On a conjecture of Clark and Ismail
Journal of Approximation Theory
The Hardy-Littlewood function: an exercise in slowly convergent series
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the conference on orthogonal functions and related topics held in honor of Olav Njåstad
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The function Q(x):=@?"n"="1(1/n)sin(x/n) was introduced by Hardy and Littlewood (1936) [5] in their study of Lambert summability, and since then it has attracted attention of many researchers. In particular, this function has made a surprising appearance in the recent disproof by Alzer, Berg and Koumandos (2005) [3] of a conjecture by Clark and Ismail (2003) [14]. More precisely, Alzer et al. have shown that the Clark and Ismail conjecture is true if and only if Q(x)=-@p/2 for all x0. It is known that Q(x) is unbounded in the domain x@?(0,~) from above and below, which disproves the Clark and Ismail conjecture, and at the same time raises a natural question of whether we can exhibit at least one point x for which Q(x)