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
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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