## Why $2R\sigma\sqrt{T+logT+1}=\tilde{\mathcal{O}}(\sigma\sqrt{T})$?

On page 17 on the paper Online Learning with Predictable Sequences, we find a regret of an algorithm equal to

$$\text{Reg}_T=\frac{R^2}{\eta}+\frac{\eta}{2}\sigma^2(T+logT+1)$$ where $$T$$ is the entire time, $$\eta$$ is learning rate, $$R$$ is constant, and $$\sigma^2$$ is the variance of data. The above using arithmetic mean-geometric mean inequality can be lower bounded by

$$2R\sigma\sqrt{T+logT+1}\leq \frac{R^2}{\eta}+\frac{\eta}{2}\sigma^2(T+logT+1)$$

why the left hand side is $$\tilde{\mathcal{O}}(\sigma\sqrt{T})$$?