In this work, a new generalized derivative operator M m α,β,λ 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 extensivelyinvestigated Fox-Wright p Ψ q function and generalized polylogarithm functions defined by M m α,β,λ f(z) = F α,β f(z) * D m λ f(z) = z + ∞ n=2 Γ (β)n m (n + λ − 1)! Γ [α(n − 1) + β]λ!(n − 1)! a n z n , where m ∈ N 0 = {0, 1, 2, 3,. . .} and min{ Re (α), Re (β)} > 0. By making use of M m α,β,λ f(z), a class of analytic functions is introduced. The sharp upper bound for the nonlinear |a 2 a 4 − a 2 3 | (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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