Best Possible Bounds for Yang Mean Using Generalized Logarithmic Mean. (10th April 2016) - Qian, Wei-Mao; Chu, Yu-Ming
Abstract : We prove that the double inequalityL p ( a, b ) < U ( a, b ) < L q ( a, b ) holds for alla, b > 0 witha ≠ b if and only ifp ≤ p 0 andq ≥ 2 and find several sharp inequalities involving the trigonometric, hyperbolic, and inverse trigonometric functions, wherep 0 = 0.5451 ⋯ is the unique solution of the equation( p + 1 ) 1 / p = 2 π / 2 on the interval( 0, ∞ ), U ( a, b ) = ( a - b ) / [ 2 arctan ⁡ ( ( a - b ) / 2 a b ) ], andL p ( a, b ) = [ ( a p + 1 - b p + 1 ) / ( ( p + 1 ) ( a - b ) ) ] 1 / p ( p ≠ - 1, 0 ), L - 1 ( a, b ) = ( a - b ) / ( log ⁡ a - log ⁡ b ) andL 0 ( a, b ) = ( a a / b b ) 1 / ( a - b ) / e are the Yang, andp th generalized logarithmic means ofa andb, respectively.