arXiv subjects

J. S. Aujla

Publications and source records attributed to J. S. Aujla.

3 recordsLinked to original sources

arXiv2016

A New Class of Operator Monotone Functions via Operator Means

In this paper, we obtain a new class of functions, which is developed via the Hermite--Hadamard inequality for convex functions. The well-known one-one correspondence between the class of operator monotone functions and operator connections declares that the obtained class represents the weighted logarithmic means. We shall also consider weighted identric mean and some relationships between various operator means. Among many things, we extended the weighted arithmetic--geometric operator mean inequality as $A\#_{t}B\leq A\ell_t B\leq \frac{1}{2}(A\#_{t}B + A\nabla_{t} B)\le A\nabla_tB$ and $A\#_{t}B\leq A\mathcal{I}_t B\leq A\nabla_{t} B$ involving the considered operator means.

math.FA↗

arXiv2012

A General Double Inequality Related to Operator Means and Positive Linear Maps

Let $A,B\in \mathbb{B}(\mathscr{H})$ be such that $0<b_{1}I \leq A \leq a_{1}I$ and $0<b_{2}I \leq B \leq a_{2}I$ for some scalars $0<b_{i}< a_{i},\;\; i=1,2$ and $Φ:\mathbb{B}(\mathscr{H})\rightarrow\mathbb{B}(\mathscr{K})$ be a positive linear map. We show that for any operator mean $σ$ with the representing function $f$, the double inequality $$ ω^{1-α}(Φ(A)#_αΦ(B))\le (ωΦ(A))\nabla_αΦ(B)\leq \fracαμΦ(AσB) $$ holds, where $μ=\frac{a_{1}b_{1}(f(b_{2}a_{1}^{-1})-f(a_{2}b_{1}^{-1}))}{b_{1}b_{2}-a_{1}a_{2}}, $ $ν=\frac{a_{1}a_{2}f(b_{2}a_{1}^{-1})-b_{1}b_{2}f(a_{2}b_{1}^{-1})}{a_{1}a_{2}-b_{1}b_{2}}, $ $ω=\frac{αν}{(1-α)μ}$ and $#_α$ ($\nabla_α$, resp.) is the weighted geometric (arithmetic, resp.) mean for $α\in (0,1)$. As applications, we present several generalized operator inequalities including Diaz--Metcalf and reverse Ando type inequalities. We also give some related inequalities involving Hadamard product and operator means.

math.FA↗

arXiv2011

Non-commutative Callebaut inequality

We present an operator version of the Callebaut inequality involving the interpolation paths and apply it to the weighted operator geometric means. We also establish a matrix version of the Callebaut inequality and as a consequence obtain an inequality including the Hadamard product of matrices.

math.FA↗