Successive coefficients of convex functions

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Abstract

It is shown that for $f$ analytic and convex in $z\in D=\{z:|z|<1\}$ and given by $f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n}$, the difference of coefficients $||a_{3}|-|a_{2}||\le 25/48$ and $||a_{4}|-|a_{3}||\le 25/48$ . Both inequalities are sharp.

Coefficient bounds & claims (3)

Machine-extracted from the paper text - useful for cross-referencing, not a verified fact.

coefficient_bound

||a_3| - |a_2|| ≤ 25/48 for class C (sharp) [Theorem (Section 4)]

coefficient_bound

||a_4| - |a_3|| ≤ 25/48 for class C (sharp) [Theorem (Section 4)]

function_family

Class C: f in A with Re(1 + z*f''(z)/f'(z)) > 0 for z in D

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