Let \((\Omega,\Sigma,\mu)\) be a complete \(\sigma\)-finite measure space and let \(L^0(\Omega)\) denote the class of measurable functions on \(\Omega\). If \((X,\|\cdot\|_X)\), \((Y,\|\cdot\|_Y)\) are Banach spaces of functions in \(L^0(\Omega)\), then \(M(X,Y)\), the space of pointwise multipliers, is defined by
\[ M(X,Y)= \{y\in L^0(W): xy\in Y\text{ for }x\in X\}, \]
with an associated functional
\[ \| y\|_{M(X,Y)}= \sup\{\| xy\|_Y: x\in X,\;\| x\|_X\leq 1\}. \]
If \(\Phi: [0,\infty)\to [0,\infty)\) is a Young (or an Orlicz) function so that \(\Phi(0)= 0\), \(\Phi\) is non-decreasing and convex, then the Orlicz space \(L^\Phi(\Omega)\) is defined by \(L^\Phi(\Omega)= \{f\in L^0(\Omega): I_\Phi(f/\lambda)< \infty\) for some \(\lambda\}\), where
\[ I_\Phi(f)= \int_\Omega\Phi(|f(t)|)\, d\mu(t), \]
with the norm \(\|\cdot\|_\Phi\) defined by \(\| f\|_\Phi= \inf\{\lambda> 0: I_\Phi(f/\lambda)\leq 1\}\). Statements of the main theorem of this paper indicate that: if \(\Phi, \Phi_1, \Phi_2\) are Young functions with inverses \(\Phi, \Phi^{-1}_1, \Phi^{-1}_2\), and there is a finite constant \(C> 0\) such that \(\Phi(u)\leq C\Phi^{-1}_1(u) \Phi^{-1}_2(u)\) for \(u> 0\), then \(\| f\|_{\Phi_2}\leq C\| f\|_{ML(\Phi_1,\Phi)}\), where \(ML(\Phi_1,\Phi)=M(L^{\Phi_1}, L^\Phi)\).
The introduction of the paper includes statements that the main theorem extends similar results of L.Maligranda and L.E.Persson [Indag.Math.51, No.3, 323–338 (1989; Zbl 0704.46018)].