Proof of a conjecture of Granath on optimal bounds of the Landau constants

We study the asymptotic expansion for the Landau constants GnGn, πGn∼ln(16N)+γ+∑k=1∞αkNkasn→∞, where N=n+1N=n+1, and γγ is Euler’s constant. We show that the signs of the coefficients αkαk demonstrate a periodic behavior such that (−1)l(l+1)2αl+1<0 for all ll. We further prove a conjecture of Granath which states that (−1)l(l+1)2εl(N)<0 for l=0,1,2,…l=0,1,2,… and n=0,1,2,…n=0,1,2,…, εl(N)εl(N) being the error due to truncation at the llth order term. Consequently, we also obtain the sharp bounds up to arbitrary orders of the formln(16N)+γ+∑k=1pαkNk<πGn