Let \(K\subset\mathbb{C}\) be a compact set. By \(K^c\) denote the complement of \(K\). In the paper under review, the author investigates boundedness properties of some universal functions. Let \((a_n)_{n\in\mathbb{N}}\) be an unbounded sequence in \(\mathbb{C}\). Birkhoff's result asserts that there exists a function \(f\in H(\mathbb{C})\) such that for every compact set \(K\), with \(K^c\) connected, and every function \(g\) continuous on \(K\) and holomorphic in the interior of \(K\), there exists an increasing subsequence \((n_k)\) of natural numbers such that \[ \sup_{z\in K}| f(z+a_{n_k})-g(z)|\rightarrow 0\quad\text{as}\quad n\rightarrow +\infty. \] \textit{W. Luh} proved an analogue result replacing the ``translation'' sequence \((f(z+a_{n_k}))\) by the ``multiplicative'' sequence \((f(a_{n_k}z))\) [Complex Variables, Theory Appl. 31, No. 1, 87--96 (1996; Zbl 0869.30022)]. It is well known that universal functions in the sense of Birkhoff may be bounded on every line. The author proves that such a result does not hold for the multiplicative universal functions.