A note on generalized Young's inequality for Orlicz functions (Q1280021)

Let \(\varphi\) be an Orlicz function such that \(a= \liminf_{t\to\infty} \varphi(t)/t>0\), \(\varphi^*\) -- the function complementary to \(\varphi\) in the sense of Young, and let \(\overline\varphi(t)= (\varphi^*)^*(t)\) be the convex minorant of \(\varphi\). Let \(q\) be the left derivative of \(\varphi^*\). It is proved that if \(a=\infty\) then \(\varphi(q(s))= \overline\varphi(q(s))\) for every \(s>0\), and if \(0<a<\infty\), then \(\varphi(q(s))+ \varphi^*(s)= s\) \(q(s)\) for \(0<s<a\).