Recent zbMATH articles in MSC 30C55 https://www.zbmath.org/atom/cc/30C55 2022-05-16T20:40:13.078697Z Werkzeug Convolution conditions for two subclasses of analytic functions defined by Jackson $$q$$-difference operator https://www.zbmath.org/1483.30036 2022-05-16T20:40:13.078697Z "El-Emam, Fatma Z." https://www.zbmath.org/authors/?q=ai:el-emam.fatma-z Summary: By using Jackson $$q$$-derivative, some characterizations in terms of convolutions for two classes of analytic functions in the open unit disc are given. Also, coefficient conditions and inclusion properties for functions in these classes are found. On the coefficients of certain subclasses of harmonic univalent mappings with nonzero pole https://www.zbmath.org/1483.31001 2022-05-16T20:40:13.078697Z "Bhowmik, Bappaditya" https://www.zbmath.org/authors/?q=ai:bhowmik.bappaditya "Majee, Santana" https://www.zbmath.org/authors/?q=ai:majee.santana Summary: Let $$Co(p), p\in (0,1]$$ be the class of all meromorphic univalent functions $$\varphi$$ defined in the open unit disc $$\mathbb{D}$$ with normalizations $$\varphi (0)=0=\varphi^{\prime} (0)-1$$ and having simple pole at $$z=p\in (0,1]$$ such that the complement of $$\varphi (\mathbb{D})$$ is a convex domain. The class $$Co(p)$$ is called the class of concave univalent functions. Let $$S_H^0 (p)$$ be the class of all sense preserving univalent harmonic mappings $$f$$ defined on $$\mathbb{D}$$ having simple pole at $$z=p\in (0,1)$$ with the normalizations $$f(0)=f_z (0)-1=0$$ and $$f_{\bar{z}}(0)=0$$. We first derive the exact regions of variability for the second Taylor coefficients of $$h$$ where $$f=h+\overline{g}\in S_H^0 (p)$$ with $$h-g\in Co(p)$$. Next we consider the class $$S_H^0 (1)$$ of all sense preserving univalent harmonic mappings $$f$$ in $$\mathbb{D}$$ having simple pole at $$z=1$$ with the same normalizations as above. We derive exact regions of variability for the coefficients of $$h$$ where $$f=h+\overline{g}\in S_H^0 (1)$$ satisfying $$h-e^{2i\theta}g\in Co(1)$$ with dilatation $$g^{\prime} (z)/h^{\prime} (z)=e^{-2i\theta}z$$, for some $$\theta, 0\leq \theta <\pi$$.