A family of inequalities originating from coding of messages

AbstractThis paper presents 96 new inequalities with common structure, all elementary to state but many not elementary to prove. For example, if n is a positive integer and a=ł(a1,…,an) and b=(b1,…,bn) are arbitrary vectors in R+n=[0,∞)n, and ρ(mi,j) is the spectral radius of an n×n matrix with elements mi,j, then∑i,jmin((aiaj),(bibj))⩽∑i,jmin((aibj),(biaj)),[2mm]∑i,jmax((ai+aj),(bi+bj))⩾∑i,jmax((ai+bj),(bi+aj)),ρ(min((aiaj),(bibj)))⩽ρ(min((aibj),(biaj))),[-4mm]∑i,jmin((aiaj),(bibj))xixj⩽∑i,jmin((aibj),(biaj))xixj,forallrealxi,i=1,…,n,∫∫log[(f(x)+f(y))(g(x)+g(y))]dμ(x)dμ(y)⩽∫∫log[(f(x)+g(y))(g(x)+f(y))]dμ(x)dμ(y).The second inequality is obtained from the first inequality (which is due to G. Zbăganu [A new inequality with applications in measure and information theories, in: Proceedings of the Romanian Academy, Series A1 (1), 2000, pp. 15–19]) by replacing min with max, and × with +, and by reversing the direction of the inequality. The third inequality is obtained from the first by replacing the summation by the spectral radius. The fourth inequality is obtained from the first by taking each summand as a coefficient in a quadratic form. The fifth inequality is obtained from the first by replacing both outer summations by products, min by ×, × by +, and the non-negative vectors a, b by non-negative measurable functions f, g. The proofs of these inequalities are mysteriously diverse.A nice generalization of the first inequality is proved: Let ∗ be one of the four operations +, ×, min and max on an appropriate interval J of R. Let a,b∈Jn. Denote by a∗a the n×n matrix ai,j=ai∗aj. Then the matrix a∗a is more different from b∗b than a∗b is from b∗a. Precisely, if ∣A∣=∑1⩽i,j⩽n∣ai,j∣, then ∥a∗a−b∗b∥⩾∥a∗b−b∗a∥