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?
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 $q$:
$$
\frac{\vartheta f}{f} = \frac{1}{2\pi i}\frac{df/d\tau}{f} = q \frac{df/dq}{f} = T [f] .
$$
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.
Plug. These two examples copied from the appendix of arXiv:2005.10733
You could also look at the numerous refs in OEIS A263916 on the classic Faber polynomials, related to the Newton identities and symmetric functions/polynomials, defined by
$$ -\ln(1 + b(1) x + b(2) x^2 + ...) = \sum_{n>=1} F_n(b(1),...,b(n)) \; x^n/n,$$
so
$$x \; D_x \ln(f(x)) = \sum_{n>=1} -F_n(b(1),...,b(n)) \; x^n .$$
