Some negative results on multiplier sequences of the first kind

It is proved that the composition of a polynomial with a multiplier sequence of the first kind may lead to a diminishing of the number of real roots of this polynomial and that the reciprocals of the moments of a nonnegative function on [0, 1] need not form a multiplier sequence of the first kind. On the basis of these facts one establishes the inaccuracy of the solution, obtained by T. Craven and G. Csordas, to S. Karlin's problem on the characterization of linear transformations that do not increase the number of zeros and also the incorrectness of M. Kostova's certain results, given in L. Iliev's monograph Laguerre entire functions (Sofia, 1987).Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 3, pp. 305–309, March, 1992.