Comparison between power difference means and Heron means

Power difference means and Heron means are well-known numerical means with parameters. Their comparison in the positive definite sense is studied. More precisely, for a power difference mean M with each fixed parameter we try to determine the exact parameter range for Heron means majorizing M (in the positive definite sense). Since this order is known to determine validity of (unitarily invariant) norm inequalities between corresponding matrix power difference means and matrix Heron means, we obtain an abundance of very precise norm inequalities between these two matrix means.