Sharp bounds for sums of coefficients of inverses of convex functions (Q2467663)

Let \(f\) be a holomorphic and injective function in the open unit disc \(\mathbb{D}\) normalized by \(f(0)=f'(0)-1=0.\) Also, let \(F\) be the inverse of \(f,\) \(k\) be a positive integer and let \(A_{n,k}\) be the Taylor coefficients defined by \[ (F(w))^k= \sum_{n=k}^\infty A_{n,k}w^n. \] In this paper the authors conjecture that for any \(n\geq2\) \[ \sum_{k=1}^n \left| A_{n,k} \right| \leq 2^{n-1}, \] and prove its weaker form \[ \left| \sum_{k=1}^n A_{n,k} \right| \leq 2^{n-1}. \] The last inequality is sharp as the function \(f(z)=z/(1+z)\) shows.