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 > 0 $, $ 0 < p\leq t < 1 $ and $ m\in\mathbb{N^{+}} $, then
$ \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*} $
and
$ \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*} $
for $ b\geq a $, and the inequalities are reversed for $ b\leq a $. As applications, we obtained some inequalities for operators and determinants.
Citation: Yonghui Ren. Generalizations of AM-GM-HM means inequalities[J]. AIMS Mathematics, 2023, 8(12): 29925-29931. doi: 10.3934/math.20231530
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 > 0 $, $ 0 < p\leq t < 1 $ and $ m\in\mathbb{N^{+}} $, then
$ \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*} $
and
$ \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*} $
for $ b\geq a $, and the inequalities are reversed for $ b\leq a $. As applications, we obtained some inequalities for operators and determinants.
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Yonghui Ren. Generalizations of AM-GM-HM means inequalities[J]. AIMS Mathematics, 2023, 8(12): 29925-29931. doi: 10.3934/math.20231530
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