An Application of Λ-method on Inequalities of Shafer-fink’s Type

In this article λ-method of Mitrinović-Vasić [1] is applied to improve the upper bound for the arc sin function of L. Zhu [4]. 1. Inequalities of Shafer-Fink’s type D. S. Mitrinović in [1] considered the lower bound of the arc sin function, which belongs to R. E. Shafer. Namely, the following statement is true. Theorem 1.1 For 0 ≤ x ≤ 1 the following inequalities are true: 3x 2 + √ 1− x2 ≤ 6( √ 1 + x− √ 1− x) 4 + √ 1 + x+ √ 1− x ≤ arc sinx . (1) A.M. Fink proved the following statement in [2] . Theorem 1.2 For 0 ≤ x ≤ 1 the following inequalities are true: 3x 2 + √ 1− x2 ≤ arc sinx ≤ πx 2 + √ 1− x2 . (2) B. J. Malešević proved the following statement in [3]. Theorem 1.3 For 0 ≤ x ≤ 1 the following inequalities are true: 3x 2 + √ 1− x2 ≤ arc sinx ≤ π π − 2 x 2 π − 2 + √ 1− x2 ≤ πx 2 + √ 1− x2 . (3) The main result of the article [3] can be formulated with the next statement. Proposition 1.4 In the family of the functions: fb(x) = (b+ 1)x b + √ 1− x2 (0 ≤ x ≤ 1), (4) according to the parameter b > 0, the function f2(x) is the greatest lower bound of the arc sin x function and the function f2/(π−2)(x) is the least upper bound of the arc sin x function. Research partially supported by the MNTRS, Serbia, Grant No. 144020.