Special problem: Let $G(z)$ be a probability generating function(pgf, the $z$ can be seen as real number or complex number), that is $$G(z) = \sum\limits_{i = 0}^\infty {{p_i}{z^i}} ,(\left| z \right| \leqslant 1;\sum\limits_{i = 0}^\infty {{p_i} = 1;} 0 \leqslant {p_i} \leqslant 1).$$ the logarithm of $G(z)$ is $$\ln G(z) = \sum\limits_{i = 0}^\infty {{q_i}{z^i}} ,(\left| z \right| \leqslant 1).$$ If ${p_0} > {p_1} > {p_2} > \cdots$ , then show that ${{q_i}}$ is absolutely convergent, namely $\sum\limits_{i = 0}^\infty {\left| {{q_i}} \right|} < \infty $, or give a counterexample.
I want to use Faa Di Bruno Formula (http://en.wikipedia.org/wiki/Faà_di_Bruno's_formula) to get the expression of $q_i$. then to estimate $\sum\limits_{i = 0}^\infty {\left| {{q_i}} \right|}$.
Remark: The problem above is a special case of my Problem which is connected to probability generating function：
Let $G(z)$ be a probability generating function, that is $$G(z) = \sum\limits_{i = 0}^\infty {{p_i}{z^i}} ,(\left| z \right| \leqslant 1;\sum\limits_{i = 0}^\infty {{p_i} = 1;} 0 \leqslant {p_i} \leqslant 1).$$
Problem 1. Under which necessary and sufficient condition on $p_i$, the logarithm of $G(z)$ $$\ln G(z) = \sum\limits_{i = 0}^\infty {{q_i}{z^i}} ,(\left| z \right| \leqslant 1).$$ is absolutely convergent, namely $\sum\limits_{i = 0}^\infty {\left| {{q_i}} \right|} < \infty $?
My conjecture is that if $G(z)$ has no zeros then $\sum\limits_{i = 0}^\infty {\left| {{q_i}} \right|} < \infty $. If this conjecture is not ture, can somebody give me a counterexample.
The Problem 1 is similar to an open problem：
Problem 2. Let $f(t) = \sum\limits_{k = 0}^\infty {{a_k}{e^{kit}}}$, we use the norm $\left\| f \right\| = \sum\limits_{k = 0}^\infty {\left| {{a_k}} \right| } $. Under which conditions on $f$ is the sequence $\left\| {{f^n}} \right\|$ bounded?
This problem are firstly proposed by Beurling(1938), quoted by Henry(1953). Hedstrom(1966) call the problem "Norms of powers of absolutely convergent Fourier series". It relates to complex-valued probabilities, see Baishanski(1999).
Problem 3. If some $q_i$ are negative, under which necessary and sufficient condition on $q_i$, $$\exp (\sum\limits_{i = 0}^\infty {{q_i}{z^i}} ) = \exp [\sum\limits_{i = 1}^\infty {{q_i}({z^i}} - 1)]$$ is a pgf?
Lévy(1937) prove that $P(z) ={e^{\sum\limits_{i = 1}^m {{q _i}({z^i}-1)} }}{\rm{, }}(\left| z \right| \le 1)$ is a pgf when a term with a sufficiently small negative coefficient is preceded by one term with positive coefficient and followed by at least two terms with positive coefficients as well(see Johnson(2005), p393-394), namely ${q_1} > 0,{q_{m - 1}} > 0,{q_m} > 0$. For $m=4$, van Harn(1987) give four inequalities to ensure $$P(z) = {e^{a(z-1) - b({z^2}-1) + c({z^3}-1) + d({z^4}-1)}}{\rm{, }}(\left| z \right| \le 1)$$ is a pgf, namely $a,b,c,d > 0$ and $ b \le \min \{ \frac{{{a^2}}}{3},\frac{c}{a},\frac{{ad}}{{2c}},\frac{{{c^2}}}{{3d}}\} $.
Problem 4. The $\frac{1}{k}($ for each $ k \in \mathbb{N})$ power of $G(z)$ is $$\sqrt[k]{{G(z)}} = \sum\limits_{i = 0}^\infty {q_i^{(k)}{z^i}} ,(\left| z \right| \le 1).$$ If some ${q_i^{(k)}}$ are negative, under which necessary and sufficient condition on $p_i$, $\sqrt[k]{{G(z)}}$ is absolutely convergent, namely $\sum\limits_{i = 0}^\infty {\left| {q_i^{(k)}} \right|} < \infty $?
When ${p_0} \ge {p_1} \ge {p_2} \ge \cdots \ge 0$ in Problem 4, this is Szekely's Discrete Convex Theorem (See Theorem 2.3.1 of Kerns(2004), page29). This problem appear firstly in Székely(2005). Székely(2005) discuss the condition that $\sqrt[k]{{G(z)}}$ is absolute convergence, when $G(z)$ is pgf of Bernoulli distibution. It is related to Negative probability(http://en.wikipedia.org/wiki/Negative_probability). Since the absolute convergence of $\sqrt[k]{{G(z)}}$ and $\ln G(z)$ are not equiavalent, so Problem 1 and Problem 4 are different.
Problem 1 can be seen as inverse problem of Problem 3, Problem 4 can be seen as inverse problem of Problem 2.
[1]Baishanski, B. (1999). Norms of powers and a central limit theorem for complex-valued probabilities. In Analysis of Divergence (pp. 523-543). Birkhäuser Boston.
[2]Beurling, A. (1938, August). Sur les intégrales de Fourier absolument convergentes et leur applicationa une transformation fonctionnelle. In Ninth Scandinavian Mathematical Congress (pp. 345-366).
[3]Hedstrom, G. W. (1967). Norms of powers of absolutely convergent Fourier series in several variables. The Michigan Mathematical Journal, 14(4), 493-495.
[4]Helson, H., & Beurling, A. (1953). Fourier-Stieltjes transforms with bounded powers. Mathematica Scandinavica, 1, 120-126.
[5]Kerns, G. J. (2004). Signed Measures in Exchangeability and Infinite Divisibility (Doctoral dissertation, Bowling Green State University).
[6]Johnson, N. L., Kemp, A. W., Kotz S. (2005). Univariate Discrete Distributions, 3ed. Wiley, New Jersey.
[7]Lévy, P. (1937). Sur les exponentielles de polynômes et sur l’arithmétique des produits de lois de Poisson. Annales scientifiques de l’École Normale Supérieure, 54, 231–292.
[8]Székely, G. J. (2005). Half of a coin: negative probabilities. Wilmott Magazine, 66-68.
[9]van Harn K.(1978). Classifying infinitely divisible distributions by functional equations. Amsterdam: Mathematisch.