This paper gives another new proof method of analytic inequality involving n variables. As its applications, we give proof of some known-well inequalities and prove five new analytic inequalities. 1. monotonicity on special variables Throughout the paper R denotes the set of real numbers and R+ denotes the set of strictly positive real numbers. Let n ≥ 2, n ∈ N. The arithmetic mean A(x) and the...

1. We just write out (i). Recall that X1 ∪ X2 = {x : x ∈ X1 or x ∈ X2}. Hence x ∈ X1 =⇒ x ∈ X1 ∪ X2, so that X1 ⊆ X1 ∪ X2. By symmetry, X1 ⊆ X1 ∪ X2. Finally, suppose that Y is a set such that X1 ⊆ Y and X2 ⊆ Y . Let x ∈ X1 ∪X2. Then x ∈ X1 or x ∈ X2. If x ∈ X1, then x ∈ Y since X1 ⊆ Y . Similarly, if x ∈ X2, then x ∈ Y . Thus in either case x ∈ Y , so that X1 ∪X2 ⊆ Y . 2. (i) Injective, not su...

The main purpose of this paper is to derive a new (p, q)-atomic decomposition on the multi-parameter Hardy space H(X1 ×X2) for 0 < p0 < p ≤ 1 for some p0 and all 1 < q < ∞, where X1 × X2 is the product of two spaces of homogeneous type in the sense of Coifman and Weiss. This decomposition converges in both L(X1 × X2) (for 1 < q < ∞) and Hardy space H(X1 × X2) (for 0 < p ≤ 1). As an application,...

= 1 or −1 according as j is or is not a quadratic residue mod p. A multivariable generalization of Theorem 1.1 follows. Theorem 1.1 is a special case of Theorem 1.2 with x3 = · · · = xp = 0. Theorem 1.2. Let p be an odd prime and p = (−1)(p−1)/2p. Then there exist integer polynomials R(x1, x2, . . . , xp) and S(x1, x2, . . . , xp) such that 4 · det(circ(x1, x2, . . . , xp)) x1 + x2 + · · ·+ xp ...

From now on X denotes a set and S denotes a family of subsets of X. Now we state the propositions: (1) Let us consider sets X1, X2, a family S1 of subsets of X1, and a family S2 of subsets of X2. Then {a×b, where a is an element of S1, b is an element of S2 : a ∈ S1 and b ∈ S2} = {s, where s is a subset of X1 ×X2 : there exist sets a, b such that a ∈ S1 and b ∈ S2 and s = a × b}. Proof: {a × b,...

This is the Cayley surface when N = 3. The next few are as follows. x4 = x1x3 + 1 2 x2 2 − x1 x2 + 1 4 x1 4 x5 = x1x4 + x2x3 − x1 x3 − x1x2 2 + x1 x2 − 1 5 x1 5 x6 = x1x5 + x2x4 + 1 2 x3 2 − x1 x4 − 2x1x2x3 − 1 3 x2 3 + x1 x3 + 3 2 x1 x2 2 − x1 x2 + 1 6 x1 . Since the first term in (1) is −xN and this is the only occurrence of this variable, these hypersurfaces are polynomial graphs over the re...