On the Ninth Coefficient of the Inverse of a Convex Function

We consider the inverse function z=g(w)=w+b2w2+⋯ of a normalized convex univalent function w=f(z)=z+a2z2+⋯ on the unit disk in the complex plane. So far, it is known that |bn|≤1 for n=2,3,⋯,8. On the other hand, the inequality |bn|≤1 is not valid for n=10. It is conjectured that |b9|≤1. The present...

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Bibliographic Details
Published in:Mathematics (Basel) 2021-04, Vol.9 (7), p.706
Main Author: Sugawa, Toshiyuki
Format: Article
Language:English
Subjects:
coefficient estimates
Conflicts of interest
convex functions
Food science
inverse function
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Summary:We consider the inverse function z=g(w)=w+b2w2+⋯ of a normalized convex univalent function w=f(z)=z+a2z2+⋯ on the unit disk in the complex plane. So far, it is known that |bn|≤1 for n=2,3,⋯,8. On the other hand, the inequality |bn|≤1 is not valid for n=10. It is conjectured that |b9|≤1. The present paper offers the estimate |b9|
ISSN:2227-7390
2227-7390
DOI:10.3390/math9070706