\(q\)-moduli of continuity in \(H^{p}(\mathbb D)\), \(p>0\) and an inequality of Hardy and Littlewood (Q696866)

The authors discuss some approximation properties of analytic functions on the unit disc \(\mathbb D\subset\mathbb C\), \(f(z)\in H^p(\mathbb D) \), \(p>0\). Following Tamrazov, the \(q\)-modulus of continuity is defined as \(\widetilde\omega_m(\delta, f)_p:=\sup_{0<t<\delta}\|\nabla^m_qf(e^{it})\|_p\), where \(\nabla^m_q\) is the \(q\)-difference operator. In the considered situation divided differences are based on an equidistant partition of \(\mathbb D\). The Bernstein-Nikol'ski-Stechkin-type (BNS-type) inequality is given: For algebraic polynomials \(P_n\) of order at most \(n\) there holds \(\|P_n^{(m)}\|_p\lesssim n^m\|\nabla_q^mP_n\|_p\), \(q=e^{i/n}\). Theorem of Jackson-type in \(H_p\) is proved: Given \(f\in H^p\), and \(m\in\mathbb N\), there is a polynomial \(R_n\) of degree \(n>m\) such that \(\|f-R_n\|_{H^p}\lesssim\widetilde\omega_m(n^{-1}, f)_p\). The \(K_m\)-functional is defined by \(K_m(\delta, f)_p:= \inf_{g^{(m)}\in H^p}\{\|f-g\|_{H^p} + \delta\|g^{(m)}\|_{H^p}\}.\) The equivalence of \(\widetilde\omega(\delta, f)_p\) and \(K_m\)-functional in \(H^p\) is established. Essential use is made of the inequality of BNS-type and inequality of Jackson-type. The Hardy-Littlewood-type theorem on the growth of fractional derivatives is proved: Let \(f\) be an analytic function on \(\mathbb D\), \(f\in H^p\). Then \(\|f^{(\alpha)}(re^{it})\|_p\lesssim(1-r)^{-\alpha}K_\alpha((1-r)^\alpha, f)_p\), \(0<r<1\), \(\alpha,p>0\). Let \(\omega(t)\) be a nondecreasing, continuous function on \([0,1]\) with \(\omega(0)=0\) and \(\int_0^\delta\frac{\omega(t)}{t} dt\lesssim\omega(\delta)\). Then, \(\|f^{(\alpha)}(re^{it})\|_p\lesssim(1-r)^{-\alpha}\omega(1-r)\), \(r\to 1-\), implies \(f\in H^p\) and \(K_\alpha(\delta^\alpha, f)_p\lesssim\omega(\delta)\). From here two consequences follow: The equivalence \(f^{(\alpha)}\in H^p\Leftrightarrow K_\alpha(\delta^\alpha,f)_p=O(\delta^\alpha)\) takes place. For \(P_n(z)=\sum_{k=0}^nz^k\) there holds the BNS-type inequality \(\|P_n^{(\alpha)}\|_p\lesssim n^\alpha K_\alpha(n^{-\alpha},P_n)_p\).