BOUNDING THE CONVEX COMBINATION OF ARITHMETIC AND INTEGRAL MEANS IN TERMS OF ONE-PARAMETER HARMONIC AND GEOMETRIC MEANS

In the article, we find the best possible parameters lambda(1), mu(1), lambda(2) and mu(2) on the interval [0, 1/2] such that the double inequalities H(a, b; lambda(1)) < alpha A(a, b) + (1 - alpha)T(a, b) < H(a, b; mu(1)), G(a, b; lambda(2)) < alpha A(a, b) + (1 - alpha)T(a, b) < G(a, b; mu(2)) hold for all alpha is an element of[0, 1] and a, b > 0 with a not equal b, where A(a, b) = (a + b)/2, = 2 integral(pi/2)(0) a(cos2 theta) b(sin2 theta) d theta/pi, H(a, b; lambda) = 2[lambda a + (1 - lambda)b][lambda b + (1 - lambda)a]/(a + b), G(a, b; mu) = root[mu a + (1 -mu)b][mu b + (1 - mu)a] are the arithmetic, integral, one-parameter harmonic and one-parameter geometric means of a and b, respectively.