Logarithmic Mean

In mathematics, the logarithmic mean is a function of two non-negative numbers which is equal to their difference divided by the logarithm of their quotient. In symbols:

M_lm(x,y) = lim(xi,eta)->(x,y) (xi-eta)/(Inxi-lneta)

= {(0 if x=0 vv y=0),(x if x=y),((y-x)/(lny-lnx) else):}

for the positive numbers x,y. This measure is useful in engineering problems involving heat and mass transfer. (Source: Wikipedia)

Please express your views of this topic Logarithmic Series by commenting on blog.

Theorem for logarithmic mean

Theorem 1:-

The inequalities,

 alphaA(a,b)+(1-alpha)G(a,b)
hold for all positive real numbers a and b with a≠b if and only if alpha<=2/3 and beta>=2/e=0.7357 .

Theorem 2:-

Let a and b be real numbers with a≠b. If 0
[G(a,b)]A(a,b)<[L(a,b)]I(a,b)<[A(a,b)]G(a,b).(1.9)

And if a,b>=e , then

[A(a,b)]G(a,b)<[I(a,b)]L(a,b)<[G(a,b)]A(a,b).(1.10)

Theorem 3:-

For all positive real numbers a and b with a≠b, we have

Mp(a,b)<12 a="" b="" p="">
with the best possible parameter p=log2/(1+log2)=0.40938.

for α,β,γin(0,1) with α+β+γ=1, what is the larger value of p and q is the smaller value of the double inequality



Lp(a,b)0 with a≠b.

Example problem for logarithmic mean:-

Problem 1:-

For α,β,γ in (0,1) in logarithmic mean α+β+γ=1,p is the larger value and q is the smaller value these types of value is double inequality. Lp(a,b)0 with a≠b?

Solution:-

In these equations Lp(a,b), A(a,b), G(a,b), and H(a,b) denoted as generalized logarithmic mean, arithmetic mean, geometric mean, and harmonic means of two positive numbers a and b.

For p inR the generalized logarithmic mean Lp(a,b) of two positive numbers a and b with a≠b is defined by



Lp(a,b)={([(a^p+1-b^p+1)/(p+1)(a-b)]^1/p,p!=0,p!=-1),(1/e(b^b/a^a)^1/(b-a),p=0),(b-a/logb-loga,p=-1):}



It is well known that Lp(a,b) is continuous and strictly increasing with respect to p inR for fixed a,b>0 with a≠b. The logarithmic mean of monotonicity  and the inequalities of the logarithmic mean.

Let A(a,b)=(a+b)/2 , I(a,b)=(1/e)(b^b/a^a)^1/(b-a) , L(a,b)=(b-a)/(logb-loga) , G(a,b)=sqrtab , and H(a,b)=(2ab)/(a+b) be the arithmetic mean, identric mean, logarithmic mean, geometric mean, and harmonic mean value of  a and b two positive numbers with a≠b,

Then

min{a,b}
For p inR , the pth power mean M_p(a,b) of two positive numbers a and b with a≠b is defined by

 M_p(a,b)=((a^p+b^p)/2)^1/p(p!=0), M_0(a,b)=(ab)^1/2.

M_log2/log_3(a,b)<2 a="" b="" p="">
for all a,b>0 with a≠b.


For α in(0,1), in p is the larger value and q is the smaller value such that,

M_p(a,b)
for all a,b>0 with a≠b.