On some matrix mean inequalities via the log-convexity property

Abstract

Let $f:[0,1]\longrightarrow[0,+\infty)$ be a log-convex function, $0\leq \mu \leq \frac{1}{2}\leq\tau\leq 1$ and $m$ be a positive integer. Then by using the Jensen's inequality we prove that

\begin{eqnarray*}
\frac{\mu^m}{\tau}\Big{(}f^m(1)\nabla_\tau f^m(0)-f^m(\tau)\Big{)}+r_m\Big{(}f(0)^{\frac{m}{2}}-f^{\frac{m}{2}}(\tau)\Big{)}^2\leq (f(1)\nabla_\mu f(0))^m-f^m(\mu)
\end{eqnarray*}
and
\begin{eqnarray*}
\frac{(1-\tau)^m}{1-\mu}\Big{(}f^m(1)\nabla_\mu f^m(0)-f^m(\mu)\Big{)}+r^{\prime}_m\Big{(}f(1)^{\frac{m}{2}}-f^{\frac{m}{2}}(\mu)\Big{)}^2\leq (f(1)\nabla_\tau f(0))^m-f^m(\tau),
\end{eqnarray*}
where $\nabla_\mu$ stands for the weighted arithmetic mean and $r_m,r^{\prime}_m$ are two positive numbers. Futhermore, by selecting some appropriate log-convex functions, we obtain some new refinements of certain classical inequalities between the difference arithmetic-power, arithmetic-harmonic and arithmetic-geometric means for scalars and matrices as well as matrix norm and determinant. The importance of the results obtained is twofold; the results themselves and the way in which they extend many of known results in the literature.