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 x_i∈[a,b],∀i=1..n, a=x₁, b=x_n

The natural logarithm is monotonically increasing function:

Let's consider h=(b-a) ⁄ n

Or

Now let's consider lim_h→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 ∀α_i, ∑α_i=1

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