An Application of Hölder’s Inequality for Convolutions

Let Ap(n) be the class of analytic and multivalent functions f(z) in the open unit disk U. Furthermore, let Sp(n, α) and Tp(n, α) be the subclasses of Ap(n) consisting of multivalent starlike functions f(z) of order α and multivalent convex functions f(z) of order α, respectively. Using the coefficient inequalities for f(z) to be in Sp(n, α) and Tp(n, α), new subclasses S∗ p (n, α) and T ∗ p (n, α) are introduced. Applying the Hölder inequality, some interesting properties of generalizations of convolutions (or Hadamard products) for functions f(z) in the classes S∗ p (n, α) and T ∗ p (n, α) are considered.