Bounds for extreme zeros of some classical orthogonal polynomials

We derive upper bounds for the smallest zero and lower bounds for the largest zero of Laguerre, Jacobi and Gegenbauer polynomials. Our approach uses mixed three term recurrence relations satisfied by polynomials corresponding to different parameter(s) within the same classical family. We prove that interlacing properties of the zeros impose restrictions on the possible location of common zeros of the polynomials involved and deduce strict bounds for the extreme zeros of polynomials belonging to each of these three classical families. We show numerically that the bounds generated by our method improve known lower (upper) bounds for the largest (smallest) zeros of polynomials in these families, notably in the case of Jacobi and Gegenbauer polynomials

Inequalities for Extreme Zeros of Some Classical Orthogonal and q-orthogonal Polynomials

Let {pn}1 n=0 be a sequence of orthogonal polynomials. We briefly review properties of pn that have been used to derive upper and lower bounds for the largest and smallest zero of pn. Bounds for the extreme zeros of Laguerre, Jacobi and Gegenbauer polynomials that have been obtained using different approaches are numerically compared and new bounds for extreme zeros of q-Laguerre and little q-Jacobi polynomials are proved

The intervention effects of a community-based hypertension control programme in two rural South African towns: the CORIS study

The objective of the hypertension programme of the Coronary Risk Factor Study (CORlS) was to evaluate the effectiveness of the first 4 years of community-based intervention. The hypertension intervention model comprised a blood pressure station where the whole population was screened for hypertension, non-drug management was provided and hypertensives were monitored after referral to general practitioners for drug therapy. Two levels of intervention were maintained: in the high-intensity intervention town (N =2 278) hypertensives were actively followed up, and in the low-intensity intervention town (N =2 620) no active follow-up procedure existed. A third town acted as control (N= 2 290).In the cohort which was hypertensive at baseline, the net decreases in systolic blood pressure (mean ± SE) after correction for changes in the control town were 0,5 ± 2,2 mmHg (men) and 4,5 ± 2,2 mmHg (women) in the low-intensity intervention town, and 5,6 ± 2,3 mmHg (men) and 7,5 ± 2,2 mmHg (women) in the high-intensity intervention town. The net decrease in diastolic blood pressure was 3,4 ± 1,2 mmHg (men) and 4,4 ± 1,1 mmHg (women) in the low-intensity intervention town, and 6,1 ± 1,2 mmHg (men) and 5,9 ± 1,1 mmHg (women) in the high-intensity intervention town. These reductions were statistically significant with one exception. The changes in the total population in the 3 communities after 4 years of intervention were similar to those found in the hypertensive cohort.Decreases in mean blood pressure were accompanied by marked increases in the proportion of hypertensives on drug treatment and the proportion under control (< 160/95 mmHg).Distribution curves of blood pressure indicated a large effect in the subgroup above the cut-off point for hypertension; however, the entire curve also shifted to the left, indicating, in addition, benefit to the whole population. An increase in the appropriate knowledge and action for hypertension control was observed in the intervention towns compared with the control town.The CORlS community-based hypertension control programme successfully reduced the risk for cardiovascular diseases in the intervention towns compared with the control town

Interlacing properties and bounds for zeros of 2ϕ1 hypergeometric and little q-Jacobi polynomials

Please read abstract in the article.http://link.springer.com/journal/111392017-05-30hb201