Denote by $${{\cal P}_{\log}}$$ the set of all non-constant Pick functions f whose logarithmic derivatives f′/f also belong to the Pick class. Let $${\cal U}({\rm{\Lambda}})$$ be the family of functions z · f(z), where $$f \in {{\cal P}_{\log}}$$ and f is holomorphic on Λ ≔ ℂ [1, + √). Important examples of functions in $${\cal U}({\rm{\Lambda}})$$ are the classical polylogarithms $$L{i_\alpha}(z): = \sum\nolimits_{k = 1}^\infty {{z^k}} /{k^\alpha}$$ for α ≥ 0; see [5](2015). In this note we prove that every $$\varphi \in {\cal U}({\rm{\Lambda}})$$ is universally starlike, i.e., φ maps every circular domain in Λ containing the origin one-to-one onto a starlike domain. Furthermore, we show that every non-constant function $$f \in {{\cal P}_{\log}}$$ belongs to the Hardy space Hp on the upper half-plane for some constant p = p(f) > 1, unless f is proportional to some function (a − z)−θ with a ∊ ℝ and 0 <θ ≤ 1. Finally we derive a necessary and sufficient condition on a real-valued function υ for which there exists $$f \in {{\cal P}_{\log}}$$ such that υ (x) = limε↓0 lim f(x + iε) for almost all x ∊ ℝ.

The Ramanujan sequence $ \{\theta_{n}\}_{n \geq 0}$, defined as $$ \theta_{0}= \frac{1}{2} \ , \ \ \ \theta_{n} = \left(\ \ \frac{e^{n}}{2} - \sum_{k=0}^{n-1} \frac{n^{k}}{k !} \ \ \right) \cdot \frac{n !}{n^{n}} \ , \ \ n \geq 1 \ ,$$ has been studied on many occasions and in many different contexts. J.Adell and P.Jodra (2008) and S. Koumandos (2013) showed, respectively, that the sequences $\{\theta_{n}\}_{n \geq 0}$ and $\{4/135 - n \cdot (\theta_{n}- 1/3 )\}_{n \geq 0}$ are completely monotone. In the present paper we establish that the sequence $\{(n+1)(\theta_{n}- 1/3 )\}_{n \geq 0}$ is also completely monotone. Furthermore, we prove that the analytic function $(\theta_{1}- 1/3 )^{-1} \sum_{n=1}^{\infty} (\theta_{n}- 1/3 ) \cdot z^{n} / n^{\alpha} $ is universally starlike for every $ \alpha \geq 1 $ in the slit domain $ \mathbb{C} \setminus [1,\infty)$. This seems to be the first result putting the Ramanujan sequence into the context of analytic univalent functions and is a step towards a previous stronger conjecture, proposed by S.Ruscheweyh, L.Salinas and T.Sugawa in 2009, namely that the function $(\theta_{1}- 1/3 )^{-1} \sum_{n=1}^{\infty} (\theta_{n}- 1/3 ) \cdot z^{n} $ is universally convex.

We show that a certain interpolation type inequality for polynomials in the unit disk, generalizing Bernstein's inequality, is actually contained in an older, even more general one. We also discuss the cases of equality.

The Ramanujan sequence {θ_n}_n ≥ 0, defined as θ_0= 1/2 , θ_n = ( e^n/2 - ∑_k=0^n-1n^k/k !) ·n !/n^n , n ≥ 1 , has been studied on many occasions and in many different contexts. J.Adell and P.Jodra (2008) and S. Koumandos (2013) showed, respectively, that the sequences {θ_n}_n ≥ 0 and {4/135 - n · (θ_n- 1/3 )}_n ≥ 0 are completely monotone. In the present paper we establish that the sequence {(n+1)(θ_n- 1/3 )}_n ≥ 0 is also completely monotone. Furthermore, we prove that the analytic function (θ_1- 1/3 )^-1∑_n=1^∞ (θ_n- 1/3 ) · z^n / n^α is universally starlike for every α≥ 1 in the slit domain ℂ∖ [1,∞). This seems to be the first result putting the Ramanujan sequence into the context of analytic univalent functions and is a step towards a previous stronger conjecture, proposed by S.Ruscheweyh, L.Salinas and T.Sugawa in 2009, namely that the function (θ_1- 1/3 )^-1∑_n=1^∞ (θ_n- 1/3 ) · z^n is universally convex.

A deep result of J. Lewis (1983) shows that the polylogarithms Li-alpha(z) := Sigma(infinity)(k = 1) z(k)/k(alpha) map the open unit disk D centered at the origin one-to-one onto convex domains for all alpha >= 0. In the present paper this result is generalized to the so-called universal convexity and universal starlikeness (with respect to the origin) in the slit-domain Lambda := C \ [1,infinity), introduced by S. Ruscheweyh, L. Salinas and T. Sugawa (2009). This settles a conjecture made in that work and proves, in particular, that Li-alpha(z) maps an arbitrary open disk or half-plane in. one-to-one onto a convex domain for every alpha >= 1.

