Let (hn)n≥0 be the Haar system of functions on [0,1]. The paper contains the proof of the estimate ∫01|∑k=0nεkakhk|2log|∑k=0nεkakhk|ds≤∫01|∑k=0nakhk|2log|e2∑k=0nakhk|ds, for n=0,1,2,…. Here (an)n≥0 is an arbitrary sequence with values in a given Hilbert space H and...