Generalizations of the Alexander integral operator for Analytic Multivalent Functions

Abstract

Let T_p,n be a subclass of analytic multivalent functions of the form f(z) = z^p + a_p+nz^p+n + a_p+n+1z^p+n+1 + . . .
for every z in the open unit disc U. Applying the fractional calculus (fractional integral and fractional derivative), A^−λ _p,nf(z) and A^λ _p,nf(z) which are generalizations of the Alexander integral operator are introduced. The object of present paper is to discuss some interesting properties of A^−λ _p,nf(z) and A^λ _p,nf(z). Also, some simple examples of results for A^−λ _p,nf(z) and A^λ _p,nf(z) are shown. To give some simple examples is very important for the research of mathematics.

Key words and phrases. Analytic function, fractional derivative, fractional integral, Alexander integral operator,dominant, subordination.

2010 Mathematics Subject Classification. 30C45.