Strong Higher Derivations on Ultraprime Banach Algebras

In this paper we show that if {$H_n$} is a continuous strong higher derivation of order n on an ultraprime Banach algebra with a constant c, then $c||H_1||^2{\leq}4||H_2||$ and for each $1{\leq}l$ < n $$c^2||H_1||\;||H_{n-l}{\leq}6||H_n||+\frac{3}{2}\sum_{\array{i+j+k=n\\i,j,k{\geq}1}}||H_i||\;||H_j||\;||H_k||+\frac{3}{2}\sum_{\array{i+k=n\\i{\neq}l,\;n-1}}||H_i||\;||H_k|| $$ and for a strong higher derivation {$H_n$} of order n on a prime ring A we also show that if [$H_n$(x),x]=0 for all $x{\in}A$ and for every $n{\geq}1$, then A is commutative or $H_n=0$ for every $n{\geq}1$.