An investigation of lower bounds on the quantitiesWn(t)=∫&OHgr;|&ohgr;|2n dVforn⩾1for the incompressible three-dimensional (3D) Euler equations has led us to consider a set of spatially averaged weighted “eigenvalues,”&lgr;S(n)(t)and&lgr;P(n)(t), of the strain matrixSand the Hessian matrix of the pressureP={p,ij}, respectively. It is shown that these obey the simple inequality,&lgr;˙S(n)+f(&thgr;n)(&lgr;S(n))2+&lgr;P(n)⩾0, wheref(&thgr;n)=1−tan2 &thgr;n.The&thgr;nare spatially averaged weighted angles between the vorticity vector&ohgr;and the vortex stretching vector&sgr;=&ohgr;⋅&bnabla;u. The weighting in the averaging process highlights regions of large vorticity. This is the angle considered by Tsinober, Kit, and Dracos in their analysis of data from turbulent grid flow experiments in which they noted a tendency toward alignment between&ohgr;and&sgr;. The Burgers vortex turns out to be a sharp solution of this inequality with a corresponding angle&thgr;n=0, giving rise to exponential growth inWn.Some special solutions for cases where&thgr;nmoves between&thgr;n=0and&thgr;n=&pgr;/2are displayed. The work of Ohkitani and Kishiba on the alignment in 3-D Euler flows between&ohgr;and the third eigenvector ofPat maximum enstrophy is also particularly relevant and is applied to the modified pressure matrixQ={p,ij−3&dgr;ijp,ii}in the limitn→∞. The finite time blow-up problem is discussed in this context. In an Appendix it is shown that an identical inequality holds for the barotropic compressible Euler equations where&zgr;=&ohgr;/&rgr; andWn(t)=∫&OHgr;&rgr;|&zgr;|2n dVreplace&ohgr;andWn,respectively. ©1997 American Institute of Physics.