On Certain Exact Differential Subordinations Involving Convex Dominants

Let h be a non-vanishing convex univalent function and p be an analytic function in D. We consider the differential subordination psi(i)(p(z), zp' (z)) < h(z) with the admissible functions psi(1) := (beta p(z)+gamma)(-alpha) ((beta p(z)+gamma)/beta(1-alpha) + zp' (z)) and psi(2) := 1/root gamma beta arctan (root beta/gamma p(1-alpha) (z)) + (1-alpha/beta p(2(1-alpha))(z)+gamma zp' (z)/p(alpha)(z). The objective of this paper is to find the dominants, preferably the best dominant (say q) of the solution of the above differential subordination satisfying psi(i)(q(z), nzq'(z)) = h(z). Furthermore, we show that psi(i)(q(z), nzq' (z)) = h(z) is an exact differential equation and q is a convex univalent function in D. In addition, we estimate the sharp lower bound of Re p for different choices of h and derive a univalence criterion for functions in H (class of analytic normalized functions) as an application to our results.