Applications of Fractional Derivatives for p-Valently α-Convex Functions of Order β

Let Ap denote the class of p-valently functions f(z) in the open unit disc U with f(j)(0)=0(j=0,1,2,…,p−1). In general, we write f(z)∈Ap by f(z)=zp+∑k=1∞ap+kzp+k(p∈N). For f(z)∈Ap,p-valently α-convex functions of order β in U are introduced. Some interesting properties for such functions can be seen. In this paper, we introduce fractional derivatives Dzj+λf(z) for f(z)∈Ap with j=0,1,2,…,p−1 and 0≤λ<1. Applying the fractional derivatives Dzj+λf(z), we would like to generalize p-valently α-convex functions of order β in U. Some results for such functions f(z)∈Ap are discussed with example functions. Further, a conjecture for our research in this paper is given with an example function.