Extremal properties of Chebyshev polynomials

Using methods of geometric function theory, we get new extremal properties of Chebyshev polynomials. The exact estimates of coefficients, covering and distortion theorems for polynomials with real coefficients and curved majorants on the interval are obtained. In each case, the extremal is Chebyshov polynomial of second, third or fourth kind. These theorems refine some classical results for algebraic polynomials with constraints on the the interval. As a corollary, we get the following analog of Schur's inequality
$$ \max\{|P(x)|:x\in[-1,1]\}\le(2n+1)\max\{|P(x)\sqrt{(1+x)/2}|:x\in [-1,1]\} $$
where $P(x)$ is the polynomial of degree $n$ with real coefficients. The equality holds for Chebyshev polynomial of the third kind.