On sharp bounds of certain close-to-convex functions

We derive general formula for the fourth coefficient of the functions belonging to the Carathéodory class involving the parameters lying in the open unit disk. Further, we obtain sharp upper bounds of initial inverse coefficients for certain close-to-convex functions satisfying any one of the inequalities: \(\text {Re}((1-z)f^{'}(z))>0,\text {Re}((1-z^2)f^{'}(z))>0,\text {Re}((1-z+z^2)f^{'}(z))>0\) $\text{Re}((1-z)f^{{}^{\prime }}(z))>0,\text{Re}((1-z^2)f^{{}^{\prime }}(z))>0,\text{Re}((1-z+z^2)f^{{}^{\prime }}(z))>0$ and \(\text {Re}((1-z)^2f^{'}(z))>0\) $\text{Re}({(1-z)}^2{f}^{{}^{\prime }}(z))>0$ .