Monotonicity of the speed for biased random walk on Galton-Watson tree

Ben Arous, Fribergh and Sidoravicius \cite{GAV2014} proved that speed of biased random walk $R{W}_{\lambda }$ on a Galton-Watson tree without leaves is strictly decreasing for $\lambda \le \frac{m_1}{1160},$ where $m_1$ is minimal degree of the Galton-Watson tree. And A"{\i}d'{e}kon \cite{EA2013} improved this result to $\lambda \le \frac{1}{2}\mathrm{.}$ In this paper, we prove that for the $R{W}_{\lambda }$ on a Galton-Watson tree without leaves, its speed is strictly decreasing for $\lambda \in [0,\frac{m_1}{1+\sqrt{1-\frac{1}{m_1}}}]$ when $m_1\ge 2;$ and we owe the proof to A"{\i}d'{e}kon \cite{EA2013}.