Generalizations of AM-GM-HM means inequalities

Abstract

<abstract><p>In this paper, we showed some generalized refinements and reverses of arithmetic-geometric-harmonic means (AM-GM-HM) inequalities due to Sababheh [J. Math. Inequal. 12 (2018), 901–920]. Among other results, it was shown that if $ a, b &gt; 0 $, $ 0 &lt; p\leq t &lt; 1 $ and $ m\in\mathbb{N^{+}} $, then</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} \frac{(a\nabla_{p}b)^{m}-(a!_{p}b)^{m}}{(a\nabla_{ t}b)^{m}-(a!_{ t}b)^{m}}\leq\frac{p(1-p)}{ t(1- t)} \end{align*} $\end{document} </tex-math></disp-formula></p> <p>and</p> <p><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \begin{align*} \frac{(a\sharp _{p} b)^{m}-(a!_{p}b)^{m}}{(a\sharp _{ t} b)^{m}-(a!_{ t}b)^{m}}\leq\frac{p(1-p)}{ t(1- t)} \end{align*} $\end{document} </tex-math></disp-formula></p> <p>for $ b\geq a $, and the inequalities are reversed for $ b\leq a $. As applications, we obtained some inequalities for operators and determinants.</p></abstract>