Estimates for Neuman–sándor Mean by Power Means and Their Relative Errors

For a,b > 0 with a = b , let NS (a,b) denote the Neuman-Sándor mean defined by NS (a,b) = a−b 2arcsinh a−b a+b and Ap (a,b) , Lp (a,b) denote the r -order power and Lehmer means. Based on our earlier worker [27], we prove that αpAp < NS < Ap and Ap < NS βpAp holds if and only if p 4/3 and p p0 , respectively, where αp = ( 21/p−1 ) / ln(1+ √ 2) if p ∈ [1/4/3,∞), βp = ⎪⎨ ⎪⎩ NS (1,x0)/Ap (1,x0) if p ∈ (1, p0], 21/p−1/ ln ( 1+ √ 2 ) if p ∈ (0,1], ∞ if p ∈ (−∞,0] are the best constants, here x0 is the unique root of the equation NS (1,x) = A(1,x)A2 (1,x) Lp0−1 (1,x) on (0,1) , and p → αpAp is decreasing on (0,∞) . Also, we have α4/3A4/3 < Ap0 < NS < A4/3 < α −1 4/3Ap0 . Our results clearly are generations of known ones. Mathematics subject classification (2010): 26E60, 26D05.