On the uniqueness of the solution of Gahov equation for the functions in the Janowski classes

Let $\Delta=\{(\alpha,\beta)\in R^2 : \alpha+\beta >0,\alpha\leq1,\beta\leq1\}$ and $(\alpha,\beta)\in\Delta$. Janowski class $S^*[\alpha,\beta]$ is the class of the functions f holomorphic in $D$ and so that $f(0)=f'(0)-1=0$ and $\zeta f'(\zeta)/f(\zeta)\prec(1+\beta\zeta)/(1-\alpha\zeta)$, $\zeta \in D$. Let $\tilde{S}^*[\alpha,\beta]$ be the subclass of with the condition $f''(0)=0$ defining the zero root of the Gahov equation $f''(\zeta)/f'(\zeta)=2\bar{\zeta}/(1-|\zeta|^2)$. The domain of uniqueness for the family $\tilde{S}^*[\alpha,\beta]$, $(\alpha,\beta)\in\Delta$, is the set $\tilde{\Delta} \subset \Delta$ such that for any $(\alpha,\beta)\in \tilde{\Delta}$ and $f\in\tilde{S}^*[\alpha,\beta]$ the Gahov equation has the unique root. The maximal (on inclusion) domain of uniqueness for the family $\tilde{S}^*[\alpha,\beta]$, $(\alpha,\beta)\in\Delta$, is find. Let $\Delta'=\Delta'_0 \cup \Delta_1 \cup \Delta'_2$, where $\Delta'_0=\{(\alpha,\beta) \in \Delta : |2\beta-3\alpha| \leq 3, 3(\alpha+\beta)\leq 2\}$, $\Delta_1=\{(\alpha,\beta) \in \Delta : 2\beta-3\alpha > 3\}$ и $\Delta'_2=\{(\alpha,\beta) \in \Delta : 2\beta-3\alpha < -3, \alpha<\alpha(\beta),\beta \in(-1,-1/5)\}$, and $\alpha(\beta)=1-(1-\beta)^3/[(1+\beta)^2-16\beta]$, $\beta \in (-1,0)$. Theorem. The set is the maximal domain of uniqueness for the family of the classes $\tilde{S}^*[\alpha,\beta]$, $(\alpha,\beta)\in\Delta$. Thus, the article adduces the full and complete solution for the problem posed and particularly solved in 1998 by the second author. The impact of the result: 1) two-parametrical series of the uniqueness conditions is obtained; 2) new property of the well-known classes of the univalent functions is established. The proving method is based on the use of the Schwarz lemma, the calculation of the sharp constant in the estimate of the left-hand side of Gahov equation, and the analysis of the dependence of the constant mentioned on the parameters.