An asymptotic numeric problem.

Question

Given a large enough integer $N$ is there always a $c\in(0,1)$ such that $$(N+ N^{1-c}){c\ln(e N)}>\ln( N+( N)^{1-c})(N+2 N^c)$$ holds?

What is this $c$ explicitly (at least a close approximation is fine)?

Answer 1

No; for any $c \in (0,1)$ the left hand side is $cN\ln(N) + o(N\ln(N))$, and the right hand side is $N\ln(N) + o(N\ln(N))$, so the right hand side will dominate for large enough $N$.