I'm interested in the following operator $T$, close relative of the standard logarithmic derivative:


$$f(x)\to Tf(x)=\frac{\text{d}(\log {f})}{\text{d}(\log {x})}=\frac{xf'}{f},$$


where $f$ is an increasing, positive $C^{\infty}(\mathbb R^+)$ function.
Does anyone know whether this pops up, say, in functional analysis or probability? If so, in which context?



On the operator $f\to xf'/f$
You see this in the discussion of modular forms and related topics. When complex variable $\tau$ is in the upper half-plane $\operatorname{Im} \tau > 0$, the related complex variable $q = e^{2\pi i \tau}$ is in the (punctured) unit disk $0 < |q| < 1$.  

An important derivation in this setting [call it say $\vartheta$] is defined as follows. If $f$ is a function of $q$, equivalently a function of $\tau$ with period $1$,
$$
\vartheta f = \frac{1}{2\pi i}\frac{d}{d\tau} f \qquad\text{in terms of }\tau
$$
or
$$
\vartheta f = q\frac{d}{dq} f \qquad\text{in terms of } q
$$
Thus, the logarithmic derivative in terms of $\tau$ is essentially your operator $T$ in terms of $q$:
$$
\frac{\vartheta f}{f} = \frac{1}{2\pi i}\frac{df/d\tau}{f} = q \frac{df/dq}{f} = T [f] .
$$




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Some random examples (i)  

$$
E\_2 = 24\frac{\vartheta \eta}{\eta} = 24 \;T [\eta]
$$
where $\eta$ is the Dedekind eta function and $E\_2$ is an Eisenstein series:
$$
\eta(q) = q^{1/24}\prod\_{n=1}^\infty(1-q^n)
\\
E\_2(q) = 1 - 24\sum\_{k=1}^\infty \sigma(k) q^k
$$  

And (ii)  

$$
T[j\_{3B}](\tau) = \frac{1}{2}E\_2(\tau) - \frac{3}{2}E\_2(3\tau)
$$
where
$$
j\_{3B}(\tau) = \frac{\eta(\tau)^{12}}{\eta(3\tau)^{12}}
$$
is a Hauptmodul for modular curve $X\_0(3)$. See [A030182](http://oeis.org/A030182).




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Plug. These two examples copied from the appendix of [arXiv:2005.10733](https://arxiv.org/abs/2005.10733)


