I think the result goes back to Polya, see <a href="http://www.ams.org/journals/tran/1960-095-02/S0002-9947-1960-0119247-6/">"Some theorems on stable processes"</a> by Blumenthal and Getoor. Another reference is Paul Levy "Sur une application de la d&eacute;riv&eacute;e d’ordre non entier au calcul des probabilit&eacute;s" page 1118 of <a href="http://gallica.bnf.fr/ark:/12148/bpt6k31295.r=comptes%20rendus%20academie%20des%20sciences%201923?rk=42918;4">CRAS 1923</a>.

It might be good to also give the gist of Levy's very simple argument: if the Fourier transform was positive then the exponent would be the log-moment generating function of a random variable $X$. But if $m>1$, the second derivative at the origin vanishes and thus also the variance of $X$. The positive function would be a delta function which is a contradiction.