
Sharp bounds for SándorYang means in terms of some bivariate means
XU Huizuo
2017, (4):
4151.
doi: 10.3969/j.issn.10005641.2017.04.004
This paper deals with the inequalities involving SándorYang means derived from the SchwabBorchardt mean using the method of real analysis. The convex com binations of the arithmetic mean A(a,b) and quadratic Q(a,b) (or contraharmonic mean C(a,b)) for the SándorYang means S_{QA}(a,b) and S_{QA}(a,b) are disscused. The main results obtained are the sharp bounds of the two convex combinations, namely, the best possible parameters α_{1}, α_{2}, α_{3}, α_{4}, β_{1}, β_{2}, β_{3}, β_{4} ∈ (0, 1), such that the double inequalities
α_{1}Q(a,b) + (1α_{1})A(a,b) < S_{QA}(a,b) < β_{1}Q(a,b) + (1β_{1})A(a,b),
α_{2}Q(a,b) + (1α_{2})A(a,b) < S_{QA}(a,b) < β_{2}Q(a,b) + (1β_{2})A(a,b),
α_{3}C(a,b) + (1α_{3})A(a,b) < S_{QA}(a,b) < β_{3}C(a,b) + (1β_{3})A(a,b),
α_{4}C(a,b) + (1α_{4})A(a,b) < S_{QA}(a,b) < β_{4}C(a,b) + (1 β_{4})A(a,b)
hold for all a, b > 0 and a≠b. Here A(a,b), Q(a,b) and C(a,b) denote respectively the classical arithmetic, quadratic, contraharmonic means of a and b, S_{QA}(a,b) and S_{QA}(a,b) are two SándorYang means derived from the SchwabBorchardt mean.
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