Geometric behavior of a class of algebraic differential equations in a complex domain using a majorization concept

Abstract

In this paper, a type of complex algebraic differential equations (CADEs) is considered formulating by \[\alpha [\varphi(z) \varphi” (z) +(\varphi' (z))^2]+ a_m \varphi^m(z)+a_{m-1} \varphi^{m-1}(z)+...+ a_1 \varphi(z)+ a_0=0.\] The conformal analysis (angle-preserving) of the CADEs is investigated. We present sufficient conditions to obtain analytic solutions of the CADEs. We show that these solutions are subordinated to analytic convex functions in terms of $e^z.$ Moreover, we investigate the connection estimates (coefficient bounds) of CADEs by employing the majorization method. We achieve that the coefficients bound are optimized by Bernoulli numbers.

Keywords

• analytic function
• subordination and superordination
• univalent function
• open unit disk
• algebraic differential equations
• majorization method