Abstract:
We establish an inequality which involves a non-negative function defined on the vertices of a finite $m$-ary regular rooted tree. The inequality may be thought of as relating an interaction energy defined on the free vertices of the tree summed over automorphisms of the tree, to a product of sums of powers of the function over vertices at certain levels of the tree. Conjugate powers arise naturally in the inequality, indeed, H\"{o}lder's inequality is a key tool in the proof which uses induction on subgroups of the automorphism group of the tree.

Abstract:
We prove: Let be real numbers with . Then we have for all real numbers : with the best possible exponents and . The left-hand side of (0.1) with is a discrete version of an integral inequality due to E.A. Milne [1]. Moreover, we present a matrix analogue of (0.1).

Abstract:
We introduce a class of spaces of infinite matrices similar to the class of Lebesgue spaces Lp(T), 1≤p≤∞, and we prove matriceal versions of Hölder inequality.

Abstract:
We prove the following let , and be real numbers, and let be positive real numbers with . The inequalities hold for all real numbers if and only if and . Furthermore, we provide a matrix version. The first inequality (with and ) is a discrete counterpart of an integral inequality published by E. A. Milne in 1925.

Abstract:
In 1994, Talagrand showed a generalization of the celebrated KKL theorem. In this work, we prove that the converse of this generalization also holds. Namely, for any sequence of numbers $00$, it is possible to find a roughly balanced Boolean function $f$ such that $\textrm{Inf}_j[f] < a_j$ for every $1 \le j \le n$.

Abstract:
We prove the following let α , β , a > 0 , and b > 0 be real numbers, and let w j ( j = 1 , … , n ; n ≥ 2 ) be positive real numbers with w 1 + … + w n = 1 . The inequalities α ∑ j = 1 n w j / ( 1 p j a ) ≤ ∑ j = 1 n w j / ( 1 p j ) ∑ j = 1 n w j / ( 1 + p j ) ≤ β ∑ j = 1 n w j / ( 1 p j b ) hold for all real numbers p j ∈ [ 0 , 1 ) ( j = 1 , … , n ) if and only if α ≤ min ( 1 , a / 2 ) and β ≥ max ( 1 , ( 1 min 1 ≤ j ≤ n w j / 2 ) b ) . Furthermore, we provide a matrix version. The first inequality (with α = 1 and a = 2 ) is a discrete counterpart of an integral inequality published by E. A. Milne in 1925.