FEKETE-SZEGO problem for a class of analytic functinos defined by convolution

Let $g$ and $h$ be two fixed normalized analytic functions and $\phi$ be starlike with respect to $1,$ whose range is symmetric with respect to the real axis. Let $\mathcal{M}^{\alpha,\beta}_{g,h}(\phi)$ be the class of analytic functions $f(z)=z+a_2z^2+a_3z^3+\ldots$, satisfying the subordination $$\left(\frac{(f*g)(z)}{z}\right)^\alpha \left(\frac{(f*h)(z)}{z}\right)^{\beta}\prec \phi(z),$$ where $\alpha$ and $\beta$ are real numbers and are not zero simultaneously. In the present investigation, sharp upper bounds of the Fekete-Szego functional $|a_3-\mu a_2^2|$ for functions belonging to the class $\mathcal{M}^{\alpha,\beta}_{g,h}(\phi)$ are obtained and certain applications are also discussed.