Let ${\mathcal{ST}}_L\left(s\right)$ and ${\mathcal{CV}}_L\left(s\right)$ denote the family of analytic and normalized functions f in the unit disk $\mathbb{D}:=\left\{z:\left|z\right|<1\right\}$ , such that the quantity $z{f}^{\prime }\left(z\right)/f\left(z\right)$ or $1+z{f}^{″}\left(z\right)/f^{\prime }\left(z\right)$ respectively are lying in the region bounded by the limaçon ${\left[{(u-1)}^2+v^2-s^4\right]}^2=4s^2\left[{\left(u-1+s^2\right)}^2+v^2\right],$ where $0<s\le 1/\sqrt{2}$ . The limaçon of Pascal is a curve that possesses properties which qualify it for the several applications in mathematics, statistics (hypothesis testing problem) but also in mechanics (fluid processing applications, known limaçon technology is employed to extract electrical power from low-grade heat, etc.). In this paper we present some results concerning the behavior of f on the classes ${\mathcal{ST}}_L\left(s\right)$ or ${\mathcal{CV}}_L\left(s\right)$ . Some appropriate examples are given.

We consider a family of analytic and normalized functions with the property that $zf'(z)/f(z)$z f ′ ( z ) / f ( z ) (or $1+zf''(z)/f'(z)$1 + z f ″ ( z ) / f ′ ( z ) ) lies in a domain bounded by a right branch of a hyperbola $\rho =\rho (s)= ( 2 \cos \frac{\varphi }{s} )^{-s}$ρ = ρ ( s ) = ( 2 cos φ s ) − s , where $0< s\le 1$0 < s ≤ 1 and $|\varphi |<(\pi s)/2$| φ | < ( π s ) / 2 . A comprehensive characteristic of that families and relations with the well known families of univalent functions are presented. Some relevant examples are indicated.

Let 𝓐 denote the family of analytic functionsfwithf(0) =f′(0) – 1 = 0, in the open unit disk Δ. We consider a class $$\begin{array}{} \displaystyle \mathcal{S}^{\ast}_{cs}(\alpha):=\left\{f\in\mathcal{A} : \left(\frac{zf'(z)}{f(z)}-1\right)\prec \frac{z}{1+\left(\alpha-1\right) z-\alpha z^2}, \,\, z\in \Delta\right\}, \end{array}$$ where 0 ≤α≤ 1/2, and ≺ is the subordination relation. The methods and techniques of geometric function theory are used to get characteristics of the functions in this class. Further, the sharp inequality for the logarithmic coefficientsγ_noff∈ $\begin{array}{} \mathcal{S}^{\ast}_{cs} \end{array}$ (α): $$\begin{array}{} \displaystyle \sum_{n=1}^{\infty}\left|\gamma_n\right|^2 \leq \frac{1}{4\left(1+\alpha\right)^2}\left(\frac{\pi^2}{6}-2 \mathrm{Li}_2\left(-\alpha\right)+ \mathrm{Li}_2\left(\alpha^2\right)\right), \end{array}$$ where Li₂denotes the dilogarithm function are investigated.

We consider a family of analytic and normalized functions that are related to the domains ℍ(s), with a right branch of a hyperbolas H(s) as a boundary. The hyperbola H(s) is given by the relation 1 ρ = 2 cos ⁡ φ s s ( 0 < s ≤ 1 , | φ | < ( π s ) / 2 ) . $\begin{array}{} \frac{1}{\rho}=\left( 2\cos\frac{\varphi}{s}\right)^s\quad (0 \lt s\le 1,\, |\varphi| \lt (\pi s)/2). \end{array}$ We mainly study a coefficient problem of the families of functions for which zf′/f or 1 + zf″/f′ map the unit disk onto a subset of ℍ(s) . We find coefficients bounds, solve Fekete-Szegö problem and estimate the Hankel determinant.

We find an unifying approach to the analytic representation of the domain bounded by a generalized Pascal snail. Special cases as the Pascal snail, Both leminiscate, conchoid of the Sluze and a disc are included. The behaviour of functions related to generalized Pascal snail is studied.

Let $\mathcal{S}^{\ast}_{L}(\lambda)$ be the class of functions $f$, analytic in the unit disc $\Delta=\{z:|z|<1\}$, with the normalization $f(0)=f'(0)-1=0$, which satisfy the condition\begin{equation*}\frac{zf'(z)}{f(z)}\prec \left(1+z\right)^{\lambda},\end{equation*}where $\prec$ is the subordination relation. The class $\mathcal{S}^{\ast}_{L}(\lambda)$ is a subfamily of the known class of strongly starlike functions of order $\lambda$. In this paper,the relations between $\mathcal{S}^{\ast}_{L}(\lambda)$ and other classes geometrically defined are considered. Also, we obtain some characteristics such as, bounds for coefficients, radius of convexity, the Fekete-Szeg\"{o} inequality, logarithmic coefficients and the second Hankel determinant inequality for functions belonging to this class. The univalent functions $f$ which satisfy the condition\begin{equation*}\Re\left\{1+\frac{zf''(z)}{f'(z)}\right\}<1+\frac{\lambda}{2},\qquad(z \in \Delta)\end{equation*}are also considered here.