Inequalities for algebraic polynomials on an ellipse

The paper presents new solutions to two classical problems of approximation theory. The first problem is to find the polynomial that deviates least from zero on an ellipse. The second one is to find the exact upper bound of the uniform norm on an ellipse with foci $\pm 1$ of the derivative of an algebraic polynomial with real coefficients normalized on the segment $[- 1,1]$.