فهرست نشریات علمی دسترسی آزاد ایران

‎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:‎\begin{align*}‎‎{\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}\\‎‎{\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}\\‎‎{\mathbf g_m}(\boldsymbol\kappa;\mathbf{B})&={\mathbf g_2}(k_1,N';B_1,{\mathbf g_{m-1}}(\boldsymbol\kappa';\mathbf{B}'))‎‎\end{align*}‎‎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.