A Note on Generating Functions for Hausdorff Moment Sequences

For functions $f$ whose Taylor coefficients at the origin form a Hausdorff moment sequence we study the behaviour of $w(y):=|f(\gamma+iy)|$ for $y>0$ ($\gamma\leq1$ fixed).


Introduction and statement of the results
A sequence {a k } k≥0 of non-negative real numbers, a 0 = 1, is called a Hausdorff moment sequence if there is a probability measure 1 µ on [0, 1] such that a k = 1 0 t k dµ(t), k ≥ 0, or, equivalently, and F is its generating function. It is well known (Hausdorff [2]) that a sequence {a k } k≥0 with a 0 = 1 is a Hausdorff moment sequence if and only if it is completely monotone i.e. ∆ n a k := ∆ n−1 a k − ∆ n−1 a k+1 ≥ 0, k ≥ 0, n ≥ 1, where ∆ 0 is the identity operator: ∆ 0 a = a.
Let T denote the set of such generating functions F . They are analytic in the slit domain Λ := C \ [1, ∞) and also belong to the set of Pick functions P (−∞, 1) (see Donoghue [1] for more information on Pick functions).
Wirths [5] has shown that f ∈ T implies that the function zf (z) is univalent in the half-plane Re z < 1, and recently the theory of universally prestarlike mappings has been developed, showing a close link to T , see [4].
Many classical functions belong to T or are closely related to it. We mention only the polylogarithms where Li α (z)/z ∈ T and which we are going to study somewhat closer in the sequel.
The main result in this note is This relation does not hold, in general, for γ > 1.
Theorem 1.1 has the following immediate consequence.
In the case γ = 0 Theorem 1.1 admits a slight generalization. It is wellknown and easy to verify that T is invariant under the Hadamard product: if then also And therefore, under the same assumption, For the polylogarithms and 0 < α ≤ β it is clear that Li β = Li α * Li β−α so that we get This result can also be obtained and even strengthened using Corollary 1.2 and the deeper relation recently established in [4]. For a certain subset of T we can go one step beyond Corollary 1.2, as far as the behaviour of |f (iy)| for y > 0 is concerned.
Fundamental for the proof of Theorem 1.5 is the following result, which is based on a general theorem in [4].
One can show that the conclusion of Theorem 1.6 is not generally valid for f ∈ T . However, for the functions g α (z) : for which the assumptions of Theorem 1.5 are fulfilled. Thus both, Theorem 1.5 and Theorem 1.6, apply to g α .

Proofs
We first note that the convex set T satisfies the condition of the main theorem in [3], which for the present case can be stated as follows: Lemma 2.1. Let λ 1 , λ 2 be two continuous linear functionals on T and assume that 0 ∈ λ 2 (T ). Then the range of the functional Proof of Theorem 1.1 First we note that it is enough to prove (1.1) for γ = 1 only. This is because f ∈ T implies f (z − δ)/f (−δ) ∈ T for all δ > 0. In Lemma 2.1 we choose λ j (f ) := f (1 + iy j ), j = 1, 2. Since Im f (z) > 0 for f ∈ T and Im z > 0, it is clear that 0 ∈ λ 2 (T ). Lemma 2.1 now implies that for the proof of Theorem 1.1 we only need to show that the expression is located in the half-plane {w : Re w ≥ 1}. To simplify this expression we set κ := (1 − ρ)/ρ, τ := y 1 /y 2 . Then our claim is Re q(κ, y, τ, t 1 , t 2 ) ≥ 1, κ ≥ 0, y > 0, t 1 , t 2 , τ ∈ [0, 1], where q(κ, y, τ, t 1 , t 2 ) = Note that by symmetry we may assume that t 1 ≤ t 2 . For fixed y, τ, t 1 , t 2 the values of w(κ) := q(κ, y, τ, t 1 , t 2 ), κ ≥ 0, form a circular arc connecting the points w(0) = v(t 1 ) and It is easily checked, that under our assumptions for y and τ the function Re v(t) increases with t ∈ [0, 1], and, in particular, Re v(t) ≥ Re v(0) = 1. This implies 1 ≤ Re w(0) ≤ Re w(∞). We will prove that Re w ′ (0) ≥ 0. Once this done a simple geometric consideration shows that under these circumstances the circular arc w(κ), κ ≥ 0, cannot leave the half-plane {w : Re w ≥ 1}, which then completes the proof of (1.1).
Proof of Theorem 1.6 We have The condition is immediately fulfilled if t > x. Otherwise we are left with This requires that σ(t)/σ(t/x) increases with t. Taking logarithms and differentiating w.r.t. the variable t, we find as a necessary and sufficient condition for (2.1) that tσ ′ (t)/σ(t) decreases for t increasing. The result follows now from Lemma 2.2.