Browsing by Author "Ruscheweyh, Stephan"

For functions f whose Taylor coefficients at the origin form a Hausdorff moment sequence we study the behaviour of w(y) := |f(γ + iy)| for y > 0 (γ ≤ 1 fixed)

The nth cyclic function is defined by φₙ(z)=∑ᵥ₌₀∞ zⁿᵛ/(nν)! (z∈ℂ, 2≤n∈ℕ). We prove that if k is an integer with 1≤k≤n−1, then (n−k)! φₙ^(k)(x)xⁿ⁻ᵏ⁻ᵅ < φₙ(x) < (n−k)! φₙ^(k)(x)xⁿ⁻ᵏ⁻ᵝ holds for all positive real numbers x with the best possible constants α=1 and β=(2n−k)/n.

D. Brannan's conjecture says that for 0 < 𝛼 ,𝛽≤1 0<α,β≤1, ∣𝑥∣=1 ∣x∣=1, and 𝑛 ∈ 𝑁 n∈N one has ∣𝐴2𝑛−1 (𝛼,𝛽,𝑥)∣≤∣𝐴2𝑛−1(𝛼,𝛽,1)∣ ∣A 2n−1 (α,β,x)∣≤∣A 2n−1 (α,β,1) ∣, where (1+𝑥𝑧 ) 𝛼 (1−𝑧) 𝛽 = ∑ 𝑛 = 0 ∞ 𝐴 𝑛 (𝛼,𝛽,𝑥) 𝑧 𝑛 (1−z) β (1+xz) α =∑ n=0 ∞ A n (α,β,x)z n .We prove this for the case 𝛼=𝛽 α=β, and also prove a differentiated version of the Brannan conjecture. This has applications to estimates for Gegenbauer polynomials and also to coefficient estimates for univalent functions in the unit disk that are ‘starlike with respect to a boundary point’. The latter application has previously been conjectured by H. Silverman and E. Silvia. The proofs make use of various properties of the Gauss hypergeometric function.

We give a new and more manageable characterization for Cyclic Pólya Frequency functions of order 3 (CPF3). Our result also improves present knowledge concerning smoothness properties in CPF. In particular, a conjecture of Mairhuber, Schoenberg, and Williamson, On variation-diminishing transformations on the circle, Rend. Circ. Mat. Palermo (2) 8 (1959), 1-30, about discontinuous CPF functions is established.

An analytic function f (z) in the unit disc D is called stable if sn(f,·)/f ≺ 1/f holds for all for n ∈ N0. Here sn stands for the nth partial sum of the Taylor expansion about the origin of f , and ≺ denotes the subordination of analytic functions in D. We prove that (1 − z)λ, λ ∈ [−1, 1], are stable. The stability of √(1 + z)/(1 − z) turns out to be equivalent to a famous result of Vietoris on non-negative trigonometric sums. We discuss some generalizations of these results, and related conjectures, always with an eye on applications to positivity results for trigonometric and other polynomials