On the operator $f\to xf'/f$ 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?
 A: 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
A: 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
$$x \; D_x \ln(f(x)) = \sum_{n>=1} -F_n(b(1),...,b(n)) \; x^n .$$
