Inequalities

[Dudeconomics]

Hi folks! I'm struggling to prove the following inequality: theta>0 such that

With restrictions beta>1 and 0<lambda<1. Epsilon is a real-valued parameter. So far I've tried a lot of manipulations, including using the Bernoulli's inequality. I've tried data simulation using several values for the parameters, and all simulations indicated theta>0. Any hint? Thanks!

 

[tkhunny]

Consider Binomial Expansion?

 

[Dudeconomics]

Hi tkhunny! Using Bernoulli's inequality I got that the upper part of the fraction is greater then -(1-epsilon)(beta -1) for 0<epsilon<1, which is a negative number. So it doesn't help.
I consider the binomial expansional, but I'm interested in this problem when epsilon is a real number (especialy 0<1-epsilon<1) and not only positive integers, so I don't think that the binomial expansion can help me either. Am I wrong in this case? Any other hint or known inequality that I can use to solve it? Thanks

 

[tkhunny]

Ah! Didn't see that you mentioned Bernoulli in the first post. Sorry about that. That's pretty much just the first two terms of the Binomial Expansion.