Inequalities for the constants of Landau and Lebesgue

摘要

The constants of Landau and Lebesgue are defined for all integers n⩾0 byGn=∑k=0n116k2kk2andLn=12π∫−ππsin((n+12)t)sin(12t)dt,respectively. We establish sharp inequalities for Gn and Ln/2 in terms of the logarithmic derivative of the gamma function. Further, we prove that the sequence (ΔGn) is completely monotonic, we provide best possible upper and lower bounds for the ratios (Gn−1+Gn+1)/Gn and (L(n−1)/2+L(n+1)/2)/Ln/2, and we present sharp bounds for Ln/2/Gn and Ln/2−Gn.