Exponents for three-dimensional simultaneous Diophantine approximations

Let $Θ= (θ_1,θ_2,θ_3)\in \mathbb{R}^3$. Suppose that $1,θ_1,θ_2,θ_3$ are linearly independent over $\mathbb{Z}$. For Diophantine exponents $α(Θ) = \sup \{γ>0:\,\,\, \limsup_{t\to +\infty} t^γψ_Θ(t) <+\infty \} ,$ $β(Θ) = \sup \{γ>0:\,\,\, \liminf_{t\to +\infty} t^γψ_Θ(t) <+\infty\}$ we prove $β(Θ) \ge {1/2} ({α(Θ)}/{1-α(Θ)} +\sqrt{{α(Θ)}/{1-α(Θ)})^2 +{4α(Θ)}/{1-α(Θ)}}) α(Θ)$

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