Highly differentiable boolean functions require super-linear DeMorgan formulas

A boolean function $f(x_1,\ldots,x_n)$ is $m$-differentiable ($m < n$) if for every assignment of constants $0/1$ to any subset of its $m$ variables, the resulting subfunction depends of all its remaining $n-m$ variables. Let $L(f)$ denote the smallest leaf-size of a De Morgan formula computing $f$.

V.A. Malyshev (1967) proved the following lower bound.

If $f(x_1,\ldots,x_n)$ is $m$-differentiable, then \[ L(f)\geq \frac{m}{2}\cdot \frac{\log_2 m}{\log_2\log_2 m}. \]