The n-Valent Convexity of Frasin Integral Operators

Let f_i, i 2 {1, 2, . . . ,k}, be an analytic function on the unit disk in the complex plane of the form f_i(z) = zⁿ + a_i,n+1zⁿ⁺¹ + . . . , n 𝜖 ℕ = {1, 2, . . .}. We consider the following Frasin integral operator:

$$ {G}_n(z)=\underset{0}{\overset{z}{\int }}n{\xi}^{\left(n-1\right)}{\left(\frac{f_1^{\prime}\left(\xi \right)}{n{\xi}^{\left(n-1\right)}}\right)}^{\alpha_1}\cdots {\left(\frac{f_k^{\prime}\left(\xi \right)}{n{\xi}^{\left(n-1\right)}}\right)}^{\alpha_k} d\xi . $$

We establish a sufficient condition under which this integral operator is n-valent convex and obtain other interesting results.