In this thesis , we first introduce some (1-\\mu)a #43;\\mu b , \\] \\[ a\\sharp_{\\mu}b=a^{\\frac{1}{2}}(a^{-\\frac{1}{2}}ba^{-\\frac{1}{2}})^{\\mu}a^{\\frac{1}{2}} , \\] \\[ a!_{\\mu}b=[(1-\\mu)a^{-1} #43;\\mu b^{-1}]^{-1} . \\] In the case $\\mu=\\frac{1}{2}$ , we omit $\\mu$ . The following relation is well known : \\[ a!_{\\mu}b\\le a\\sharp_{\\mu}b\\le a\\bigtriangledown_{\\mu}b . \\] The inequality $a\\sharp b\\le a\\bigtriangledown b$ is called arithmetic geometric mean inequality .
