Extensions for certain subordination relations

Abstract

For some complex number ? which has a positive real part, a certain subordination relation concerned with the Bernardi integral operator I? was proven by D. J. Hallenbeck and St. Ruscheweyh (Proc. Amer. Math. Soc. 52(1975), 191-195). By considering the analyticity of the functions defined by the Bernardi integral operator I? for some non-zero complex number ? with Re ? ? 0, an extension for certain subordination relation are discussed.