Monday, October 4, 2010

On an inequality

Few days ago I recalled my PhD work. I didn't make to defend it for various reasons (maybe one day), but this isn't the reason of this blog post. The reason is an interesting inequality I used in my work. It looks like, for any integrable function f(x) > 0, ∀x∈[a,b]
This is, more or less, a generalised form of the inequality of arithmetic and geometric means. E.g. for any xi∈[a,b],∀i=1..n, a=x1, b=xn
The natural logarithm is monotonically increasing function:
Let's consider h=(b-a) ⁄ n
Or
Now let's consider limh→0 and the fact that the limit keeps the inequality:
The natural logarithm function is continuous, so:
As a result
Considering that the natural exponential function is also monotonically increasing function we receive the original inequality.

Another proof is the fact that the natural logarithm function is concave, i.e. for ∀αi, ∑αi=1

Now, if we consider αi=Δxi ⁄ (b-a)=h ⁄ (b-a) we will receive the same result.