Hyperstability of a logarithmic-type functional equation on restricted domains

Let $X$ be a real normed vector space and $f:(0,\infty )\to X$. In this paper, we prove the hyperstability of the logarithmic functional equation

$f\left(x^y\right)?yf\left(x\right)=0$

on $\Gamma$ of Lebesgue measure zero. More precisely, we prove that if $f:(0,\infty )\to X$ satisfies

$6f\left(x^y\right)?yf\left(x\right)6\le ?(x,y)$

for all $(x,y)\in {\Gamma }^{+}?\{(x,y):y>\alpha \left(x\right)\}[\mathrm{resp.}{\Gamma }^{?}?\{(x,y):0<y<\alpha \left(x\right)\}]$ of Lebesgue measure zero, where $\alpha :(0,\infty )\to (0,\infty )$ is an arbitrary given function and $?:(0,\infty )\times (0,\infty )\to [0,\infty )$ satisfies the condition $?(x,y)∕y\to 0$ as $y\to \infty$ [resp. $y\to 0$], then $f$ satisfies the functional equation $f\left(x^y\right)=yf\left(x\right)$ for all $x>0$ and$y>0$.