*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=x*,

_{1}*b=x*

_{n}The natural logarithm is monotonically increasing function:

Let's consider

*h=(b-a) ⁄ n*

Or

Now let's consider

*lim*and the fact that the limit keeps the inequality:

_{h→0}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}=1Now, if we consider

*α*we will receive the same result.

_{i}=Δx_{i}⁄ (b-a)=h ⁄ (b-a)