Sharp llogl inequalities for differentially subordinated martingales and harmonic functions (Q733350)

Let \((x_n)_{n\in N_0}\) and \((y_n)_{n\in N_0}\) be martingales adapted to the same filtration and taking values in a separable Hilbert space. Assuming that \((x_n)\) is differentially subordinate to \((y_n)\), i.e., \(|x_n-x_{n-1}|\leq |y_n-y_{n-1}|\) a.s. for any \(n\in N_0\), the author considers the inequality \[ E|x_n|\leq K E|y_n|\log |y_n|+L, \;\;n\in N_0 \] and, for every positive \(K>0\), determines the best constant \(L=L(K)\). It is shown that in the ranges \(K<2\) and \(K\geq 2\) the expressions for the optimal \(L\) are different, and while for \(K<2\) the inequality is strict, for \(K\geq 2\) it may hold as equality for some nontrivial martingales. A similar problem is also addressed for differentially subordinated harmonic functions taking values in a separable Hilbert space. However, in the latter case the results are not that complete as for martingales.