COMPARISONS OF HEINZ OPERATOR MEANS WITH DIFFERENT PARAMETERS - Burqan, Aliaa Abed Al-Jawwad (Malaysian Journal of Science, 2019)
This article aims to present new comparisons of Heinz operator means with different parameters by the help of appropriate scalar comparisons and the monotonicity principle for bounded self-adjoint Hilbert space operators. In particular, for any positive operators, we establish the inequality <a href="https://www.codecogs.com/eqnedit.php?latex=H{τ&space;}(A,B)≤&space;\left&space;(&space;1-\frac{(1-2μ&space;)^{2}}{(1-2τ&space;)^{2}}&space;\right&space;)(A\bigtriangledown&space;B)+\frac{(1-2μ&space;)^{2}}{(1-2τ&space;)^{2}}&space;H{μ&space;}(A,B)," target="_blank"><img title="H{τ }(A,B)≤ \left ( 1-\frac{(1-2μ )^{2}}{(1-2τ )^{2}} \right )(A\bigtriangledown B)+\frac{(1-2μ )^{2}}{(1-2τ )^{2}} H{μ }(A,B)," src="https://latex.codecogs.com/gif.latex?H{τ&space;}(A,B)≤&space;\left&space;(&space;1-\frac{(1-2μ&space;)^{2}}{(1-2τ&space;)^{2}}&space;\right&space;)(A\bigtriangledown&space;B)+\frac{(1-2μ&space;)^{2}}{(1-2τ&space;)^{2}}&space;H{μ&space;}(A,B)," alt=""> where satisfying <a href="https://www.codecogs.com/eqnedit.php?latex=τ&space; eq&space;\frac{1}{2},\left&space;|&space;τ&space;-\frac{1}{2}&space;\right&space;|≥&space;\left&space;|&space;μ&space;-\frac{1}{2}&space;\right&space;|&space;and&space;\left&space;|&space;2τ&space;-1&space;\right&space;|+\left&space;|&space;2μ&space;-1&space;\right&space;|≤&space;1" target="_blank"><img title="τ eq \frac{1}{2},\left | τ -\frac{1}{2} \right |≥ \left | μ -\frac{1}{2} \right | and \left | 2τ -1 \right |+\left | 2μ -1 \right |≤ 1" src="https://latex.codecogs.com/gif.latex?τ&space; eq&space;\frac{1}{2},\left&space;|&space;τ&space;-\frac{1}{2}&space;\right&space;|≥&space;\left&space;|&space;μ&space;-\frac{1}{2}&space;\right&space;|&space;and&space;\left&space;|&space;2τ&space;-1&space;\right&space;|+\left&space;|&space;2μ&space;-1&space;\right&space;|≤&space;1" alt="">