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Hi, If $P_n(x) $ is unnormalized shifted Legendre polynomial, and $g_{n,m}(x) = \int_0^x P_n(x_1)x_1^m dx_1, n>m $ then what is the upper bound $ |g_{n,m} (x)|_{max} , x\in (0,1) $ as a function of $m$ and $n$ ? A loose upper bound is of course $ \frac{1}{m+1}$, but i am looking for a tighter bound as a function of $n$. Thank you for your answers.

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