On completeness of a system of functions that are close to power functions

We obtain necessary and sufficient conditions for the completeness of the system $\{f_n(x)=x^{\lambda_n}[1+\varepsilon_n(x)]\}$ in $L_p[0, a]$, where $\varepsilon_n(x)\in L_p[0, a]$, $\varlimsup_{n\to\infty}\frac{\ln m_n}{\lambda_n}<0$, $m_n = \|\varepsilon_n(x)\|_{L_p[0, a]}$, $n=1,2,\dots$; $1<p<\infty$.
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