Abstract
Fora,b>0witha≠b, the Schwab-Borchardt meanSB(a,b)is defined asSB(a,b)={b2-a2/cos-1(a/b) ifa<b,a2-b2/cosh-1(a/b) ifa>b. In this paper, we find the greatest values ofα1andα2and the least values ofβ1andβ2in[0,1/2]such thatH(α1a+(1-α1)b,α1b+(1-α1)a)<SAH(a,b)<H(β1a+(1-β1)b,β1b+(1-β1)a)andH(α2a+(1-α2)b,α2b+(1-α2)a)<SHA(a,b)<H(β2a+(1-β2)b,β2b+(1-β2)a). Similarly, we also find the greatest values ofα3andα4and the least values ofβ3andβ4in[1/2,1]such thatC(α3a+(1-α3)b,α3b+(1-α3)a)<SCA(a,b)<C(β3a+(1-β3)b,β3b+(1-β3)a)andC(α4a+(1-α4)b,α4b+(1-α4)a)<SAC(a,b)<C(β4a+(1-β4)b,β4b+(1-β4)a). Here,H(a,b)=2ab/(a+b),A(a,b)=(a+b)/2, andC(a,b)=(a2+b2)/(a+b)are the harmonic, arithmetic, and contraharmonic means, respectively, andSHA(a,b)=SB(H,A),SAH(a,b)=SB(A,H),SCA(a,b)=SB(C,A), andSAC(a,b)=SB(A,C)are four Neuman means derived from the Schwab-Borchardt mean.
Highlights
B > 0 with a ≠ b, the Schwab-Borchardt mean SB(a, b) is defined as {{{{{ √b2 cos−1− a2 (a/b) if a < b, SB (a, b) = {{{{{ √a2 − b2 { cosh−1 (a/b) if a > b. (1)It is well known that the mean SB(a, b) is strictly increasing in both a and b, nonsymmetric, and homogeneous of degree 1 in its variables
We find the greatest values of α1 and α2 and the least values of β1 and β2 in [0, 1/2] such that H(α1a+(1−
By Lemma 1 and (15) we know that φ(x) is strictly increasing on (0, π/2)
Summary
The second author of this paper reviewed two elegant papers [14, 15] by Neuman and found that the bivariate means SAH, SHA, SCA, and SAC, derived from the SchwabBorchardt mean are very interesting. They are defined as follows: SAH = SB (A, H) , SHA = SB (H, A) , (4). We call the means SAH, SHA, SCA, and SAC, defined in (4) the Neuman means. The bounding quantities are arithmetic convex, geometric convex, and harmonic convex combinations of their generating means. < SAC < C (β4a + (1 − β4) b, β4b + (1 − β4) a) holds for all a, b > 0 with a ≠ b if and only if α4 ≤ [1 + √3√3/π − 1]/2 and β4 ≥ (3 + √6)/6
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