Asymptotics of Landau constants with optimal error bounds

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We study the asymptotic expansion for the Landau constants Gₙ πGₙ N + γ+4 2 + ∑ₛ₌₁∞ β₂ₛN²ˢ, n→ ∞, where N=n+3/4, γ=0.5772 is Euler's constant, and (-1)ˢ⁺¹β₂ₛ are positive rational numbers, given explicitly in an iterative manner. We show that the error due to truncation is bounded in absolute value by, and of the same sign as, the first neglected term for all nonnegative n. Consequently, we obtain optimal sharp bounds up to arbitrary orders of the form N+γ+4 2+∑ₛ₌₁²ᵐβ₂ₛN²ˢ< πGₙ < N+γ+4 2+∑ₛ₌₁²ᵏ⁻¹β₂ₛN²ˢ for all n=0,1,2,, m=1,2,, and k=1,2,. The results are proved by approximating the coefficients β₂ₛ with the Gauss hypergeometric functions involved, and by using the second order difference equation satisfied by Gₙ, as well as an integral representation of the constants ρₖ=(-1)ᵏ⁺¹β₂ₖ/(2k-1)!.