Sharp One-Parameter Mean Bounds for Yang Mean

Abstract EN

We prove that the double inequality J α ( a , b ) < U ( a , b ) < J β ( a , b ) holds for all a , b > 0 with a ≠ b if and only if α ≤ 2 / ( π - 2 ) = 0.8187 ⋯ and β ≥ 3 / 2 , where U ( a , b ) = ( a - b ) / [ 2 arctan ( ( a - b ) / 2 a b ) ] , and J p ( a , b ) = p ( a p + 1 - b p + 1 ) / [ ( p + 1 ) ( a p - b p ) ] ( p ≠ 0 , - 1 ) , J 0 ( a , b ) = ( a - b ) / ( log a - log b ) , and J - 1 ( a , b ) = a b ( log a - log b ) / ( a - b ) are the Yang and p th one-parameter means of a and b , respectively.