Second Hankel determinant for a class defined by modified Mittag-Leffler with generalized polylogarithm functions

Volume 18, Issue 4, pp 453--459 http://dx.doi.org/10.22436/jmcs.018.04.06 Publication Date: December 12, 2018       Article History

Authors

M. N. M. Pauzi - School of Modelling and Data Science (Previously: School of Mathematical Sciences), Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600 Selangor D.E., Malaysia. M. Darus - School of Modelling and Data Science (Previously: School of Mathematical Sciences), Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600 Selangor D.E., Malaysia. S. Siregar - Department of Science and Biotechnology, Faculty of Engineering and Life Sciences, Universiti Selangor, Batang Berjuntai, Bestari, Jaya 45600, Selangor D.E., Malaysia.


Abstract

In this work, a new generalized derivative operator \( \mathfrak{M}_{\alpha,\beta,\lambda}^{m}\) is introduced. This operator obtained by using convolution (or Hadamard product) between the linear operator of the generalized Mittag-Leffler function in terms of the extensively-investigated Fox-Wright \(_{p}\Psi_{q}\) function and generalized polylogarithm functions defined by \[ \mathfrak{M}_{\alpha,\beta,\lambda}^{m}f(z)=\mathfrak{F}_{\alpha,\beta}f(z)*\mathfrak{D}_{\lambda}^{m}f(z) = z+\sum_{n=2}^{\infty}\frac{\Gamma(\beta)n^{m}(n+\lambda-1)!}{\Gamma[\alpha(n-1)+\beta]\lambda ! (n-1)!}a_{n}z^{n}, \] where \(m \in \mathbb{N}_{0} = \{0,1,2,3,\ldots\}\) and \(\min\{Re(\alpha),Re(\beta)\}>0\). By making use of \(\mathfrak{M}_{\alpha,\beta,\lambda}^{m}f(z)\), a class of analytic functions is introduced. The sharp upper bound for the nonlinear \(|a_{2}a_{4}-a_{3}^{2}|\) (also called the second Hankel functional) is obtained. Relevant connections of the results presented here with those given in earlier works are also indicated.


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