Approximation by rational functions with polynomials of positive coefficients as the denominators

The present paper proves that if f(x)∈C _0,1] changes its sign exactly l times at0 1 2 < … l <1 in(0, 1), then there exists a p_n(x)∈II _n(+), such that $$\left| {f(x) - \frac{{\rho (x)}} {{p_n (x)}}} \right| \leqslant C\omega _\phi (f,n^{ - 1/2} ),$$ where ?(x) is defined by $$\rho (x) = \left\{ \begin{gathered} \prod\limits_{i = 1}^l {(x - y_i ), if f(x) \geqslant 0} for x \in (y\iota ,1), \hfill \\ - \prod\limits_{i = 1}^l {(x - y_i ), if f(x) < 0 for x \in (y\iota ,1).} \hfill \\ \end{gathered} \right.$$ which improves and generalizes the result of 7].