Sharp Power Mean Bounds for the One-Parameter Harmonic Mean

Abstract EN

We present the best possible parameters α = α ( r ) and β = β ( r ) such that the double inequality M α ( a , b ) < H r ( a , b ) < M β ( a , b ) holds for all r ∈ ( 0 , 1 / 2 ) and a , b > 0 with a ≠ b , where M p ( a , b ) = [ ( a p + b p ) / 2 ] 1 / p ( p ≠ 0 ) and M 0 ( a , b ) = a b and H r ( a , b ) = 2 [ r a + ( 1 - r ) b ] [ r b + ( 1 - r ) a ] / ( a + b ) are the power and one-parameter harmonic means of a and b , respectively.