}.
\]
Since $(q^{-1} \sharp p) q (q^{-1} \sharp p) = p$, we have
\begin{eqnarray*}
q \natural_{1-t} p &=& (q^{-1} \sharp p)^{1-t} q (q^{-1} \sharp p)^{1-t} = (q^{-1} \sharp p)^{-t}[(q^{-1} \sharp p) q (q^{-1} \sharp p)](q^{-1} \sharp p)^{-t} \\
&= &
(p^{-1} \sharp q)^t p (p^{-1} \sharp q)^t = p \natural_t q.
\end{eqnarray*}
\medskip
\item By direct computation, we have
\begin{align*}
p^{-1} \sharp (p \natural_t q) &= p^{-1/2}\left(p^{1/2}(p \natural_t q)p^{1/2}\right)^{1/2}p^{-1/2}\\
&= p^{-1/2}\left(p^{1/2}(p^{-1} \sharp q)^{t} p (p^{-1} \sharp q)^{t}p^{1/2}\right)^{1/2}p^{-1/2}\\
&= p^{-1/2}\left([p^{1/2}(p^{-1} \sharp q)^{t} p^{1/2}][p^{1/2} (p^{-1} \sharp q)^{t}p^{1/2}\right])^{1/2}p^{-1/2}\\
&= p^{-1/2}\left(p^{1/2}(p^{-1} \sharp q)^{t} p^{1/2}\right)p^{-1/2}\\
&= (p^{-1} \sharp q)^{t}.
\end{align*}
Then $(q \natural_t p)^{-1} \sharp q= q\sharp (q \natural_t p)^{-1} = (q^{-1} \sharp (q \natural_t p))^{-1} = [(q^{-1} \sharp p)^t]^{-1} = (p^{-1} \sharp q)^t$.
\medskip
\item By (2), we have $c_t = (p^{-1} \sharp q)^t$ and thus $c_t p c_t = p \natural_t q$ by definition.
Since $c_t^{-1} = (q^{-1} \sharp p)^t$, we have ${c_t^{-1}qc_t^{-1}} = q \natural_t p$.
\medskip
\item 
Let $c_i = (p^{-1} \sharp q)^i$ for all $i \in [0, 1]$. Then $p\natural_i q = c_i pc_i$ by (3). It follows that 
\begin{equation}\label{c_{i+j}}
c_{i+j} = c_ic_j,
\quad c_{ij} = c_i^j\qquad \mbox{for all } j \in [0, 1],\quad 0 \le i+j \le1.
\end{equation}
We first assume $r \le s$. Since $c_rpc_r, c_spc_s \in P$ and 
\[
c_spc_s = c_{s-r}c_rpc_rc_{s-r}= c_{s-r}[(c_rpc_r)^{-1} ]^{-1}c_{s-r},
\]
we have $(c_rpc_r)^{-1} \sharp (c_spc_s)  = c_{s-r}$ by Lemma \ref{lem:equ}.
Then
\begin{align*}
(p \natural_r q) \natural_t (p \natural_s q) &= (c_rpc_r) \natural_t (c_spc_s)\\
&= [(c_rpc_r)^{-1} \sharp (c_spc_s)]^t (c_rpc_r)[(c_rpc_r)^{-1} \sharp (c_spc_s)]^t \\
&= c_{s-r}^t (c_rpc_r)c_{s-r}^t \\
&= c_{t(s-r)} (c_rpc_r)c_{t(s-r)}  \\
&= c_{t(s-r)+r}pc_{t(s-r)+r}  \\
&= c_{(1-t)r+ts} p c_{(1-t)r+ts} \\
&= p \natural_{(1-t)r+ts} q.
\end{align*}
If $r \ge s$, then by (1) we have 
\[
(p \natural_r q) \natural_t (p \natural_s q) = (p \natural_s q) \natural_{1-t} (p \natural_r q) = p \natural_{[1-(1-t)]s + (1-t)r} q = p \natural_{(1-t)r+ts} q.\qedhere
\]
\end{enumerate}
\end{proof}


\begin{proposition}\label{natural-r+s}
Let $p, q \in P$. Then
\begin{equation}\label{eq:natural-r+s}
p \natural_{r+s} q = (p \natural_r q) \natural_{\frac {s}{1-r}} q, \qquad \mbox{for all }\, 0 \le r, s, r+s \le 1.
\end{equation}
\end{proposition}



\begin{proof}
According to Proposition \ref{T:t-natural-G} (4), we have (by assigning $s = 1$ and $t = \frac{s}{1-r}$, where the first $s$ is the $s$ in Proposition \ref{T:t-natural-G} (4) and the second $s$ is the $s$ in this proposition), 
\[
 (p \natural_r q) \natural_{\frac {s}{1-r}} q =  (p \natural_r q) \natural_t (p \natural_1 q) = p \natural_{(1-t)r+t} q = p \natural_{r+(1-r)t} q = p \natural_{r+s} q. \qedhere
\]
\end{proof}


The following figures illustrate both sides of Proposition~\ref{T:t-natural-G} (4).

\begin{minipage}{\linewidth}
      \centering
      \begin{minipage}{0.45\linewidth}
          \begin{figure}[H]
              \includegraphics[width=\linewidth]{SpectralMean5.pdf}
              \caption{$(p \natural_r q) \natural_t (p \natural_s q)$}
          \end{figure}
      \end{minipage}
      \hspace{0.05\linewidth}
      \begin{minipage}{0.45\linewidth}
          \begin{figure}[H]
              \includegraphics[width=\linewidth]{SpectralMean6.pdf}
              \caption{$p \natural_{(1-t)r+ts} q$}
          \end{figure}
      \end{minipage}
  \end{minipage}

\vspace{.5cm}



Proposition \ref{sharp-r+s} and Proposition \ref{natural-r+s} provide  information about how the curves behave. Proposition \ref{sharp-r+s} is a consequence of the fact that the $t$-geometric mean joining $p$ and $q$ is a geodesic.


Finally, we extend Theorem \ref{T:log-majorization} to symmetric space by using Theorem \ref{T:log-majorization} itself and applying Kostant's theorem as in the proof of  \cite [Theorem 3.5]{LLT14}.

\begin{theorem} \label{T:sharp-natural-G}
Let $p, q \in P$. Then
\[
	{p \sharp_t q \prec_G p \natural_t q}, \quad  \mbox{for all } \, 0 \le t \le 1.
\]
\end{theorem}


\begin{proof}

A result of Kostant \cite[Theorem 3.1]{Ko73} asserts that for any given
$f, g \in G$, {$f\prec_G g$} if and only if $|\pi(f)|\le |\pi(g)|$ for any finite dimensional representation $\pi$ of $G$, where $|\pi(g)|$ is the spectral radius of $\pi(g)$.
Let $\pi: G\to {\rm GL}(V)$ be any finite dimensional representation of $G$.
There exists an inner product on $V$ such that $\pi(z)$ is positive definite