An Application of Λ-method on Shafer-fink’s Inequality

In the paper λ-method Mitrinovi´c-Vasi´c is applied aiming to improve Fink's inequality, and Shafer's inequality for arcus sinus function is observed. In monography [1, p. 247] Shafer's inequality is stated: (1) 3x 2 + √ 1 − x 2 ≤ asin x (0 ≤ x ≤ 1). The equality holds only for x = 0. In paper [2] Fink has proved the inequality: (2) asin x ≤ πx 2 + √ 1 − x 2 (0 ≤ x ≤ 1). The equality holds at both ends of the interval x = 0 and x = 1. Let us notice that from the inequality (1) and (2) the function g(x) = asin x is bounded by the corresponding functions from the two-parameters family of functions: (3) Φ a,b (x) = ax b + √ 1 − x 2 (0 ≤ x ≤ 1), for some values of parameters a, b > 0. For the values of parameters a, b > 0 the family Φ a,b (x) is the family of raising convex functions on variable x on interval (0, 1). Let us apply λ-method Mitrinovi´c-Vasi´c [1] on considered two-parameters family Φ a,b in order to determine the bound of function g(x) under the following conditions: (4) Φ a,b (0) = g(0) and d dx Φ a,b (0) = d dx g(0). It follows that a = b + 1. In that way we get one-parameter subfamily: (5) f b (x) = Φ b+1,b (x) = (b + 1)x b + √ 1 − x 2