Iranian Mathematical Society Journal of the Iranian Mathematical Society 2717-1612 5 1 2024 01 01 A note on arithmetic-geometric-harmonic mean inequality of several positive operators 33 43 184321 10.30504/jims.2023.415016.1144 EN F. Mirzapour Department of Mathematics, Faculty of science, University of Zanjan, Zanjan, Iran. 0000-0001-7477-6475 A. Morassaei Department of Mathematics, Faculty of science, University of Zanjan, Zanjan, Iran Journal Article 2023 09 05 ‎Suppose that $B_1,\cdots,B_m$ are positive operators on a Hilbert space $\mathcal{H}$‎. ‎In this paper we generalize the weighted arithmetic‎, ‎geometric and harmonic means as follows‎:<br /><br />‎\begin{align*}‎<br /><br />‎{\mathbf a_m}(\boldsymbol\kappa;\mathbf{B})&={\mathbf a_2}(k_1,N';B_1,{\mathbf a_{m-1}}(\boldsymbol\kappa';\mathbf{B}'))=\frac{k_1B_1+\cdots+k_mB_m}{N}\\‎<br /><br />‎{\mathbf h_m}(\boldsymbol\kappa;\mathbf{B})&={\mathbf h_2}(k_1,N';B_1,{\mathbf h_{m-1}}(\boldsymbol\kappa';\mathbf{B}'))=\left(\frac{k_1B_1^{-1}+\cdots+k_mB_m^{-1}}{N}\right)^{-1}\\‎<br /><br />‎{\mathbf g_m}(\boldsymbol\kappa;\mathbf{B})&={\mathbf g_2}(k_1,N';B_1,{\mathbf g_{m-1}}(\boldsymbol\kappa';\mathbf{B}'))‎<br /><br />‎\end{align*}‎<br /><br />‎where $\boldsymbol\kappa=(k_1,\cdots,k_m)‎, ‎N=k_1+\cdots+k_m‎, ‎\boldsymbol\kappa'=(k_2,\cdots‎, ‎k_m)$ and $N'=k_2+\cdots+k_m$‎. ‎We show that the arithmetic-geometric-harmonic mean inequality holds‎. ‎Also we investigate nine property of the geometric mean‎. https://jims.ims.ir/article_184321_3573da7c02b254a2b7901dca40a5cc02.pdf