I came across the following inequality, dubbed the "Neoclassical Inequality" which holds uniformly in $p\geq 1$ and $n\in\mathbb N$:

$$\frac{1}{p^2}\sum_{j=0}^n\frac{a^{\frac{j}p}b^{\frac{n-j}p}}{\frac{j}p!\frac{n-j}p!}\leq \frac{(a+b)^{\frac{n}p}}{\frac{n}p!}$$

Notice that if $p=1$, we get the binomial theorem. As well, the inequality is homogenous in $a$ and $b$ so wlog replace $a$ with $x$ and $b$ with $(1-x)$.

My question concerns the proof of this inequality, found in the paper 'Differential equations driven by rough signals', Rev. Mat. Iberoamericana 14 (1998) 215-310, on page 254, theorem 2.2.3. Also see Keisuke Hara and Masanori Hino, Fractional order Taylor’s series and the neo-classical inequality.

Without going into too much detail (see the first paper above for the gory details), the idea of the proof is to define functions $F_{\theta_j}(x,v)$ which more-or-less correspond to individual terms of the sum and reduce the proof of the inequality to showing that

$$\sum_{j=0}^nF_{\theta_j}(x,v)\leq 1 \qquad \qquad \ \ (\star)$$

holds for all $v>1/n$ and all $x$. Now comes the key idea: we can prove $(\star)$ by showing that

$$\left(\frac{\partial}{\partial x}(x(1-x))\frac{\partial}{\partial x}-\frac{\partial}{\partial v}\right)F_\theta\geq 0,$$

so that by the maximum principle for sub-parabolic functions, we can conclude that any positive linear combination of the $F_\theta$ attains its maximum over the region $v>1/n,x\in(0,1)$ on its parabolic boundary. In particular, we get $(\star)$.

This is a technique that's new to me. I was wondering where else one can use such maximum principles to prove inequalities. It looks like the general idea is to find an appropriate parabolic differential operator for the inequality which makes the terms in the inequality a subsolution. More importantly, I was wondering when one should consider the inspiration to apply such a technique to proving an inequality.


As far as I know this inequality was first proved by Terry Lyons in 90-s by standard method of Lagrange multipliers.

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