Sharp results on linear combination of simple expressions of analytic functions

In this paper we use methods from the theory of differential subordinations to study the linear combination $azf^{`}insert ignore into journalissuearticles values(z); + bf^{`}insert ignore into journalissuearticles values(z); + c\frac{finsert ignore into journalissuearticles values(z);}{z}$ and give sharp bounds over the module, the argument and the real part of $\alpha f^{`}insert ignore into journalissuearticles values(z); + \beta\frac{finsert ignore into journalissuearticles values(z);}{z}$.