| Research Abstract |
In what follows, a capital letter means a bounded linear operator on a Hulbert space. Furuta inequality (1987) asserts that if A <greater than or equal> B <greater than or equal> 0, then for r
(*)(A^<r/>A^pA^<r/>)^<1/> <greater than or equal> (A^<r/>B^pA^<r/>)^<1/>
holds for p <greater than or equal> 0 and q <greater than or equal> 1 with (l+r)q <greater than or equal> p+r. Furuta inequality yields the famous Lowner-Heinz one (1934), that is, A <greater than or equal> B <greater than or equal> 0 ensures A^p <greater than or equal> B^p for 1 <greater than or equal> p <greater than or equal> 0 when we put r = 0 in (*). We obtained a lot of applications of Furuta inequality in the following three branches, (a) operator ibnequalities, (b) norm inequalities and (c) operator equations. We cite some of them as follows : (alpha_1) relative operator entropy, (alpha_2) Ando-Hiai log majorization, (alpha_3) Aluthge transformation, (b_1) Heinz-Kato inequality, (b_2) Kosaki trace inequality , (c_1) Pedersen-Takesaki operator equation. Recently we obtained a one page simplified proof of generalized Furuta inequality and equivalence relations among this generalized Furuta inequality and related operator functions. We have to consider some relations among Kantrovich inequality, Holder-McCarthy inequality, Reid inequality, Jensen inequality and Furuta inequality. Further applications and developments of Furuta inequality will be expected
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