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 $RW_\lambda$ on a Galton-Watson tree without leaves is strictly decreasing for $\lambda\leq \frac{m_1}{1160},$ where $m_1$ is minimal degree of the Galton-Watson tree. And A\"{\i}d\'{e}kon \cite{EA2013} improved this result to $\lambda\leq \frac{1}{2}.$ In this paper, we prove that for the $RW_{\lambda}$ on a Galton-Watson tree without leaves, its speed is strictly decreasing for $\lambda\in \left[0,\frac{m_1}{1+\sqrt{1-\frac{1}{m_1}}}\right]$ when $m_1\geq 2;$ and we owe the proof to A\"{\i}d\'{e}kon \cite{EA2013}.