arXiv subjects
Publications and source records attributed to Ali Morassaei.
Let $ f:(0,\infty)\rightarrow \Bbb{R} $ be a completely monotonic function. In this paper, we present some properties of this functions and several new classes of completely monotonic functions. We also give some special functions such that its have completely monotonic condition.
In this paper, we prove that every continuous $h$-mid-convex with suitable conditions on $h$ is $h$-convex function. Also, we extend Ostrowski theorem, Blumberg-Sierpinski theorem, Bernstein-Doetsch theorem, Mehdi theorem.
In this paper, we give the Alzer inequality for Hilbert space operators as follows: Let $A, B$ be two selfadjoint operators on a Hilbert space $\mathcal H$ such that $0 < A, B \le \frac{1}{2}I$, where $I$ is identity operator on $\mathcal H$. Also, assume that $A \nabla_λB:=(1-λ)A+λB$ and $A \sharp_λB:=A^{\frac{1}{2}}\left(A^{-\frac{1}{2}}BA^{-\frac{1}{2}}\right)^λA^{\frac{1}{2}}$ are arithmetic and geometric means of $A, B$, respectively, where $0 < λ< 1$. We show that if $A$ and $B$ are commuting, then $$ B'~\nabla_λ~A' - B'~\sharp_λ~A' \le A~\nabla_λ~B - A~\sharp_λ~B\,, $$ where $A':=I-A$, $B':=I-B$ and $0 < λ\le \frac{1}{2}$. Also, we state an open problem for an extension of Alzer inequality.
In this paper, we establish some reverses of the operator entropy inequalities under certain conditions by using the Mond-Pečarić method. In particular, we present {\tiny \begin{align*} f&\left[\int_T(A_s\natural_{p+1}B_s)dμ(s)+t_0\left(I_{\mathscr H}-\int_TA_s\natural_pB_sdμ(s)\right)\right]-γ_ff(t_0)\left(I_{\mathscr H}-\int_TA_s\natural_pB_sdμ(s)\right)\nonumber\\ &\le γ_f\widetilde{S}_p^f(\mathbf{A}|\mathbf{B})\,, \end{align*}} where $T$ is a locally compact Hausdorff space and $μ$ is a Radon measure on $T$, $0<m A_s \leq B_s \leq M A_s\,\,(s\in T)$ for some positive real numbers $m, M$ such that $m<1<M$, $\int_TA_s=\int_TB_s=I_{\mathscr H}$, $f: (0,\infty) \to [0,\infty)$ be operator concave, $γ_f=\max\left\{\frac{f(t)}{μ_f t+ν_f}: m\leq t\leq M,μ_f=\frac{f(M)-f(m)}{M-m}, ν_f=\frac{Mf(m)-mf(M)}{M-m}\right\}$, $t_0\in[m,M]$, $p\in[0,1]$, and $$ \widetilde{S}_p^f(\mathbf{A}|\mathbf{B})=\int_TA_s^{\frac{1}{2}}\left(A_s^{-\frac{1}{2}}B_sA_s^{-\frac{1}{2}}\right)^p f\left(A_s^{-\frac{1}{2}}B_sA_s^{-\frac{1}{2}}\right)A_s^{\frac{1}{2}}dμ(s)\,. $$
In this paper, we employ the Mond--Pečarić method to establish some reverses of the operator Bellman inequality under certain conditions. In particular, we show \begin{equation*} δI_{\mathscr K}+\sum_{j=1}^nω_jΦ_j\left((I_{\mathscr H}-A_j)^{p}\right)\ge \left(\sum_{j=1}^nω_jΦ_j(I_{\mathscr H}-A_j)\right)^{p} \,, \end{equation*} where $A_j\,\,(1\leq j\leq n)$ are self-adjoint contraction operators with $0\leq mI_{\mathscr H}\le A_j \le MI_{\mathscr H}$, $Φ_j$ are unital positive linear maps on ${\mathbb B}({\mathscr H})$, $ω_j\in\mathbb R_+ \,\,(1\leq j\leq n)$, $0 < p < 1$ and $δ=(1-p)\left(\frac{1}{p}\frac{(1-m)^p-(1-M)^p}{M-m}\right)^{\frac{p}{p-1}}+\frac{(1-M)(1-m)^p-(1-m)(1-M)^p}{M-m}$ . We also present some refinements of the operator Bellman inequality.