# Monotonicity of the ratio of scaled functions

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.

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 $$\mathbb{R}$$.

• 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 $\pi/2 < x < \pi$.
– CMG
Commented Jun 11, 2020 at 6:34

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.