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Let's assume that $ f: V \rightarrow \mathbb{R}_{>0} $ is a monotonic, continuous and smooth function, i.e. the $n$-th derivative $ f^{(n)} $ exists for all integers $ n $, where $ V $ is a real interval. I am interested in the monotonicity of the following function: $$ g(x) = \frac{f(x)}{f(c \times x)}, \quad c\in\mathbb{R}_{>0} $$ One could answer this by differentiating $ g(x) $ and check whether $ g'(x) \geq 0 $ (or $ \leq 0 $) for all $ x \in V $. By applying the quotient rule one finds that it is sufficient to show that $ c \times f(x) f'(c \times x) - f(c \times x)f'(x) \geq 0 $ (or $ \leq 0 $) for all $ x \in V $. However, this can get rather messy at times.

I was wondering if there are any properties of $ f(x) $ that imply monotonicity of $ g(x) $.

Suppose for example that $ V = [0, \infty) $ and that $ f^{(n)}(x) > 0 $ for all $ x \in V $ and all integers $ n $. Further suppose that $ f(0) = 1 $. I could not find an example were in such a case $ g(x) $ was not monotonically increasing for $ c < 1 $ or not monotonically decreasing for $ c > 1 $.

Any reference, counter example, proof or proof idea regarding this issue would be appreciated.

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2 Answers 2

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Counter example:

Consider function $f(x) = e^{sin(x)+x}$ and $c=\frac{1}{2}$.

$f'(x) = (cos(x)+1)e^{sin(x)+x}$

$f'(x) \geq 0$ for $x \in \mathbb{R}$ so $f$ is monotonic.

$g(x) = \frac{f(x)}{f(x/2)} = \frac{e^{sin(x)+x}}{e^{sin(x/2)+x/2}} = e^{sin(x)-sin(x/2)+x/2 } $

$g'(x) = (cos(x) - \frac{1}{2}cos(x/2) + \frac{1}{2} ) e^{sin(x)-sin(x/2)+x/2}$

And function $g$ is not monotonic in $\mathbb{R}$.

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  • $\begingroup$ This is a nice example, however, it does not meet the described conditions. My conjecture asks for all the derivatives of $ f(x) $, i.e. $ f^{(n)}(x) \; \forall \; n \in \mathbb{N} $, being positive for all $ x \in [0,\infty) $. The second derivative in your example is $ f''(x) = f^{(2)}(x) = e^{\sin\left(x\right)+x}\left(\cos\left(x\right)+1\right)^{2} - e^{\sin\left(x\right)+x}\sin\left(x\right) $ which is e.g. negative for $ \pi/2 < x < \pi $. $\endgroup$
    – CMG
    Commented Jun 11, 2020 at 6:34
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You might want to look in: Biernacki, M, Krzy ̇z, J: On the monotonicity of certain functionals in the theory of analytic functions. Ann. Univ. Mariae Curie-Skłodowska, Sect. A 9, 135-147 (1955). There should be a result of that type for analytic functions. You can also find the relevant Theorem here.

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    – Community Bot
    Commented Jan 31, 2022 at 16:02

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