Absence of long-range order in one-dimensional spin systems

For a one-dimensional Ising model with interaction energy $$Eleft{ mu right} = - sumlimits_{1 leqslant i< j leqslant N} {J(j - i)} mu _iota mu _j left[ {J(k) geqslant 0,mu _iota = pm 1} right]$$ it is proved that there is no long-range order at any temperature when $$S_N = sumlimits_{k = 1}^N {kJleft( k right) = o} left( {[log N]^{{1 mathord{left/ {vphantom {1 2}} right. kern-nulldelimiterspace} 2}} } right)$$ The same result is shown to hold for the corresponding plane rotator model when $$S_N = oleft( {left[ {{{log N} mathord{left/ {vphantom {{log N} {log log N}}} right. kern-nulldelimiterspace} {log log N}}} right]} right)$$