Applications of the normalized Le Roy-type Mittag-Leffler function on partial sums of analytic functions

In the present study, our main aim is to give lower bounds for the ratio of some the normalized Le Roy-type Mittag-Leffler function and their sequences of partial sums, as well as for the quotients of the derivative of the normalized Le Roy-type Mittag-Leffler function and their partial sums. In addition, some applications related to the obtained results are given. In view of the importance of these results, their geometric interpretation of the image domain is also included.