Consider functions $f(x)$ defined and real-analytic for $x>0$ , such that
for all real $x,y > 0$ and integer $n > 0$ :
$$ sign (\frac{d^n }{d^n x} f(x) ) = sign (\frac{d^n}{d^n x} f(x+y) ) $$
In other words ; the signs of the derivatives do not change over the positive reals.
Some trivial examples are $e^x$ where the signs are always positive. And $e^{-x}$ where the signs are always alternating between positive and negative. Bernstein functions and related things are based on this.
But what about other sign patterns ?
Such as +,+,-,++,-,+,+,-,... ?
What are typical solutions ?
Are Some Patterns forbidden ? ( not existing )
I am intrested in both periodic and nonperiodic Patterns.
I assume we can generalise Bernstein's results once we understand this.