Inequality for Burkholder's martingale transform

We find the sharp constant $C=C(\tau ,p,\mathbb{E}G,\mathbb{E}F)$ of the following inequality $\mathrm{\parallel }(G^2+{\tau }^2{F}^2{)}^{1\mathrm{/}2}{\mathrm{\parallel }}_p\le C\mathrm{\parallel }F{\mathrm{\parallel }}_p,$ where $G$ is the transform of a martingale $F$ under a predictable sequence $\epsilon$ with absolute value 1, $1<p<2$, and $\tau$ is any real number.