Best approximation-preserving operators over Hardy space

Let $$T_n$$ be the linear Hadamard convolution operator acting over Hardy space $$H^q$$ , $$1\le q\le \infty $$ . We call a best approximation-preserving (BAP operator) if $$T_n(e_n)=e_n$$ where $$e_n(z):=z^n,$$ and $$\Vert T_n(f)\Vert _q\le E_n(f)_q$$ for all $$f\in H^q$$ $$E_n(f)_q$$ is approximation by algebraic polynomials of degree most $$n-1$$ in space. give necessary sufficient conditions to BAP $$H^\infty apply this result establish an exact lower bound bounded holomorphic functions. In particular, we show that Landau-type inequality $$\left| {\widehat{f}}_n\right| +c\left| {\widehat{f}}_N\right| \le E_n(f)_\infty $$c>0$$ $$n<N$$ holds every H^\infty iff $$c\le \frac{1}{2}$$ $$N\ge 2n+1$$