We show that the double domination number of an \( n \)-vertex, isolate-free graph with minimum degree \( \delta \) is bounded above by \(\frac{n(\ln(\delta + 1) + \ln \delta + 1)}{\delta}.\) This result improves a previous bound obtained by J. Harant and M. A. Henning [On double domination in graphs, \({Discuss. Math. Graph Theory}\) \({25} (2005), 29-34]\). Further, we show that for fixed \( k \) and large \( \delta \), the \( k \)-tuple domination number is at most \(\frac{n(\ln \delta + (k – 1 + o(1))\ln \ln \delta)}{\delta},\) a bound that is essentially best possible.