Hölder type matrix inequalities of Pate, Blakley, and Roy extended to the inner product of Frobenius

Suppose m, n, and k   are positive integers, and let 〈·,·〉 be the standard inner product on the spaces RpRp, p>0p>0. Recently Pate has shown that if D   is an m×nm×n non-negative real matrix, and u and v   are non-negative unit vectors in RnRn and RmRm, respectively, then〈(DDt)kDu,v〉⩾〈Du,v〉2k+1,〈(DDt)kDu,v〉⩾〈Du,v〉2k+1,with equality if and only if 〈(DDt)kDu,v〉=0, or there exists α>0α>0 such that Du=αvDu=αv and Dtv=αuDtv=αu. This extends to non-symmetric non-square matrices a 1965 result of Blakley and Roy, and resolves a special case of a graph theoretic inequality conjectured by Sidorenko. We generalize the above, obtaining pure matrix inequalities involving the Frobenius inner product, 〈·,·〉f. In particular, we show that if k is a positive integer, and D, X, and Y   are non-negative matrices that are m×n,n×p, and m×pm×p, respectively, then∑i=1p‖xi‖‖yi‖2k〈D(DtD)kX,Y〉f⩾(〈DX,Y〉f)2k+1,where X   has columns x1,x2,…,xpx1,x2,…,xp, Y   has columns y1,y2,…,ypy1,y2,…,yp, and ‖·‖‖·‖ is the 2-norm. Necessary and sufficient conditions for equality are also given.