Sufficient Conditions for λ-Spirallike and λ-Robertson Functions of Complex Order

for some real numbers λ ∈ R with |λ| < π/2 and b ∈ C. We denote this class byC α (b). Noting that the above function classes include several subclasses which have important role in the analytic and geometric function theory. From this reason, we want to state some of them. (i) Sλ 0 (b) =: Sλ(b) (b ∈ C∗; |λ| < π/2) is the class of the λ-spirallike function of complex order b introduced and studied by Al-Oboudi and Haidan [1].


Introduction and Definitions
Let A denote the class of functions of the form which are analytic in the open unit disk U = { :  ∈ C and || < 1}.A function () ∈ A is said to be the -spirallike function of complex order  and type  (0 ≤  < 1) in U, denoted by S   () if and only if for some real numbers  ∈ R with || < /2 and  ∈ C * := C \ {0}.Furthermore, a function () ∈ A is also said to be the -Robertson function of complex order  and type  (0 ≤  < 1) in U if and only if for some real numbers  ∈ R with || < /2 and  ∈ C * .We denote this class by C   ().Noting that the above function classes include several subclasses which have important role in the analytic and geometric function theory.From this reason, we want to state some of them.
) is the class of the -Robertson function of complex order  (see, [2]).
is the class of convex functions of complex order  and type  introduced and studied by the author [3], and C  (1) =: C  is said to be the convex functions class of order  (0 ≤  < 1) and was studied by Robertson [4].
In this paper, we obtain several sufficient conditions for the analytic functions () belonging to the classes S   (), C   (), S *  (), C  (), S(), C(), S *  , and C  by making use of the well-known Jack Lemma [13].

Main Result
In order to derive our main result, we have to recall here the following Jack Lemma.
Lemma 1 (see [13]).Let () be analytic in U such that where  > 1 is a real number.
Now, with the help of Lemma 1, we can prove the following result.
Proof.Define a function () by Then, () is analytic in U and (0) = 0.It follows from ( 12) that