Abstract
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|<1.617.
Highlights
An analytic function f on the unit disk D = {z ∈ C : |z| < 1} of the complex plane C is called convex if f maps D univalently onto a convex domain in C
The present paper offers the estimate |b9 | < 1.617
Let K denote the class of convex functions f normalized so that w = f (z) = z + a2 z2 + a3 z3 + · · ·
Summary
An analytic function f on the unit disk D = {z ∈ C : |z| < 1} of the complex plane C Let K denote the class of convex functions f normalized so that w = f (z) = z + a2 z2 + a3 z3 + · · · . This is sharp for every n and, the function f 0 (z) = z/(1 − z) = z + z2 + z3 + · · · satisfies the equality for all n.
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