Best Possible Bounds for Yang Mean Using Generalized Logarithmic Mean

Abstract

We prove that the double inequalityLp(a,b)<U(a,b)<Lq(a,b)holds for alla,b>0witha≠bif and only ifp≤p0andq≥2and find several sharp inequalities involving the trigonometric, hyperbolic, and inverse trigonometric functions, wherep0=0.5451⋯is the unique solution of the equation(p+1)1/p=2π/2on the interval(0,∞),U(a,b)=(a-b)/[2arctan⁡((a-b)/2ab)], andLp(a,b)=[(ap+1-bp+1)/((p+1)(a-b))]1/p  (p≠-1,0),L-1(a,b)=(a-b)/(log⁡a-log⁡b)andL0(a,b)=(aa/bb)1/(a-b)/eare the Yang, andpth generalized logarithmic means ofaandb, respectively.