Mathematics, 37th edition 2005 New Delhi.

[2] Jamal Rooin and Mehdi Hassani 'Some new inequalities between important means and applications to Ky Fan Types inequalities,(pre print)

[3] Lokesha. V Zhi-Hua Zhang, The weighted Heron Dual Mean in 'n' variables (communicated).

[4] Seppo mustonen, 'Logarithmic mean for several arguments' (pre print)

[5] Zh.-H. Zhang, Lokesha. V and Zh.-G. Xiao, The weighted Heron Mean of several positive numbers. RGMIA. Research Report collections 8(3) Article 6. (2005) Australia.

[6] Zh.-H. Zhang, Lokesha. V and Zh.-G. Xiao, The weighted Heron Mean in 'n' variables, Journal of Analysis and computation, 1(1) (2005), 57-69.

[7] Zh.-H. Zhang, Lokesha. V and Y.-D. Wu, The New Bound of the logarithmic mean, Advanced studies in contemporary Mathematics. 2(2), Sep (2005), South Korea.

V. Lokesha (1) and K.M. Nagaraja (2)

(1) Department of Mathematics Acharya Institute of Technology, Soldevanahalli, Bangalore-90, Karnataka. India E-mail: lokiv@yahoo.com

(2) Department of Mathematics Sri Krishna Institute of Technology, Chickabanavara, Bangalore-90, Karnataka. India E-mail: kmn_2406@yahoo.co.in
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Author:Lokesha, V.; Nagaraja, K.M.
Publication:Advances in Theoretical and Applied Mathematics
Article Type:Report
Geographic Code:9INDI
Date:May 1, 2007
Words:1502
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