Close-to-convex function generates remarkable solution of 2nd order complex nonlinear differential equations

الملخص EN

Consider the complex nonlinear differential equation f^'' (z)-2/z f^' (z)-2/z^2 f^3 (z)= H(z) , where P(z)= (-2)/z,Q(z)=(-2)/z^2 are complex coefficients, and H(z) be a complex function performs non- homogeneous term of given equation.

In this paper, we investigated that w(z)=(zf^')/f is a remarkable solution of given equation and belongs to hardy space H^2 ; with studying the growth of that solution by two ways ; through the maximum modulus and Brennan’s Conjecture and another by finding the supremum function of a volume of the surface area 〖K 〗_θ.

Furthermore, we discussed the solution behaviour with meromorphic coefficients properties for given equation.