A generalization of an inequality of Zygmund (Q1210454)

The author proves the following two statements and consequences thereof: Let \(g\) be a complex-valued function of \(e^{ix}\) for which \({{dg(e^{ix})} \over {dx}}\) exists. Then for any nonnegative nondecreasing convex function \(\chi\), for any \(\alpha\in\mathbb R\) and any polynomial \(p\) of degree \(n\) the relations \[ \int_ 0^{2\pi} \chi\left( {{|\text{Im}(e^{i\alpha}g(e^{i\theta})p'(g(e^{i\theta})))|} \over n} \right)\,d\theta \leq \max_ \beta \left( \int_ 0^{2\pi} \chi\bigl( | \text{Re}(p(e^{i\beta}g(e^{i\theta})))| \bigr) \,d\theta \right) \tag{1} \] and \[ \left( \int_ 0^{2\pi} (g(e^{i\theta}p'(g(e^{i\theta}))|^ q \,d\theta\right)^{1/q} \leq A_ q n\cdot\max_ \beta \left( \int_ 0^{2\pi} |\text{Re}(p(e^{i\beta} g(e^{i\theta})))|^ q \,d\theta\right)^{1/q} \tag{2} \] \((1\leq q<\infty)\) hold with equality iff \(p(z)=Az\).