Inequalities and asymptotic expansions for the constants of Landau and Lebesgue

The constants of Landau and Lebesgue are defined, for all integers n⩾0n⩾0, in order, byGn=∑k=0n116k2kk2andLn=12π∫-ππsinn+12tsin12tdt,which play important roles in the theories of complex analysis and Fourier series, respectively. Diverse inequalities and approximations for these constants have been investigated and developed by many authors. Here, in this paper, we establish new asymptotic expansions for the constants GnGn and Ln/2Ln/2 of Landau and Lebesgue, respectively, in terms of the digamma and polygamma functions. Based on our expansion for the Landau constants GnGn, we present new bounds for the Landau constants GnGn in terms of the digamma and polygamma functions. We also establish inequalities for the Lebesgue constants Ln/2Ln/2, which are applied to derive an asymptotic expansion for Ln/2Ln/2 in terms of 1/(n+1)1/(n+1). Furthermore, by giving numerical calculations to be compared, among several developed asymptotic expansions for the constants GnGn and Ln/2Ln/2, it is shown that our expansions presented here would be best ones.