On one family of holomorphic in a circle of functions with positive real part of $n$th derivative

Let $\Phi(z)=z^n+b_2z^{n+1}+b_3z^{n+2}+\cdots$ be a holomorphic in the unit circle $|z|<1$ function with $b_k\ge0$, $k=2,3,\dots$. Let $V_n(\Phi)$ be a family of functions $F(z)=z^n+a_2z^{n+1}+a_3z^{n+2}+\cdots$, for which $|a_k|\le b_k$, $k=2,3,\dots$. The radius of the greatest circle is established for which every function $F(z)\in V_n(\Phi)$ satisfies the condition $\operatorname{Re}F^{(n)}(z)>0$ .