Some variational formulas in the class \({\tilde K}_ n(E)\) and their applications (Q1366376)

Let \(K_n(E)\) denote the class of functions \(F(z)\) that are analytic on the unit disk \(E\), and for which the difference quotient \[ [F(z); z_0,\dots,z_n]= \sum^n_{m= 0} {F(z_m)\over (z_m- z_0)\cdots(z_m- z_{m+ 1})\cdots(z_m- z_n)} \] is nonzero for arbitrary distinct numbers \(z_0,\dots,z_n\in E\). A function \(F(z)\) analytic on \(E\) is called \(n\)-normed on \(E\) if its power series expansion has the form \(F(z)= z^n+ \sum^\infty_{k= 2}a_{k, n}z^{n+ k-1}\). The class of \(n\)-normed functions is denoted by \(\widetilde K_n(E)\). The author derives variational formulas of the type \(F(z,\varepsilon)= F(z)+ \varepsilon Q(z)+ o(|\varepsilon|, E)\) for functions in the class \(\widetilde K_n(E)\). These variational formulas are used to solve several extremal problems.