  Let $f$ be an operator convex function on $(0,\infty)$, and $\Phi$ be a
unital positive linear maps on $B(H)$.
  we give a complementary inequality to
  Davis-Choi-Jensen's inequality as follows \begin{equation*} f(\Phi(A))\geq
\frac{4R(A,B)}{(1+R(A,B))^2}\Phi(f(A)), \end{equation*} where $R(A,
B)=\max\{r(A^{-1}B) ,r(B^{-1}A)\}$ and $r(A)$ is the spectral radius of $A$. We
investigate the complementary inequalities related to the operator power means
and the Karcher means via unital positive linear maps,
  and obtain the following result: If $A_{1}$, $A_{2}$,\dots, $A_{n}$, are
positive definite operators in $B(H)$, and $0<m_i\leq A_i\leq M_i$, then
\begin{equation*} \Lambda( \omega;\Phi(\mathbb{A}))\geq\Phi(\Lambda( \omega;
\mathbb{A}))\geq \frac{4\hbar}{(1+\hbar)^2}~\Lambda( \omega;\Phi(\mathbb{A})),
\end{equation*} where $\hbar= \max\limits_{1\leq i\leq n} \frac{M_i}{m_i}$.
Finally, we prove that if $G(A_1,\dots,A_n)$ is the generalized geometric mean
defined by Ando-Li-Mathias for $n$ positive definite operators, then
  \begin{align*}
\Phi(G(A_1,\dots,A_n))\geq\left(\frac{2h^\frac{1}{2}}{1+h}\right)^{n-1}G(\Phi(A_1),\dots,\Phi(A_n)),
\end{align*} where $h=\max\limits_{1\leq i,j\leq n} R(A_i, A_j)$.
