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Multiplicative inequalities for weighted geometric mean in Hermitian unital Banach \(*\)-algebras


Consider the quadratic weighted geometric mean

$$\begin{aligned} x\circledS _{\nu }y:= \vert \vert yx^{-1} \vert ^{\nu }x \vert ^{2} \end{aligned}$$

for invertible elements xy in a Hermitian unital Banach \(*\) -algebra and real number \(\nu \). 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 \(\left( 1-\nu \right) \left| x\right| ^{2}+\nu \left| y\right| ^{2}\) in terms of \(x\circledS _{\nu }y,\) where \(\nu \in \left[ 0,1\right] ,\) under various assumptions for the elements xy involved. Applications for the classical weighted geometric mean

$$\begin{aligned} a\sharp _{\nu }b:=a^{1/2} ( a^{-1/2}ba^{-1/2}) ^{\nu }a^{1/2} \end{aligned}$$

of positive elements ab that satisfy the condition \(0<ka\le b\le Ka\) for certain numbers \(0<k<K,\) are also given.

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The author would like to thank the anonymous referees for their valuable comments that have been implemented in the final version of the paper.

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Correspondence to S. S. Dragomir.

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Dragomir, S.S. Multiplicative inequalities for weighted geometric mean in Hermitian unital Banach \(*\)-algebras. RACSAM 112, 1349–1365 (2018).

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  • Weighted geometric mean
  • Young’s inequality
  • Operator modulus
  • Arithmetic mean-geometric mean inequality
  • Hermitian unital Banach \(*\)-algebra

Mathematics Subject Classification

  • 47A63
  • 47A30
  • 15A60
  • 26D15
  • 26D10