On a Result of Levin and Stečkin

Abstract

The following inequality for0<p<1andan≥0originates from a study of Hardy, Littlewood, and Pólya:∑n=1∞((1/n)∑k=n∞ak)p≥cp∑n=1∞anp. Levin and Stečkin proved the previous inequality with the best constantcp=(p/(1-p))pfor0<p≤1/3. In this paper, we extend the result of Levin and Stečkin to0<p≤0.346.

On lower and upper bounds of matrices

“…Recently, the author [14] has extended the result of Levin and Stečkin to hold for 0 < p 0.346. Inequalities of type (1.5) with more general weights are also studied in [14], among which the following one for 0 < p < 1, α 1:…”

On Copson’s inequalities for $0< p<1$

“…The equivalence of the above two inequalities can be easily established following the discussions in [8,Sect. 1].…”

“…(1.4) It is noted in [12] that the constant p p in (1.4) may not be best possible and the best constant for 0 < p ≤ 1/3 was shown by Levin and Stečkin [13, Theorem 61] to be indeed (p/(1p)) p . In [8], it is shown that the constant (p/(1p)) p stays best possible for all 0 < p ≤ 0.346. It is further shown in [11] that the constant (p/(1p)) p is best possible when p = 0.35.…”

“…There exists an extensive and rich literature on extensions and generalizations of Copson's inequalities and Hardy's inequality (1.3) for p > 1. For recent developments in this direction, we refer the reader to the articles in [6][7][8][9][10][11] and the references therein. On the contrary, the case 0 < p < 1 is less known as can be seen by comparing inequalities (1.3) and (1.4).…”

“…On one hand, the constant in (1.3) is shown to be best possible for all p > 1. On the other hand, though it is known the best constant that makes inequality (1.4) valid is (p/(1p)) p when 0 < p ≤ 0.346, it is shown in [8] that the constant (p/(1p)) p fails to be best possible when 1/2 ≤ p < 1 and the best constant in these cases remains unknown.…”