Inequalities for the constants of Landau and Lebesgue

The constants of Landau and Lebesgue are defined for all integers n⩾0 byGn=∑k=0n116k2kk2 and Ln=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.