Estimates for the real taylor coefficients in one function class

Considering the class $\widetilde K^R_n(E)$ of analytic functions $F(z)=z^n+a_{2,n}z^{n+1}+a_{3,n}z^{n+2}+\cdots$ in the unit disk with $a_{m,n}\in\mathbb R$ and the nonvanishing $n$th divided difference $[F(z);z_0,\dots,z_n]$ for all $z_0,\dots,z_n\in E$ we establish that $|a_{k,n+2}|\le(k\gamma_{k,n}-1)/(\gamma_{k,n}+k-2)$, where $\gamma_{k,n}=\max|a_{k,n}|$. If $n$ is an odd number then $\gamma_{k,n}=(n+k-1)/(n+1)$.