On Jensen's inequality and Hölder's inequality for \(g\)-expectation (Q974648)

The author proves that, for \(n>1\), the \(n\)-dimensional Jensen's inequality holds for the \(g\)-expectation if and only if the generator \(g \equiv g(\cdot, t, y, z)\) defined on \(\Omega \times [0, T]\times \mathbb{R} \times \mathbb{R}^d\) is independent of \(y\) and linear in \(z\). Applications are made to Minkowski's and Hölder's inequalities.