[Math] sigmoid $\circ$ sigmoid = sigmoid

examples-counterexamplesfunction-and-relation-compositionfunctions

The question is whether the concatenation of two sigmoid function must again be a sigmoid function.

Definition. A bounded and two-times differentiable function $f:\Bbb R\to\Bbb R$ is called sigmoid function if $f'\not=0$ and there is exactly one $x\in\Bbb R$ with $f''(x)=0$.

Here is an example:

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Given two sigmoid-functions $f$ and $g$, the function $h:=f\circ g$ is obviously bounded and two times differentiable. Also $h'\not=0$ is not hard to see. I even know that there is an inflection point (i.e. a value $x$ with $h''(x)=0$). But I failed to show that there is only one!

I admit that my initial motivation for this question comes from this post. I thought this would be one way to attack it in an elegant way. Still hard, though. Also no idea if there might be counterexamples.

Best Answer

Define $g(x)=10\tanh(x/10)+\tanh(10x)$ and $f(x)=g(x-5)$.

$g(x)$:

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$g'(x):$

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$g''(x):$

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$f(g(x)):$

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Note that despite how the graphs look, all derivatives are obviously bounded.

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