Description of a class of locally pseudoconvex algebras which have an equivalent locally \(M\)-pseudoconvex topology (Q1293498)

For \(r \in ]0,1]\) consider an \(r\)-homogeneous seminorm \(p\) on a commutative topological algebra \(A\) and let \(\phi\) be an increasing function defined on \([0,\infty[\) to itself such that \(p(x^2) \leq \phi (p(x))\) for any \(x \in A\). It is proved that \(p\) is \(A\)-pseudoconvex and equivalent with a submultiplicative semi-norm. A description of a class of locally pseudoconvex algebras which have an equivalent locally \(m\)-pseudoconvex topology is also given.