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We apply the Schwarz lemma to find general formulas for the third coefficient of Carathéodory functions dependent on a parameter in the closed unit polydisk. Next we find sharp estimates of the Hankel determinant 
$H_{2,2}$ and Zalcman functional 
$J_{2,3}$ over the class 
${\mathcal{C}}{\mathcal{V}}$ of analytic functions 
$f$ normalised such that 
$\text{Re}\{(1-z^{2})f^{\prime }(z)\}>0$ for 
$z\in \mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$, that is, the subclass of the class of functions convex in the direction of the imaginary axis.
