In the book Math Problems AMM (1957), Problem 230, there is the next inequality of D. Polya:
- let $a,b>0$, $0\leq x \leq a $,
- $f(x)$ --- being not a linear function, and $f(0)=0$, $f(a)=b$, $f(x)\geq 0$, $f''(x)\geq 0$.
Then the next inequality holds true
$$
2\pi \int_0^a f(x)\sqrt{1+(f'(x))^2} \,dx \leq \pi b\sqrt{a^2+b^2}.
$$
($\pi$ of course may be droped on both sides). There is a proof and geometrical interpretation in the above mentioned book. From my point of view the proof is not direct and a bit sophisticated, as it seems.
Problem 1. Find a direct proof of this inequality based on some classical inequalities, such as Cauchy-Bunyakovskii, Holder, Jensen, Hardy and so on.
Problem 2. Find some generalisations, say in terms of two arbitrary functions, and so on.
