In the present paper, we define and study a new family of holomorphic functions which involve the Bazilevič functions and the $\lambda$-pseudo-starlike functions. We establish coefficient estimates for the first four determinants of the symmetric Toeplitz matrices $T_2\left(2\right)$, $T_2\left(3\right)$, $T_3\left(1\right)$ and $T_3\left(2\right)$ for the functions in this family. Further, we investigate several special cases and consequences of our results.

This article introduces another type of integral transformation in its general format; this general transformation is proposed as a new general integral transformation called the “Emad-Salam-Elaf Transform.” This general integral transformation is elaborated, and its fundamental functions and theorems have been proven. The eligibility of the proposed general integral transform is proven by solving ordinary differential equations of the first and second-order nature.Another type of general integral transform has been studied and proven for some elementary functions, its applicability and ability to evaluate the exact solution have been proven by utilizing the transform to find the solution of first and second order ordinary differential equations.