Best Possible Bounds for Yang Mean Using Generalized Logarithmic Mean

摘要

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 not equal 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) = root 2 pi/2 on the trival (0, infinity), U(a, b) = (a - b)/[root 2 arctan ((a-b)/root 2ab)], and L-p (a, b) = [a(p+1) - b(p+1))/((p+1)(a-b))](1/p) (p not equal -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 pth generalized logarithmic means of a and b, respectively.