Studies of convolution play an important role in Geometric Function Theory (GFT). Such studies attracted a large number of researchers in recent years. By making use of the Hadamard product (or convolution), several new and interesting subclasses of analytic and univalent functions have been introduced and investigated in the direction of well-known concepts such as the subordination and superordination inequalities, integral mean and partial sums, and so on. In this article, we apply the Hadamard product (or convolution) by utilizing some special functions. Our contribution in this paper includes defining a new linear operator in the form of the generalized Mittag-Leffler function in terms of the extensively-investigated Fox-Wright p Ψ qfunction in the right-half of the open unit disk where where (z) > 0. We then show that the new linear convolution operator is bounded in some spaces. In particular, several boundedness properties of this linear convolution operator under mappings from a weighted Bloch space into a weighted-log Bloch space are also investigated. For uniformity and convenience, the Fox-Wright p Ψ q -notation is used in our results.

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In this paper, several conditions are put in order to compose the sequence of partial sums ,  and  of the fractional operators of analytic univalent functions ,  and   of bounded turning which are bounded turning too.

Recently, the study of the fractional formal (operators, polynomials and classes of special functions) has been increased. This study not only in mathematics but extended to another topics. In this effort, we investigate a generalized integro-differential operator defined by a fractional formal (fractional differential operator) and study some its geometric properties by employing it in new subclasses of analytic univalent functions.

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