Research Article | Open Access
Sufficient Conditions for -Spirallike and -Robertson Functions of Complex Order
We obtain several sufficient conditions for -spirallike and -Robertson functions of complex order by making use of the well-known Jack Lemma.
1. Introduction and Definitions
Let denote the class of functions of the form which are analytic in the open unit disk and . A function is said to be the -spirallike function of complex order and type in , denoted by if and only if for some real numbers with and . Furthermore, a function is also said to be the -Robertson function of complex order and type in if and only if for some real numbers with and . We denote this class by .
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) is the class of the -spirallike function of complex order introduced and studied by Al-Oboudi and Haidan .(ii) is the class of the -Robertson function of complex order (see, ).(iii) is the class of starlike functions of complex order and type introduced and studied by the author , and is said to be the starlike functions class of order and was studied by Robertson .(iv) is the class of convex functions of complex order and type introduced and studied by the author , and is said to be the convex functions class of order and was studied by Robertson .(v) is known the -spirallike univalent functions class and was defined by Spacek , is said to be the starlike functions class of complex order and was studied by Nasr and Aouf , and is known the -spirallike functions class of order and was studied by Libera .(vi) is known the -Robertson type functions class and was first studied by Robertson , is called the convex functions class of complex order and was studied by Wiatrowski , Nasr and Aouf  and Aouf , and is known the -Robertson type functions class of order and was studied by Chichra .
In this paper, we obtain several sufficient conditions for the analytic functions belonging to the classes , , , , , , , and by making use of the well-known Jack Lemma .
2. Main Result
In order to derive our main result, we have to recall here the following Jack Lemma.
Lemma 1 (see ). Let be analytic in such that . Then, if attains its maximum value on circle at a point , one has where is a real number.
Now, with the help of Lemma 1, we can prove the following result.
Theorem 2. Let , and , and let be defined by If satisfies any of the following inequalities: then The powers are taken by their principal value.
Proof. Define a function by
Then, is analytic in and . It follows from (12) that
Thus, we have
We claim that in . For otherwise, by Lemma 1, there exists such that , where and . Therefore, (14)–(18) yield which contradicts our assumptions (6)–(10), respectively. Therefore, holds true for all . We finally have thus, we have
Remark 3. Taking different choices of ,, , and in Theorem 2, we obtain new sufficient conditions for functions to be in the classes , , , , , , and .
The author would like to thank the referee for his helpful comments and suggestions.
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