On the difference of inverse coefficients of convex functions

Abstract

AbstractLet f be analytic in the unit disk $${\mathbb {D}}=\{z\in {\mathbb {C}}:|z|<1 \}$$ D = { z ∈ C : | z | < 1 } , and $${\mathcal {S}}$$ S be the subclass of normalised univalent functions given by $$f(z)=z+\sum _{n=2}^{\infty }a_n z^n$$ f ( z ) = z + ∑ n = 2 ∞ a n z n for $$z\in {\mathbb {D}}$$ z ∈ D . Let F be the inverse function of f defined in some set $$|\omega |\le r_{0}(f)$$ | ω | ≤ r 0 ( f ) , and be given by $$F(\omega )=\omega +\sum _{n=2}^{\infty }A_n \omega ^n$$ F ( ω ) = ω + ∑ n = 2 ∞ A n ω n . We prove the sharp inequalities $$-1/3 \le |A_4|-|A_3| \le 1/4$$ - 1 / 3 ≤ | A 4 | - | A 3 | ≤ 1 / 4 for the class $${\mathcal {K}}\subset {\mathcal {S}}$$ K ⊂ S of convex functions, thus providing an analogue to the known sharp inequalities $$-1/3 \le |a_4|-|a_3| \le 1/4$$ - 1 / 3 ≤ | a 4 | - | a 3 | ≤ 1 / 4 , and giving another example of an invariance property amongst coefficient functionals of convex functions.