On inequalities and asymptotic expansions for the Landau constants

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

“…One is to obtain large-n asymptotic approximations for the constants, in a time period spanning from the early twentieth century [11,18] to very recently [7,12]. The other direction is to find sharper bounds of G n for all nonnegative integers n. Authors working on the sharper bounds includes Brutman [3] and Falaleev [8] (in terms of elementary functions), Alzer [2] and Cvijović and Klinowski [6] (using the digamma function), Zhao [20], Mortici [14] and Granath [9] (involving higher order terms), and Chen and Choi [5] and Chen [4] (digamma function and higher order terms). The list is by no means complete.…”

Multiple-correction and continued fraction approximation (II)

“…(1.12) Since then, many authors have made significant contributions to sharper the inequalities and the asymptotic expansions for G(n), see e.g. Alzer [2], Chen [9], Cvijović and Srivastava [13], Granath [22], Mortici [34], Nemes [36,37], Popa [38], Popa and Secelean [39], Zhao [46], Gavrea and M. Ivan [20], Chen and Choi [7,10,8], etc. To the best knowledge of the authors, the latest lower and upper bounds of G(n) along this research direction are due to Chen and Choi [8].…”

New bounds and asymptotic expansions for the constants of Landau and Lebesgue

“…The approximation of G n goes to two related directions. One is to find sharper bounds of G n for all positive integers n, and the other is to obtain large-n asymptotic approximations for the constants G n (see, e.g., [2,[4][5][6][7][8][9][10][11][12]14,17,21,22,[24][25][26][27][28][29]33,35]). Watson [33] proved the asymptotic formula …”

“…Formula (1.3) follows for example from Entry 25 in Chapter 12 in Ramanujans notebook [3], which gives a more general continued fraction formula for quotients of gamma functions, and which have several proofs published by different authors. Very recently, Granath [17] derived the asymptotic expansions for the Landau constants and related inequalities by using Brouncker's continued fraction formula. ; n 2 N;…”