A four-mean theorem and its application to pseudospectra

Let $N\ge 4$. We show that, if $x_1,\dots ,x_N$ and $y_1,\dots ,y_N$ are $N$-tuples of strictly positive numbers whose arithmetic, geometric and harmonic means agree, then $$\underset{j}{\max }{x}_j<(N-2)\underset{j}{\max }{y}_j\text{and}\underset{j}{\min }{x}_j<(N-2)\underset{j}{\min }{y}_j\mathrm{.}$$ This is used to show that, if $N\ge 4$ and $A,B$ are $N\times N$ matrices with super-identical pseudospectra, then, for every polynomial $p$, we have $$\mathrm{\parallel }p(A)\mathrm{\parallel }<\sqrt{N-2}\mathrm{\parallel }p(B)\mathrm{\parallel },$$ unless $p(A)=p(B)=0$. This improves a previously known inequality to the point of being sharp, at least for $N=4$.