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Upper and Lower Bounds for Noncommutative Perspectives of Operator Monotone Functions: the Case of Second Variable


Assume that the function \(f:[0,\infty )\rightarrow \mathbb {R}\) is operator monotone in \([0,\infty )\). We can define the perspective \(\mathcal {P}_{f}\left (B,A\right ) \) by setting

$$ \mathcal{P}_{f}\left( B,A\right) :=A^{1/2}f\left( A^{-1/2}BA^{-1/2}\right) A^{1/2}, $$

where A, B > 0. In this paper, we show among others that, if σCρ > 0, D > 0, ςQτ > 0 and 0 < nDCN for some constants ρ, σ, ς, τ, n, N, then

$$ \begin{array}{@{}rcl@{}} 0& \le& \frac{n}{N{\varsigma}^{2}}\left[ \mathcal{P}_{f}\left( {\varsigma} ,N+\sigma \right) -\mathcal{P}_{f}\left( {\varsigma} ,\sigma \right) \right] Q^{2} \\ & \leq& \mathcal{P}_{f}\left( Q,D\right) -\mathcal{P}_{f}\left( Q,C\right) \\ & \leq& \frac{N}{n\tau^{2}}\left[ \mathcal{P}_{f}\left( \tau ,n+\rho \right) -\mathcal{P}_{f}\left( \tau ,\rho \right) \right] Q^{2}. \end{array} $$

Applications for the weighted operator geometric mean and the perspective

$$ \mathcal{P}_{\ln \left( \cdot +1\right) }\left( B,A\right) :=A^{1/2}\ln \left( A^{-1/2}BA^{-1/2}+1\right) A^{1/2},~ A,B>0 $$

are also provided.

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

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Correspondence to Silvestru Sever Dragomir.

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Dragomir, S.S. Upper and Lower Bounds for Noncommutative Perspectives of Operator Monotone Functions: the Case of Second Variable. Acta Math Vietnam 47, 581–595 (2022).

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  • Noncommutative perspectives
  • Relative operator entropy
  • Operator monotone functions

Mathematics Subject Classification (2010)

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