Best Possible Bounds for Yang Mean Using Generalized Logarithmic Mean

We prove that the double inequality holds for all with if and only if and and find several sharp inequalities involving the trigonometric, hyperbolic, and inverse trigonometric functions, where is the unique solution of the equation on the interval , , and , and are the Yang, and th generalized logarithmic means of and , respectively.

Very recently, Yang [16] introduced the Yang mean of two distinct positive real numbers  and  and proved that the inequalities hold for all ,  > 0 with  ̸ = , where (, ) = √( 2 +  2 )/2 is the quadratic mean of  and .
In [18,19], the authors proved that the double inequalities  Zhou et al. [20] proved that  = 1/2 and  = log 3/(1 + log 2) = 0.6488 ⋅ ⋅ ⋅ are the best possible parameters such that the double inequality holds for all ,  > 0 with  ̸ = .The main purpose of this paper is to present the best possible parameters  and  such that the double inequality   (, ) < (, ) <   (, ) holds for all ,  > 0 with  ̸ = .As application, we derive several sharp inequalities involving the trigonometric, hyperbolic, and inverse trigonometric functions.Some complicated computations are carried out using Mathematica computer algebra system.

Lemmas
In order to prove our main result we need two lemmas, which we present in this section.

Mathematical Problems in Engineering 5
We divide the proof into four cases.

Applications
As applications of Theorem 3 in engineering problems, we present several sharp inequalities involving the trigonometric, hyperbolic, and inverse trigonometric functions in this section.