an:03871693
Zbl 0547.26010
Lorch, Lee; Newman, Donald J.
On the composition of completely monotonic functions and completely monotonic sequences and related questions
EN
J. Lond. Math. Soc., II. Ser. 28, 31-45 (1983).
00147130
1983
j
26A48 26A51
completely monotonic sequences and functions
The authors answer several previously open questions about c.m. (completely monotonic) sequences and functions. (1) If W(x) is c.m. on \([a,\infty)\) and \(\{\Delta x_ k\}\) is c.m. with \(x_ 0\geq a,\) then \(\{W(x_ k)\}_ 0^{\infty}\) is c.m. Also, the sequence \(\{\mu_ k^{\lambda}\},\quad\mu_ 0=1,\quad\mu_ k>0,\quad k=1,2,...,\) is c.m. for all \(\lambda >0\) if and only if \(\mu_ k=\exp (-\nu_ k)\) with \(\{\Delta\nu_ k\}^ c.\)m., \(\nu_ 0=0.\) (2) If V'(x) is c.m. on \((0,\infty),\) and \(\{\Delta x_ k\}_ 0^{\infty}\) is c.m., then \(\{\Delta V(x_ k)\}_ 0^{\infty}\) is c.m. (3) Partial converse of (1): Let \(W(x)>0\quad (0\leq x<\infty),\quad W'(x)<0\quad (0<x<\infty),\) and let \(W'(0^+)\) existe (finite). If \(\{W(\lambda x_ k)\}\) is c.m. for all small \(\lambda >0\) and \(x_ 0\geq 0,\) then \(\{\Delta x_ k\}\) is c.m. However, (4) if f is c.m. on \([0,\infty)\) there is a \(\phi\) (t) with \(\phi (0)=0,\quad f(\phi (t))\) c.m. on \([0,\infty),\) but \(\phi\) '(t) not c.m. on \((0,\infty).\) (5) If \(\phi (f(x))\) is c.m. on \((0,\infty)\) for all f(x) that are c.m. on \((0,\infty)\) then \(\phi\) (x) is absolutely monotonic on (0,\(\infty)\). (6) If f and g are convex and nonnegative and \(k\geq 1\) then \((f^ k+g^ k)^{1/k}\) is convex. (7) If f is monotonic of order \(N\geq 2\) then \([f(x)]^{1/(N-1)}\) is convex. (8) If f is monotonic of order N and \(\lambda >1,\) then \([f(x)]^{\lambda}\) is monotonic of order N if \(N=1,2,3\quad or\quad 4\) but not necessarily if \(N\geq 5.\) (9) If f is c.m. and \(\lambda >1\) then \([f(t)]^{\lambda}\) is monotonic of order 5. Many interesting special cases and corollaries are also given.
R.P.Boas