New Refinements of Cusa-Huygens inequality

Abstract

In the paper, we refine and extend Cusa-Huygens inequality by simple functions. In particular, we determine sharp bounds for sin(x)/x of the form (2 + cos(x))/3 − (2/3 − 2/π)Υ(x), where Υ(x) > 0 for x ∈ (0, π/2), Υ(0) = 0 and Υ(π/2) = 1, such that sin x/x and the proposed bounds coincide at x = 0 and x = π/2. The hierarchy of the obtained bounds is discussed, along with a graphical study. Also, alternative proofs of the main result are given.