Upper bounds for the distances between consecutive zeros of solutions of first order delay differential equations

By introducing a new class of sequences involving iterates of the delay, we are able to derive a sharper estimate on the ratio x(τ(t))/x(t)x(τ(t))/x(t). Based on this, we get some new upper bounds for the distance between consecutive zeros of solutions of the first order delay differential equationx′(t)+p(t)x(τ(t))=0.x′(t)+p(t)x(τ(t))=0. In particular, our results are not covered by previously known results. Some examples and a table are given to illustrate our results.