Estimates for polynomials in the unit disk with varying constant terms

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

Performing Ethnicity to Resist Marginalisation The Tangsa in Assam

Schendel, H.W. van [Promotor]Bal, E.W. [Copromotor

Stable functions and Vietoris' theorem

AbstractAn 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

Univalence criterion for meromorphic functions and Loewner chains

The object of the present paper is to obtain a more general condition for univalence of meromorphic functions in the U*. The significant relationships and relevance with other results are also given. A number of known univalent conditions would follow upon specializing the parameters involved in our main results

Inequalities for Cyclic Functions

AbstractThe nth cyclic function is defined byϕn(z)=∑ν=0∞znν(nν)!(z∈C,2⩽n∈N). We prove that if k is an integer with 1⩽k⩽n−1, then(n−k)!ϕ(k)n(x)xn−kα<ϕn(x)<(n−k)!ϕ(k)n(x)xn−kβ holds for all positive real numbers x with the best possible constantsα=1andβ=2n-k over n

Cluster Headache Impact Questionnaire (CHIQ) – a short measure of cluster headache related disability

Background Cluster headache (CH) is a severe, highly disabling primary headache disorder. However, there is little research on CH-related disability, and most of it is based on non CH-specific questionnaires. The aim of this study was to develop a short, CH-specific disability questionnaire. Methods The 8-item Cluster Headache Impact Questionnaire (CHIQ) was developed based on a literature review and patient and expert interviews. The questionnaire was tested in 254 CH patients (171 males; 47.5 ± 11.4 years; 111 chronic CH, 85 active episodic CH, 52 episodic CH in remission) from our tertiary headache center or from a German support group. Results Reliability and validity of the CHIQ was evaluated in active episodic and chronic CH patients ( n = 196). Internal consistency (Cronbach’s α = 0.88) and test-retest reliability (ICC 0.91, n = 41) were good. Factor analysis identified a single factor. Convergent validity was shown by significant correlations with the Headache Impact Test (HIT-6, r = 0.58, p < 0.001), subscales of the depression, anxiety and stress scales (DASS, r = 0.46–0.62; p < 0.001) and with CH attack frequency ( r = 0.41; p < 0.001). CHIQ scores significantly differentiated between chronic CH (25.8 ± 6.5), active episodic CH (23.3 ± 6.9) and episodic CH patients in remission (13.6 ± 11.9, p < 0.05 for all 3 comparisons). Conclusions The CHIQ is a short, reliable, valid, and easy to administer measure of CH-related disability, which makes it a useful tool for clinical use and research

Test-retest reliability of visual-evoked potential habituation

Objective Habituation of visual-evoked potentials (VEPs) is typically described as deficient interictally in migraine patients, supposedly indicating altered cortical excitability. Use of this parameter for monitoring changes over time, e.g. under treatment, requires demonstration of test-retest reliability. Methods VEPs were recorded interictally in 41 episodic migraine patients and 40 controls. N75-P100 amplitudes were measured over six consecutive blocks of 75 VEPs each. Amplitude regression slopes and block ratios were used to quantify VEP habituation. Test-retest reliability was assessed over 15 minutes and two to three weeks. Results Controls showed significantly more negative VEP habituation slopes than migraine patients (-0.210.40 vs. 0.04 +/- 0.46 mu V/block, p<0.05). Results were similar for block ratios, though, in the migraine group, VEP habituation significantly increased from test to two- to three-week retest (p<0.05). In addition, VEP habituation test-retest correlations were mostly poor both in migraine patients and controls (intraclass correlation coefficients, 15 minutes: -0.13 to 0.30, two to three weeks: 0.07 to 0.59). Conclusions Deficient VEP habituation in migraine was confirmed. However, the test-retest reliability of VEP habituation was rather weak. Therefore, we suggest that VEP habituation should be used for evaluation of cortical excitability under treatment only at the group level and only when a control group with sham treatment is included