The Research Of Cusa-Yang-Chu Type Inequalities With Two Parameters

Let (p, q) ? c(p, q) be a function defined on R2, we are mainly devoted to finding possible relationships between parameters p and q such that the double inequality holds for x ∈ (0,π/2). As a consequence, we obtain a class of new Cusa-Yang-Chu inequalities with two sharp parameters p and q. While q=2/3P+1/15, this is the Cusa-Yang-Chu type inequality with the best relationships of p or q. Our main approaches consist of the monotonicity of Up (t)=1-tp/p, four Lemmas concern-ing the function’s monotonicity, and the power series expansions of trigonometric functions. More precisely, our paper is made up of five parts as follow:In chapter 1, we briefly recall the development of trigonometric functions, the latest research results and the starting point of this article.In chapter 2, we study the monotonicity of the following function Up(t) w. r. t. t and p, respectively, That is the starting point of our main research to the Cusa-Yang-Chu type in-equalities inequalities later. Based on some trigonometric functions’power series expansions with Bernoulli numbers, we give four Lemmas to judge the involving function’s monotonicity. How to clarify the monotonicity involving function under some conditions is the main aim od this chapter.In chapter 3, in term of the consequence of chapter 2, we further focus on the monotonicity of function Tp,q( x) in x in accordance with different region of p and q, where With the monotonicity of Tp,q(x) in hand, we state and prove the main results of our research in chapter 4. Let which is a weighted qth power mean function. By using the properties of Mq, we give the necessary and sufficient condition to the goal inequality while q=2/3p+1/15, and attain the best constants of the goal inequality. It is our main work in this chapter how to realize the necessary condition of the goal inequality while q 2/3P+1/15.Chapter 5 is devoted to applications of our main results. Here, we give a few of famous inequalities as the special cases of our main conclusions.