Hankel determinant of third kind for certain subclass of multivalent analytic functions

The objective of this paper is to obtain an upper bound (not sharp) to the third order Hankel determinant for certain subclass of multivalent ($p$-valent) analytic functions, defined in the open unit disc $E$. Using the Toeplitz determinants, we may estimate the Hankel determinant of third kind for the normalized multivalent analytic functions belongng to this subclass. But, using the technique adopted by Zaprawa [1], i. e., grouping the suitable terms in order to apply Lemmas due to Hayami [2], Livingston [3] and Pommerenke [4], we observe that, the bound estimated by the method adopted by Zaprawa is more refined than using upon applying the Toeplitz determinants.