Let 1≤p<∞. We show that 'positive polynomial approximation property' holds in the space Lp(R,dμ) (or Cw0) if and only if the algebraic polynomials are dense in L2p(R,dμ) (or Cw0). If μ is not a 2p-singular measure (or w is not a singular weight), this also implies the more general 'oscillation-diminishing polynomial approximation property'.

Let 1 <= p < infinity. We show that 'positive polynomial approximation property' holds in the space L-p(R, d mu) (or C-w(0)) if and only if the algebraic polynomials are dense in L-2p(R, d mu) (or C-root w(0)). If mu is not a 2p-singular measure (or root w is not a singular weight), this also implies the more general 'oscillation-diminishing polynomial approximation property'. (C) 2012 Elsevier Inc. All rights reserved.

Given n+1 angles 0≤θ0<θ1⋯<θn≤π, we discuss various extremal problems over the class of polynomials Pn endowed with the norm |p|n=max0≤j≤n|p(eiθj)+p(e−iθj)2|.

We establish special cases of a conjecture of S.P. Robinson [S.P. Robinson, Approximate identities for certain dual classes, DPhil thesis, University of York, UK, 1996] concerning Cesàro means of certain classes of analytic functions in the unit disk. This has applications, for instance, to the so-called Kaplan classes and subordination under ‘linearly accessible’ functions.

Let \(\| \cdot\|\) be the uniform norm in the unit disk. We study the quantities \(M_n(\alpha) := \inf(\|zP(z) + \alpha\|-\alpha)\) where the infimum is taken over all polynomials \(P\) of degree \(n-1\) with \(\|P(z)\| = 1\) and \(\alpha> 0\). In a recent paper by Fournier, Letac and Ruscheweyh (Math. Nachrichten 283 (2010), 193-199) it was shown that \(\inf_{\alpha> 0} M_n(\alpha) = 1/n\). We find the exact values of \(M_n(\alpha)\) and determine corresponding extremal polynomials. The method applied uses known cases of maximal ranges of polynomials.

We give various estimates and discuss sharpness questions for a generalized Bohr radius applicable to simply connected domains of the complex plane.

Let parallel to . parallel to denote the uniform norm in the unit disk of the complex plane C. The main result in this note is as follows: For any complex polynomial P of degree at most n and any alpha is an element of C the inequality parallel to P parallel to <= (n + 1)(parallel to z P(z) + alpha parallel to - vertical bar alpha vertical bar) holds. For any alpha not equal 0 the factor n + 1 is best possible, and we determine the cases of equality. (C) 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

In this paper the theory of Hadamard product multipliers is extended from the unit disk in the complex plane to arbitrary so-called disk-like domains, i.e. such domains which are the union of disks or half-planes, all containing the origin. In such a domain, say Ω, we define (the class Rαd(Ω) of) generalized prestarlike functions of order α⩽1 and ask for Hadamard multipliers g analytic at z=0 for which f∈Rαd(Ω) implies g∗f∈Rαd(Ω). We prove that such a multiplier necessarily has to be analytic inΩ∗:={uv:u∈Ω,v∈C∖Ω}. In many cases (we prove this for all proper disks containing the origin) we actually find that Rαd(Ω∗) is the precise description of the set of all such multipliers. For these disks, Ωγ say, the domains Ωγ∗ turn out to be bounded by the outer loops of certain Limaçons of Pascal. The parameter γ is related to the characteristic q(Ωγ)=(1−γ)/(1+γ):=r/s of the disk, where r is the shortest distance of the origin to the boundary of that disk, and s the largest. Large subclasses of Rαd(Ω∗) are being explicitly determined. For the case γ=0, i.e. Ωγ=Ωγ∗=D, this result coincides with an old one by Ruscheweyh and Sheil-Small, previously conjectured by G. Pólya and I.J. Schoenberg. The notion of the characteristic of a disk (containing the origin) is then extended to general disk-like domains, and some multipliers are identified for those general classes Rαd(Ω). The previously determined class of ‘universally prestarlike functions’, defined in the slit-domain C∖[1,∞], is identified as the class of ‘universal multipliers’ for Rαd(Ω) in any disk-like domain Ω.

We introduce universally convex, starlike and prestarlike functions in the slit domain ℂ [1, ∞), and show that there exists a very close link to completely monotone sequences and Pick functions.

We study inequalities connecting the product of uniform norms of polynomials with the norm of their product. This circle of problems include the Gelfond-Mahler inequality for the unit disk and the Kneser-Borwein inequality for the segment $[-1,1]$. Furthermore, the asymptotically sharp constants are known for such inequalities over arbitrary compact sets in the complex plane. It is shown here that this best constant is smallest (namely: 2) for a disk. We also conjecture that it takes its largest value for a segment, among all compact connected sets in the plane.