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On the composition of completely monotonic functions and completely monotonic sequences and related questions. (English) Zbl 0547.26010
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.
Reviewer: R.P.Boas

26A48 Monotonic functions, generalizations
26A51 Convexity of real functions in one variable, generalizations
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