For q ∈ (0, 1) let the q-difference operator be defined as follows $$\partial _q f(z) = \frac{{f(qz) - f(z)}} {{z(q - 1)}} (z \in \mathbb{U}),$$ where \(\mathbb{U}\) denotes the open unit disk in a complex plane. Making use of the above operator the extended Ruscheweyh differential operator R _q ^λ f is defined. Applying R _q ^λ f a subfamily of analytic functions is defined. Several interesting properties of a defined family of functions are investigated.

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.

The aim of a study of the presented paper is the differential subordination involving harmonic means of the expressions \(p(z)\) , \(p(z) + zp'(z)\) , and \(p(z) + \frac{zp'(z)}{p(z)}\) when \(p\) is an analytic function in the unit disk, such that \(p(0)=1, p(z)\not \equiv 1\) . Several applications in the geometric functions theory are given.

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.