For p-valently spirallike and p-valently Robertson functions in the open unit disk U, reciprocal classes S p (α, β), and C p (α, β) are introduced. The object of the present paper is to discuss some interesting properties for functions f(z) belonging to the classes S p (α, β) and C p (α, β).

Applying the subordination principle for analytic functions in the open unit disk U, I. H. Kim and N. E. Cho (Comput. Math. Appl. 59 (2010), 2067-2073) considered some sufficient conditions for Carathéodory functions. The purpose of this paper is to discuss some sufficient conditions for the class of strongly Carathéodory functions in U.2010 Mathematics Subject Classification. 30C45.

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For analytic functions f (z) in the closed unit disk U, two boundary points z 1 and z 2 such that α = (fThe object of the present paper is to discuss some interesting conditions for f (z) to be |f ′ (z) − 1| < ρ|1 − α| in U with some examples.2010 Mathematics Subject Classification. 30C45.