OPTIMAL CONVEX COMBINATION BOUNDS OF THE CONTRAHARMONIC AND HARMONIC MEANS FOR THE WEIGHTED GEOMETRIC MEAN

We find the greatest value α and the least value β such that the double inequality αC(a, b) + (1− α)H(a, b) < S(a, b) < βC(a, b) + (1− β)H(a, b) holds for all a, b > 0 with a 6= b. Here C(a, b), H(a, b) and S(a, b) denote the contraharmonic, harmonic, and the weighted geometric means of two positive numbers a and b respectively. AMS Subject Classification: 65N15


Introduction
(1. 2) The weighted geometric mean is a special case of Geni ′ s mean which can be found in the literature [6] and is related to the identric mean in the literature [9] as follows: For more properties of the mean S(a, b) can be seen in the literature [8], [10] and [14].
In [14], it has been shown that where and also that inequality is sharp in a certain sense.
In [15], it has been shown that holds for all a, b > 0 with a = b.
In [16], the authors found the greatest value p and the least value q such that the double inequality holds for all a, b > 0 and α, β > 0 with a = b.
In [5], the authors found the greatest value α and the least value β such that the double inequality holds for all a, b > 0 with a = b.
The purpose of the present paper is to find the greatest value α and the least value β such that the double inequality holds for all a, b > 0 with a = b.
Secondly.We prove that βC(a, b) ) is the best possible upper convex combination bound of contraharmonic and harmonic means for the weighted geometric mean S(a, b).
Finally.We prove that αC(a, b) ) is the best possible lower convex combination bound of contra harmonic and harmonic means for the weighted geometric mean S(a, b).

For= 1 e ( b b a a ) 1 b
a, b > 0 with a = b the weighted geometric mean S(a, b) of a and b with weights a a+b and b a+b : −a , A(a, b) = a+b 2 , T(a, b) = 2(a 2 +ab+b 2 ) 3(a+b) and C(a, b) = a 2 +b 2 a+b be the harmonic, geometric, logarithmic, Seiffert, identric, arithmetic, centroidal and contraharmonic means of two positive real numbers a and b with a = b.Then min{a, b} < H(a, b) < G(a, b) < L(a, b) < P(a, b) < I(a, b) < A(a, b) < T (a, b) < S(a, b) < C(a, b) < max(a, b).