A property of a system of functions close to exponential functions

We consider the system $\{f_n(x)=x^{\lambda_n}[1+\varepsilon_n(x)]\}$ in the interval $[a,b]$ ($0\leqslant a<b<\infty$). Under certain conditions on $\lambda_n>0$ and $\varepsilon_n(x)$ such as the condition $\varlimsup\limits_{n\to\infty}\frac{\ln m_n}{\lambda_n}>0$, $m_n=||\varepsilon_n(x)||_{L_p[a,b]}$, we obtain a bound for the coefficients of the polynomial $P(x)=\sum c_nf_n(x)$ in terms of $||P(x)||_{L_p[a,b]}$. It is found that this bound is not valid without this condition (assuming the other conditions to remain the same).