Some generalizations of real power form for Young-type inequalities and their applications
The weighted arithmetic-geometric means inequality \[ \nu a+(1-\nu)b\geq a^\nu b^{1-\nu}\quad (a,b\geq0,\, \nu\in[0,1]) \] is known as the Young inequality. Among various generalizations of this inequality, it has been shown in [\textit{D. Choi}, Math. Inequal. Appl. 21, No. 1, 99--106 (2018; Zbl 1386.15041)] that \[ (\nu a+(1-\nu)b)^m\geq \left(a^\nu b^{1-\nu}\right)^m+(2r_0)^m\left(\left(\frac{a+b}{2}\right)^m-(ab)^{m/2}\right) \] holds for any positive integer \(m\), where \(r_0=\min\{\nu,1-\nu\}\). In the present paper, the authors utilize majorization techniques to prove the above inequality in the case where \(m\) is replaced with a real number \(p\geq 1\). Some operator versions are presented as well.
scientific article; zbMATH DE number 7613300