Let's first check the conditions on the parameters by including the function domain restrictions (implied by specifying Reals
) but omitting at first the main inequality. We'll simplify this bit and come back to the main question. There are three complicated, transcendental functions in the main term, two involving Log
and one involving ArcTanh
. We'll add their function domains to the restrictions. These are already implied, but we're helping Reduce
out by singling them out:
Reduce[{
(*(Sqrt[f+a^2/(4 b^2)]/f Log[12 x b^2 f+3 x a^2+Sqrt[
3] x Sqrt[(4 b^2 f+a^2) (4 b (((2-3 b) k)/x^2+3 b f)+3 a^2)]]-
a/(2 b f)ArcTanh[(x a)/Sqrt[4/3 (2-3 b) b k+x^2 (4 b^2 f+a^2)]]-
a/(4 b f) Log[(-2+3 b) k-3 x^2 b f])>0,*)
f > 0, b > 0, a > 0,
FunctionDomain[
Log[12 x b^2 f + 3 x a^2 +
Sqrt[
3] x Sqrt[(4 b^2 f + a^2) (4 b (((2 - 3 b) k)/x^2 + 3 b f) +
3 a^2)]], x],
FunctionDomain[
ArcTanh[(x a)/Sqrt[4/3 (2 - 3 b) b k + x^2 (4 b^2 f + a^2)]], x],
FunctionDomain[Log[(-2 + 3 b) k - 3 x^2 b f], x]},
x, Reals]
(* False *)
The result False
means there are no real x
satisfying the constraints. Adding the main inequality could only restrict the answer more, but of course, starting from "no solutions," the only possible result is still "no solutions."