Best Possible Bounds for Yang Mean Using Generalized Logarithmic Mean

Abstract EN

We prove that the double inequality L p ( a , b ) < U ( a , b ) < L q ( a , b ) holds for all a , b > 0 with a ≠ b if and only if p ≤ p 0 and q ≥ 2 and find several sharp inequalities involving the trigonometric, hyperbolic, and inverse trigonometric functions, where p 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 ) ] , and L 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 ) and L 0 ( a , b ) = ( a a / b b ) 1 / ( a - b ) / e are the Yang, and p th generalized logarithmic means of a and b , respectively.