  The notion of Ces\`aro stable function is generalized by introducing Ces\`aro
mean of type $(b-1;c)$ which give rise to a new concept of generalized Ces\`aro
stable function. As an application of generalized Ces\`aro stable functions we
also prove for a convex function of order $\lambda\in[1/2,1)$, its Ces\`aro
mean of type $(b-1;c)$ is close-to-convex of order $\lambda$. Further two
conjectures are also posed in the direction of generalized Ces\`aro stable
function. Some particular cases of these conjectures are also discussed.
