In this paper, we investigate several fuzzy differential subordinations that are connected with the Borel distribution series $\mathcal{B}\left(\lambda ,\alpha ,\beta \right)\left(z\right)$ of the Mittag-Leffler type, which involves the two-parameter Mittag-Leffler function $E_{\alpha ,\beta }\left(z\right)$. Using the above-mentioned operator $\mathcal{B}\left(\lambda ,\alpha ,\beta \right),$ we also introduce and study a class ${\mathcal{M}}_{\lambda ,\alpha ,\beta }^F\left(\eta \right)$ of holomorphic and univalent functions in the open unit disk $\Delta$. The Mittag-Leffler-type functions, which we have used in the present investigation, belong to the significantly wider family of the Fox-Wright function ${}_p{\Psi }_q\left(z\right)$, whose p numerator parameters and q denominator parameters possess a kind of symmetry behavior in the sense that it remains invariant (or unchanged) when the order of the p numerator parameters or when the order of the q denominator parameters is arbitrarily changed. Here, in this article, we have used such special functions in our study of a general Borel-type probability distribution, which may be symmetric or asymmetric. As symmetry is generally present in most works involving fuzzy sets and fuzzy systems, our usages here of fuzzy subordinations and fuzzy membership functions potentially possess local or non-local symmetry features.

In this paper we introduce a new subclass of the bi-univalent functions defined in the open unit disc and connected with a q-analogue derivative. We find estimates for the first two Taylor-Maclaurin coefficients $\left|a_2\right|$ and $\left|a_3\right|$ for functions in this subclass, and we obtain an estimation for the Fekete-Szegő problem for this function class.

The aim of this article is to make a connection between the Pascal distribution series and some subclasses of normalized analytic functions whose coeffcients are probabilities of the Pascal distribution. For these functions, for linear combinations of these functions and their derivatives, for operators defined by convolution products, and for the Alexander-type integral operator, we find simple suffcient conditions such that these mapping belong to a general class of functions defined and studied by Goodman, Rønning, and Bharati et al.

In this paper, we obtain some applications of the theory of differential subordination, differential superordination, and sandwich-type results for some subclasses of symmetric functions connected with a q-analog integral operator.

Institute of Mathematics of the Czech Academy of Sciences

In this paper, we introduce new subclasses of the function classΣof bi-univalent functions connected with aq-analogue of Bessel function and defined in the open unit disc. Furthermore, we find estimates on the first two Taylor-Maclaurin coefficientsa2anda3for functions in these new subclasses.