OPTIMAL CONVEX COMBINATION BOUNDS OF THE CONTRAHARMONIC AND HARMONIC MEANS FOR THE SEIFFERT 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) < P (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 P(a, b) denote the contraharmonic, harmonic, and the Seiffert means of two positive numbers a and b respectively. AMS Subject Classification: 26D15


Introduction
search [3,8].In particularly, many remarkable inequalities for the Seiffert mean can be found in the literature [4,6,8].( In [1], Seiffert proved for all a, b > 0 with a = b . The following bounds for the Seiffert mean P (a, b) in terms of the power mean M r (a, b) = ((a r + b r )/2) 1/r (r = 0) were presented by Jagers in [7]: for all a, b > 0 with a = b .
H ästö [8] found the sharp lower bound for the Seiffert mean as follow: for all a, b > 0 with a = b .In [3], Seiffert proved and for all a, b > 0 with a = b .In [4], the authors found the greatest value α and the least value β such that the double inequality 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 .

Theorem
The double inequality holds for all a, b > 0 with a = b if and only if α ≤ 1 π and β ≥ 5 12 .

For
a, b > 0 with a = b the Seiffert means P(a, b) was introduced by Seiffert [1, 2] as follows: P (a, b) = a − b 4 arctan a / b − π .