SHARP BOUNDS FOR TOADER-QI MEAN IN TERMS OF LOGARITHMIC AND IDENTRIC MEANS

摘要

In the article, we prove that the double inequality lambda root L(a,b)I(a,b) < TQ(a,b) < mu root L(a,b)I(a,b) holds for all a,b > 0 with a not equal b if and only if lambda <= root e/pi and mu >= 1, and give an affirmative answer to the conjecture proposed by Yang in [39], where L(a,b) = (b-a)/(logb-loga), I(a,b) = (b(b)/a(a))(1/(b-a))/e and TQ(a,b) = 2/pi integral(pi/2)(0) a(cos2 theta)b(sin2 theta) d theta are respectively the logarithmic, identric and Toader-Qi means of a and b.