By using the fractional Caputo-Fabrizio derivative, we introduce two types new high order derivations called CFD and DCF. Also, we study the existence of solutions for two such type high order fractional integro-differential equations. We illustrate our results by providing two examples.

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Let f (z) be an analytic function in the open unit disk D normalized with f (0) = 0 and f (0) = 1. With the help of subordinations, for convex functions f (z) in D, the order of close-to-convexity for f (z) is discussed with some example. MSC: Primary 30C45

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In this paper we introduce and investigate the class of T (λ, β, A, B) that we call the class of quasi-starlike and quasi-convex functions with respect to the values of the parameter λ. Also we define the class K (λ, β, µ, m, A, B) which is satisfy non-homogeneous Cauchy-Euler differential equation. We obtain coefficient bounds for the functions belonging the above classes.