# Is this continuous analogue to the AM–GM inequality true?

First let us remind ourselves of the statement of the AM–GM inequality:

Theorem: (AM–GM Inequality) For any sequence $(x_n)$ of $N\geqslant 1$ non-negative real numbers, we have

It is well known that the sum operator ∑ can be generalised so that it operates on continuous functions rather than on discrete sequences. This generalisation is the integral operator ∫.

Likewise, we can generalise the product operator to operate on continuous functions. We begin with the following property of the discrete product: (assuming that $A_k>0$). Using this we can define the continuous product (also known as the geometric integral according to Wikipedia) as follows:

assuming that the integral converges. From these definitions we can seek to generalise the AM–GM inequality to continuous functions:

Proposition: (Continuous AM–GM Inequality) For any suitably well-behaved non-negative function $f$ defined on $[a,b]$ with $a, we have: or in traditional notation:

As a simple illustration, consider the function $f(x)=1+\sin x$ and fix the lower bound $a=0$. On the following graph the $x$-axis represents the upper bound of integration $b$, and on the $y$-axis are represented the the arithmetic mean (blue) and the geometric mean (red) of $f$ on the interval $[a,b]$ where $a=0$ and $b=x$.

Clearly the claimed inequality seems to hold, and is in fact quite tight when $b\approx0$. This inequality is of interest not least because it is readily abe to produce a number of non-trivial numerical inequalities pertaining to known constants. For instance, again with the example $f(x)=1+\sin x$, set $b=\pi/2$. Then assuming the proposition holds, we have

Evaluating these integrals and rearranging, we obtain: where $G\approx 0.9159$ is Catalan's constant.

Anyway I would like to ask, firstly:

Is the claimed inequality true, and if so what is the proof and on what class of functions is it applicable?

Secondly, and this is more of a fun little challenge:

Assuming its veracity, can this inequality be used to prove remarkable numerical inequalities, e.g. $\pi < 22/7$ or $e^\pi - \pi < 20$?

$$\frac1{b-a}\int_a^b\exp(g(x))\,dx\ge\exp\left(\frac1{b-a}\int_a^b g(x)\,dx\right).\frac1{b-a}\int_a^b\exp(g(x))\,dx\ge\exp\left(\frac1{b-a}\int_a^b g(x)\,dx\right).$$
In disguise it is the case of Jensen's inequality for the convex function $$\phi(t)=\exp(t)\phi(t)=\exp(t)$$.