On strengthened weighted Carleman's inequality

Abstract

ON STRENGTHENED WEIGHTED CARLEMAN'S INEQUALITYALEKSANDRA CIZMESIJA, JOSIP PECARIC AND LARS-ERIK PERSSON In this paper we prove a new refinement of the weighted arithmetic-geometric mean inequality and apply this result in obtaining a sharpened version of the weighted Carleman's inequality.

Approximation formulas for the constant e and an improvement to a Carleman-type inequality

“…for x > 0, then b k > 0 for k ∈ N. In fact, Yang [44], Gylletberg and Yan [16], and Chen [5] presented the following recurrence relation for determining the coefficients b k in (3.11): 12) and then proved b k > 0 for k ∈ N; see also Lü et al [27]. Hu and Mortici [20] used an argument of Alzer and Berg [2] to derive an integral representation for b k , and then obtained some new properties of b k , including b k > 0 for k ∈ N. We remark that the recurrence relation of the coefficients b k given in [19…”

Padé approximant related to inequalities involving the constant e and a generalized Carleman-type inequality

“…In [4–20], some strengthened and generalized results of (1.1) and (1.2) have been given by estimating the weight coefficient . For example, Yang [17] proved that, for , and then used it to obtain the following strengthened Carleman inequality: …”

New Strengthened Carleman's Inequality and Hardy's Inequality

“…In [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16], some refined work on Carleman's inequality and Hardy's inequality had been gained. It is observing that in [3] the authors obtained the following inequalities…”