Inequalities between operator means based on the Mond-Pečarić method

As a continuation of our previous paper (Complementary inequalities to inequalities of Jensen and Ando based on Mond- Pecaric method, Linear Alg. Appl., 318 (2000), 87-107) we show further general complementary inequalities to operator inequalities on a positive linear map associated with two operator means. Let $A$ and $B$ be positive invertible operators on a Hilbert space $H$, $sigma, tau$ two operator means and $Phi $ a positive linear map. If $alpha >0$ is a given real constant, then real constants $beta $ and $beta ^0$ such that begin{; ; align*}; ; Phi (A sigma B) & geq alpha Phi (A) tau Phi (B) +beta Phi (A), \ Phi (A sigma B) & geq alpha Phi (A) tau Phi (B) +beta ^0 Phi (B) end{; ; align*}; ; are determined. As applications, we obtain general complementary estimates for the results by Aujla-Vasudeva cite{; ; AV2}; ; and J.I.Fujii cite{; ; JIF}; ; on the Hadamard product and operator means, and Mond-Pev{; ; c}; ; ari'{; ; c}; ; -v{; ; S}; ; unde-Varov{; ; s}; ; anec cite{; ; MPSV}; ; on several mixed operator means.