A generalization of two refined Young inequalities (Q891931)

The authors introduce a natural generalization of the classical Young inequality for scalars and apply their result to obtain refined Young-type inequalities for traces, determinants, and unitarily invariant norms of positive definite matrices. Their generalization asserts that if \(a\), \(b\) are positive real numbers and \(\upsilon \in [ 0,1]\), then \(\left( a^{\upsilon }b^{1-\upsilon }\right) ^{m}+r_{0}^{m}\left( a^{m/2}-b^{m/2}\right) ^{2}\leq \left( \upsilon a+\left( 1-\upsilon \right) b\right) ^{m}\) \ for \(m=1,2,\dots\), where \(r_{0}=\min \left( \upsilon ,1-\upsilon \right) \).