How can I find $\lambda_H$ and $\lambda_T$ such that $$\max_{0 \leq \lambda_H , \ \lambda_T \ \leq 1 }\left\{\frac{4.6575342 \times 10^{-4}}{2.1722965 \times 10^{-4} + \lambda_H},\frac{1.0958904 \times 10^{-2}}{3.4311896 \times 10^{-4}+\lambda_T}\right\}<1?$$
Is this problem equivalent to finding $x$ and $y$ such that $$\min_{0 \ \leq \ x , \ y \ \ \leq \ 1}\{.4664048+(2147.0588450)x,0.0313096+(91.2500009)y\} \geq 1?$$
Each of the functions is convex, their max is then convex, so the solution of the maximization lies at one the corners: (0,0), (0,1), (1,0) and (1,1). It should be (0,0).
What is not clear is the $<1$. Is it a constraint? If so it should be written as a constraint.
If you want to impose the value is $<1$ then it is just a matter of increasing the variables from zero until the point you reach 1 or one of them hits one.