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An improvement of Ozaki’s P-valent conditions
Nunokawa, M.,Cho, N. E.,Kwon, O. S.,Sokó,ł, J. Springer Business + Business Media 2016 Acta mathematica Sinica. English series Vol.32 No.4
<P>The old result due to [Ozaki, S.: On the theory of multivalent functions II. Sci. Rep. Tokyo Bunrika Daigaku Sect. A, 45-87 (1941)], says that if f(z) = z(p) + Sigma(infinity)(n=p+1) a(n)z(n) is analytic in a convex domain D and for some real alpha we have Re{exp(i alpha)f((p))(z)} > 0 in D, then f(z) is at most p-valent in D. In this paper, we consider similar problems in the unit disc D = {z is an element of C : vertical bar z vertical bar < 1}.</P>
Jaeger, Sara R.,Lee, Soh Min,Kim, Kwang-Ok,Chheang, Sok L.,Roigard, Christina M.,Ares, Gastó,n Elsevier 2018 Food quality and preference Vol.68 No.-
<P><B>Abstract</B></P> <P>Questionnaires are popular in product-focused emotion research with consumers. Ease of implementation is paramount in this regard, as is versatility. In the presented studies, focus is directed to scaling variations as an element of methodological versatility, and a comparison is performed of CATA and RATA question formats (check-all-that-apply and rate-all-that-apply, respectively). Five studies, with a range of tasted products and food/beverage names were conducted, involving 908 consumers in New Zealand, China and Korea. Emoji questionnaires, recently established as a methodological variant in product-elicited emotion research, were used. The average percentage of emoji used for describing samples was similar for CATA and RATA questions when used in Central Location Tests with tasted samples, but higher for RATA than CATA questions in online surveys. Discriminative ability of CATA and RATA questions was linked to the characteristics of the focal samples. The recommendation for method choice is to use CATA emoji-questions when samples have distinct emotional associations, whereas RATA seems better able to discriminate between samples with more similar emotional profiles. Neither CATA nor RATA emoji-questions were regarded by consumers as difficult or tedious.</P> <P><B>Highlights</B></P> <P> <UL> <LI> CATA and RATA questions were compared using emoji questionnaires. </LI> <LI> In 3 of 5 studies the average percentage of emoji use was higher in RATA than CATA questions. </LI> <LI> In 3 of 5 studies there was evidence of similar or greater sample discrimination by CATA than RATA questions. </LI> <LI> CATA and RATA questions were perceived as easy and not tedious. </LI> <LI> Choice of CATA vs RATA should be made in conjunction with sample set characteristics. </LI> </UL> </P>