On Some Inequality for the Landau Constants

Abstract

We improve several results recently established by Dejun Zhao for the Landau's constants.

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 …”

Multiple-correction and faster approximation

“…See e.g. Alzer [1], Chen [8], Cvijović and Srivastava [14], Granath [21], Mortici [30], Nemes [32,33], Popa [34], Popa and Secelean [35], Zhao [41], Gavrea and M. Ivan [18], Chen and Choi [5,7], etc. To the best knowledge of authors, the latest upper bound is due to Chen [9], who proved G(n) < c 0 + 1 π ψ n + 5 4 + 1 64(n + 3 4 ) , (n ≥ 0).…”