Multiplicative inequalities for weighted geometric mean in Hermitian unital Banach ∗-algebras.

Consider the quadratic weighted geometric meanxⓈνy:=||yx-1|νx|2<graphic></graphic>for invertible elements x,  y in a Hermitian unital Banach ∗<inline-graphic></inline-graphic>-algebra and real number ν<inline-graphic></inline-graphic>. In this paper, by utilizing some results of Tominaga, Furuichi, Liao-Wu-Zhao, Zuo-Shi-Fujii and the author, we obtain various upper and lower bounds for the positive element 1-νx2+νy2<inline-graphic></inline-graphic> in terms of xⓈνy,<inline-graphic></inline-graphic> where ν∈0,1,<inline-graphic></inline-graphic> under various assumptions for the elements x,  y involved. Applications for the classical weighted geometric meana♯νb:=a1/2(a-1/2ba-1/2)νa1/2<graphic></graphic>of positive elements a,  b that satisfy the condition 0<ka≤b≤Ka<inline-graphic></inline-graphic> for certain numbers 0<k<K,<inline-graphic></inline-graphic> are also given. [ABSTRACT FROM AUTHOR]