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

In this work, a new generalized derivative operator 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 m(alpha,beta,lambda)(m) f(z) = f(alpha,beta) f(z) * D-lambda(m) f(z) = z + Sigma(infinity)(n=2) Gamma(beta)n(m)(n + lambda - 1)!/Gamma[alpha(n - 1) + beta]lambda!(n - 1)! a(n)z(n), where in m is an element of N-0 = {0,1,2, 3, . . .} and min{ Re (alpha), Re (beta)1 > 0. By making use of m(alpha,beta,lambda)(m) f(z), a class of analytic functions is introduced. The sharp upper bound for the nonlinear vertical bar a(2)a(4) - 4(3)(2)vertical bar (also called the second Hankel functional) is obtained. Relevant connections of the results presented here with those given in earlier works are also indicated.