Optimal bounds for neuman means in terms of harmonic and contraharmonic means

For a,b > 0 with a≠b,the schwab-borchardt mean s b (a,b) is defined as s b (a,b) = {√b2- a2/c o s-1 (a/b) if a < √b2- a2cosh-1 (a/b) if a > b. in this paper,we find the greatest values of α 1 and α2 and the least values of β1 and β2 in [0,1/2] such that h (α1 a+(1 - α1) b,α1b+(1 - α1) a) < sah (a,b) < h (β1 a+(1-β1) b,β1 b+(1-β1) a) and h (α2a+(1- α2) b,α2 b+(1-α2) a) < sha (a,b) < h (β2 a+(1 - β2) b,β2 b+(1 - β2) a). similarly,we also find the greatest values of α3 and α4 and the least values of β3 and β4 in [1/2,1] such that c (α3 a+(1 - α3) b,α3 b+(1 - α3) a) < sca (a,b) < c (β3 a+(1 - β3) b,β3 b+(1 - β3) a) and c (α4 a+(1 - α4) b,α4 b+(1 - α4) a) < sac (a,b) < c (β4 a+(1 - β4) b,β4 b+(1 - β4) a). here,h (a,b) = 2ab/(a + b),a(a,b) = (a + b)/2,and c(a,b) = (a2+b2)/(a + b) are the harmonic,arithmetic,and contraharmonic means,respectively,and sha (a,b) = sb(h,a),sah (a,b) = sb (a,h),sca (a,b) = sb (c,a),and sac (a,b) = sb (a,c) are four neuman means derived from the schwab-borchardt mean. © 2013 zai-yin he et al.