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P´olya, Inequalities (second edition), Cambridge University Press, Cambridge, 1951.  T. Popoviciu, Asupra mediilor aritmetice si medie geometrice (Romanian), Gaz. Mat. Bucharest 40 (1934), 155-160.  S. Rabinowitz, Index to Mathematical Problems 1980-1984, Mathpro Press, Westford (Massachusetts), 1992.
So f is convex if and only if Hy · y > 0 for all nonzero y, that is, if H is positive definite. The bad news about this criterion is that determining whether a matrix is positive definite is not a priori an easy task: one cannot check M x · x ≥ 0 for every vector, so it seems one must compute all of the eigenvalues of M , which can be quite a headache. The good news is that there is a very nice criterion for positive definiteness of a symmetric matrix, due to Sylvester, that saves a lot of work.
IMO 1968/2) Prove that for all real numbers x1 , x2 , y1 , y2 , z1 , z2 with x1 , x2 > 0 and x1 y1 > z12 , x2 y2 > z2 , the inequality 1 1 8 ≤ + 2 2 (x1 + x2 )(y1 + y2 ) − (z1 + z2 ) x1 y1 − z1 x2 y2 − z22 is satisfied, and determine when equality holds. (Yes, you really can apply the material of this section to the IMO! 6 Constrained extrema and Lagrange multipliers In the multivariable realm, a new phenomenon emerges that we did not have to consider in the one-dimensional case: sometimes we are asked to prove an inequality in the case where the variables satisfy some constraint.