Abstract

Sufficient conditions are obtained to ensure starlikeness of positive order for analytic functions defined in the open unit disk satisfying certain third-order differential inequalities. As a consequence, conditions for starlikeness of functions defined by integral operators are obtained. Connections are also made to earlier known results.

1. Introduction

Let denote the class of analytic functions defined in the open unit disk . For and a positive integer, let and , with . For , denote by the subclass of consisting of functions starlike of order satisfying The class is the well-known subclass of starlike functions studied widely in geometric function theory.

In the sequel, we give emphasis to the class , where . Evidently for . The class was investigated by Silverman [1], who showed that coincides with for univalent functions with negative coefficients. This class has subsequently been studied in several other works (see, e.g., [2]).

The problem of determining sufficient conditions to ensure starlikeness of functions has been widely investigated. These include conditions in terms of differential inequalities; see, for example, [2–11]. Miller and Mocanu [12], Kuroki and Owa [13], and, more recently, Ali et al. [14] determined conditions for starlikeness of functions defined by an integral operator of the form or by the double integral operator

In this paper, conditions on certain third-order differential inequalities are found that would imply starlikeness of positive order. As a consequence, conditions on the kernel of certain integral operators are also obtained to ensure that the functions defined by these operators are starlike. Connections are also made to earlier known results.

Recall that an analytic function is subordinate to an analytic function in , written as , if there exists an analytic self-map of with satisfying .

The following lemmas will be required in the sequel.

Lemma 1 (see [15, Theorem 1, page 192] and see also [16, Theorem 3.1b, page 71]). Let be convex in with , and . If and then where The function is convex and is the best -dominant.

Lemma 2 (see [17] and see also [16, Theorem 3.1d, page 76]). Let h be a starlike function with . If satisfies then The function is convex and is the best -dominant.

2. Main Results

The following two results are easily obtained by simple adaptations of Theorem 2.1 and Theorem 2.6 in [13]. The proofs are therefore omitted.

Lemma 3. Let , , and . If then with an extremal function .

Lemma 4. Let , , and . If then is a starlike function of order .

Remark 5. Even though the conditions given in Lemmas 3 and 4 are sufficient to deduce , they are in fact sufficient to imply .

The above two lemmas are next used to obtain conditions in terms of a third-order differential inequality and a third-order integral operator to deduce starlikeness of of order .

Theorem 6. Let , , , and . Further let and satisfy If then . Equality is attained for .

Proof. Let A brief computation shows that Hence, (15) can be written in the subordination form as It follows from Lemma 1 that which implies Hence on using Lemma 3.
For sharpness, it is evident that the function satisfies Thus,

Theorem 7. Let , , , and . If where then satisfies .

Proof. Let satisfy From Theorem 6, the solution of (26) belongs to the class . Now (26) has the form where Equation (27) has a solution with
Note that the function in Lemma 4 satisfies . Thus replacing the appropriate parameters in the equation yields a solution This completes the proof.

The next result provides a sufficient condition for starlikeness of order involving a second-order differential inequality.

Lemma 8. Let , and with . If then with an extremal function .

Proof. Inequality (33) can be expressed in the subordination form
Writing it follows that Now Lemma 1 with yields which implies
Let Since an application of Lemma 2 shows that Therefore,
Combining (38) and (42) yields which means , whence .

Remark 9. For and , Lemma 8 reduces to [12, Lemma 2.2].

The following result gives starlikeness for a function given by a double integral operator associated with Lemma 4. The proof is analogous to Theorem 2.2 of [12] and is omitted.

Lemma 10. Let , , , and . If then satisfies .

An application of Lemma 8 yields the following sufficient condition for starlikeness in terms of a third-order differential inequality.

Theorem 11. Let , , , , and . Further let If then . Equality is attained for .

Proof. Proceeding similarly as in the proof of Lemma 8, inequality (47) can be written as Let Then a computation yields so that Hence
Applying Lemma 1 yields This implies that and thus which, in comparison with Lemma 8, gives the required result.
Further the result is sharp for which satisfies

Remark 12. For , the choice in Theorem 11 results in For , this coincides with Lemma 8 at , which was also exhibited in [13, Corollary 2.4]. Further, for , (57) gives which for and is the result given in [8, Theorem 1].

Corresponding to Theorem 11, a sufficient condition for starlikeness of order for functions defined by a triple integral operator is obtained in the following result.

Theorem 13. Let , , , , and . Further let If then satisfies .

Proof. Let satisfy From Theorem 11, we find that the solution of (62) lies in . Now (62) becomes where Equation (63) has a solution with
In view of Lemma 10, the equation has a solution This completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work presented here was supported in part by a research university grant from Universiti Sains Malaysia. The authors are thankful to the referees for their suggestions that helped improve the clarity of this paper.