Abstract
In this paper, we employ some operator techniques to establish some refinements and reverses of the Callebaut inequality involving the geometric mean and Hadamard product under some mild conditions. In particular, we show
where \(A_j, B_j\in {\mathbb {B}}({\mathscr {H}})\,\,(1\le j\le n)\) are positive operators such that \(0<m{^\prime } \le B_j\le m <M \le A_j\le M{^\prime }\,\,(1\le j\le n)\), either \(1\ge t\ge s>{\frac{1}{2}}\) or \(0\le t\le s<\frac{1}{2}\), \(r{^\prime }=\min \left\{ \frac{t-s}{t-1/2},\frac{s-1/2}{t-1/2}\right\} \) and \(K(t,2)=\frac{(t+1)^2}{4t}\,\,(t>0)\).
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The author would like to sincerely thank the anonymous referee for some useful comments and suggestions. The author also would like to thank the Tusi Mathematical Research Group (TMRG).
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Communicated by Poom Kumam.
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Bakherad, M. Some Reversed and Refined Callebaut Inequalities Via Kontorovich Constant. Bull. Malays. Math. Sci. Soc. 41, 765–777 (2018). https://doi.org/10.1007/s40840-016-0364-9
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DOI: https://doi.org/10.1007/s40840-016-0364-9
Keywords
- Callebaut inequality
- Cauchy–Schwarz inequality
- Hadamard product
- Operator geometric mean
- Kontorovich constant