Let $\begin{array}{} \mathcal{S}^*_B \end{array}$ be the class of normalized starlike functions associated with a function related to the Bell numbers. By establishing bounds on some coefficient functionals for the family of functions with positive real part, we derive for functions in the class $\begin{array}{} \mathcal{S}^*_B \end{array}$ several sharp coefficient bounds on the first six coefficients and also further sharp bounds on the corresponding Hankel determinants. Bounds on the first three consecutive higher-order Schwarzian derivatives for functions in the class $\begin{array}{} \mathcal{S}^*_B \end{array}$ are investigated.

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.

For analytic functionsfin the unit disk $${\mathbb {D}}$$D normalized by $$f(0)=0$$f(0)=0 and $$f'(0)=1$$f′(0)=1 satisfying in $${\mathbb {D}}$$D respectively the conditions $${{\,\mathrm{Re}\,}}\{ (1-z)f'(z) \}> 0,\ {{\,\mathrm{Re}\,}}\{ (1-z^2)f'(z) \}> 0,\ {{\,\mathrm{Re}\,}}\{ (1-z+z^2)f'(z) \}> 0,\ {{\,\mathrm{Re}\,}}\{ (1-z)^2f'(z) \} > 0,$$Re{(1-z)f′(z)}>0,Re{(1-z2)f′(z)}>0,Re{(1-z+z2)f′(z)}>0,Re{(1-z)2f′(z)}>0, the sharp upper bound of the third logarithmic coefficient in case when $$f''(0)$$f′′(0) is real was computed.

Let − 1 ≤ B < A ≤ 1 . The condition on β is determined so that 1 + β z p ′ ( z ) / p k ( z ) ≺ ( 1 + A z ) / ( 1 + B z ) ( − 1 < k ≤ 3 ) implies p ( z ) ≺ 1 + z . Similarly, the condition on β is determined so that 1 + β z p ′ ( z ) / p n ( z ) or p ( z ) + β z p ′ ( z ) / p n ( z ) ≺ 1 + z ( n = 0 , 1 , 2 ) implies p ( z ) ≺ ( 1 + A z ) / ( 1 + B z ) or 1 + z . In addition to that, the condition on β is derived so that p ( z ) ≺ ( 1 + A z ) / ( 1 + B z ) when p ( z ) + β z p ′ ( z ) / p ( z ) ≺ 1 + z . A few more problems of the similar flavor are also considered. MSC:30C80, 30C45.

In the present article, we consider certain subfamilies of analytic functions connected with the cardioid domain in the region of the unit disk. The purpose of this article is to investigate the estimates of the third Hankel determinant for these families. Further, the same bounds have been investigated for two-fold and three-fold symmetric functions.