Crossed ladders and power means

Crossed ladders and power means ∗

Given to positive numbers a and b, we show how we easily can construct the power mean Pk(a, b) of order k for the cases k = −2, −1, −1/2, 0, 1/2, 1 and 2. This is done by observing that the power means correspond to certain distances in the crossed ladders problem. The so-called crossed ladders problem, of unknown origin, has been discussed in the literature at least since 1895 (see [3, p. 62-6...

Optimal inequalities for the power, harmonic and logarithmic means

For all $a,b>0$, the following two optimal inequalities are presented: $H^{alpha}(a,b)L^{1-alpha}(a,b)geq M_{frac{1-4alpha}{3}}(a,b)$ for $alphain[frac{1}{4},1)$, and $ H^{alpha}(a,b)L^{1-alpha}(a,b)leq M_{frac{1-4alpha}{3}}(a,b)$ for $alphain(0,frac{3sqrt{5}-5}{40}]$. Here, $H(a,b)$, $L(a,b)$, and $M_p(a,b)$ denote the harmonic, logarithmic, and power means of order $p$ of two positive numbers...

The optimal rubbling number of ladders, prisms and Möbius-ladders

A pebbling move on a graph removes two pebbles at a vertex and adds one pebble at an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed. In this new move, one pebble each is removed at vertices v and w adjacent to a vertex u, and an extra pebble is added at vertex u. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that v...

Optimal Inequalities for Power Means

Weighted Power Means of Operators

Let σ1,σ2, · · · ,σn be positive real numbers satisfying n ∑ i=1 σi = 1 and A1,A2, · · · ,An be positive operators. Let Mσ,γ(A) = ( n ∑ i=1 σiA γ i ) 1/γ, γ > 0. It has been shown that lim γ→0+ Mσ,γ exists. Some known inequalities have also been generalized.