Some Inequalities on T 3 Tree

The article proves several inequalities derived from nodal multiplication on T3 tree. The proved inequalities are helpful to estimate certain quantities related with the T3 tree as well as examples of proving an inequality embedded with the floor functions.


Introduction
The T 3 tree, which first appeared in [1] and was formerly introduced in [2], is a perfect complete binary tree that is considered to be a new tool to study integers.
The tree can reveal many new properties of integers such as the symmetric properties discovered in [3] and [4], the genetic property found in [5], and other properties introduced in [6] and [7].The tree also shows its big potentiality in factorization of big semiprimes, as seen in [8] and [9].A recent study found several inequalities related with estimation of multiplication on the tree.This article introduces the main results.

Preliminaries
This section lists for later sections the necessary preliminaries, which include definitions, notations and lemmas.

Definitions and Notations
Symbol T 3 is the T 3 tree that was introduced in [1] and [2] and symbol ( ) sition also by default begins at zero.Symbol x     is the floor function, an integer function of real number x that satisfies inequality ⇒ means conclusion B can be derived from condition A.
For convenience in deduction of a formula, comments are inserted by symbols that express their related mathematical foundations.For example, the following deduction ( ) ( ) means that, lemma (L) supports the step from B to C, and proposition (P) supports the step from C to D.

Lemmas
Lemma 1. (See in [1]) T 3 tree has the following fundamental properties.(P1).Every node is an odd integer and every odd integer bigger than 1 must be on the T 3 tree.Odd integer N with ).On level k with 0,1, k =  , there are 2 k nodes starting by whereas, if Lemma 2. (See in [10]) Let α and x be a positive real numbers; then it holds ( ) Particularly, if α is a positive integer, say n α = , then it yields ( )

Main Results with Proofs
Proposition 1.For positive integer k and real number then when and when and thus Similarly, when , , N α be a node of T 3 and n be an integer with 0 m n ≤ ≤ ; then it holds Thus for arbitrary integer Proof.Considering that ( ) and Consider in ( 7) it knows (3) and ( 4) hold and consequently ( 5) and ( 6) hold.
and thus for arbitrary integer Consequently, it yields and Proof.The condition that ( ) which says (11) holds.
Likewise, by definition of the floor function and referring to the Inequalities (5), ( 6) and (10), it yields , 2 which is the (12).
Similarly, the Inequalities (10) and the definition of the floor function lead to which is just the (13). and Proof.By Lemma 2 and Proposition 1, it holds when 0 s m ≤ ≤

Conclusion
The T 3 tree is emerging its value in studying integers.A lot of equations and inequalities will be research objectives.Since most of the inequalities on the T 3 tree are in the form of floor functions, their proofs are often skillful.The inequalities proved in this article are not only quite useful for knowing the T 3 tree, but also excellent samples for proving inequalities with the floor functions.Hope it helpful to the readers of interests.

N
is by default the node at position j on level k of T 3 , where 0 k ≥ and 0 2 1 k j ≤ ≤ − .Number of the level by default begins at zero and index of the po-X.B. Wang DOI: 10.4236/apm.2018.88043712 Advances in Pure Mathematics

Proposition 5 .
Let ( ) , m N α and ( ) , n N β be nodes of T 3 with 0 m n ≤ ≤ ; then it holds for integer 0 s m ≤ ≤ ( ). Multiplication of arbitrary two nodes of T 3 , say ( )