Abstract
We apply the general theory of differential subordination to obtain certian interesting criteria for -valent starlikeness and strong starlikeness. Some applications of these results are also discussed.
1. Introduction
Let be the class of functions of the form which are analytic in the open unit disk .
Let be the class of functions of the form which are analytic in . If satisfies , then we say that is a Carathรฉodory function.
With a view to recalling the principle of subordination between analytic functions, let the functions and be analytic in . Then we say that the function is subordinate to if there exists a Schwarz function , analytic in with such that We denote this subordination by In particular, if the function is univalent in , the above subordination is equivalent to
For and , a function is said to be in the class if it satisfies Also, we write , the class of strongly starlike -valent functions of order in . , the class of Janowski starlike -valent function, , the class of -valent starlike function, and , the class of -valent starlike function of order .
For Carathรฉodory functions, Miller [1] obtained certain sufficient conditions applying the differential inequalities. Recently, Nunokawa et al. [2] have given some improvement of result by Miller [1]. Recently Ravichandran and Jayamala [3] studied some subordination results for Carathรฉodory functions. In this paper by extending the result of Ravichandran and Jayamala [3], we find sufficient conditions for the subordination to hold for given and criteria for -valent starlikeness. Our results include results obtained by Nunokawa et al. [2]. We also give some criteria for -valently starlikeness and strong starlikeness.
To prove our result we need the following lemma due to Miller and Mocanu [4].
Lemma 1.1 (see [4, Theoremโ3.4h, page 132]). Let be analytic and univalent in the unit disk and and let be analytic in a domain containing with when . Set Suppose that(i) is starlike univalent in ,(ii) for .If is analytic in with, , , and then and is the best dominant.
2. Application of Differential Subordination
By making use of Lemma 1.1, we first prove the following theorem.
Theorem 2.1. Let and be a positive real number. Let be convex univalent in and . If satisfies where then and is the best dominant of (2.1).
Proof. Let
Then clearly and are analytic in and . Also let
Since is convex univalent, is starlike univalent. Therefore is starlike univalent in , and
for .
From (2.1)โ(2.6) we see that
Therefore, by applying Lemma 1.1, we conclude that and is the best dominant of (2.1). The proof of the theorem is complete.
By taking as real and in Theorem 2.1, we get the following corollary.
Corollary 2.2. Let , ,โ, be real number such that and . If satisfies where then and is the best dominant of (2.8).
Corollary 2.3. Let , . If satisfies in and where then
Proof. Let , then and (2.11) can be written as Taking in Corollary 2.2 and using (2.14), we have
By taking and in Corollary 2.3, we get the following result of Padmanabhan [5].
Corollary 2.4. Let and then
Theorem 2.5. Let and . Let be convex univalent in and satisfy If satisfies then and is the best dominant of (2.19)
Proof. By setting and it can be easily observed that and are analytic in and that .
Also, by letting
we find that is starlike univalent in and that
The differential subordination
becomes
Now, the result follows as an application of Lemma 1.1.
Theorem 2.6. Let , and be complex numbers, . Let be univalent in and satisfy the following conditions for : (1)let be starlike,(2).If satisfies then and is the best dominant.
Proof. The proof of this theorem is much akin to the proof of Theorem 2.5 and hence can be omitted.
Remark 2.7. By taking ,โ,โ,โ, and in Theorem 2.5 we get the result of Nunokawa et al. [2] which was proved by a different method.
Remark 2.8. For the choices of in Theorem 2.5, we get the result of [3, Theoremโโ1, pageโโ192] and for in Theorem 2.6 we get the result of [3, Theoremโโ2, pageโโ194].
Corollary 2.9. Let ,โ and . If satisfies in , then implies Also, implies
Proof. By taking ,โ, ,โ, and in Theorem 2.6, we get the first part.
Proof of the second part follows, by setting ,โ,โ, and .
For ,โ, , and ,โ in Theorem 2.5, we have the following result.
Corollary 2.10. If satisfies , and
where
then
One notes that if , then is the exterior of the parabola given by
with its vertex as (see [5, 6]).
By taking in Corollary 2.10, we obtain the following.
Corollary 2.11. If satisfies , , and then Region has been shown shaded in Figure 1.
Letting ,โ,โ, and in Theorem 2.5, we get the following.
Corollary 2.12. If satisfies ,โ, and where for some , then
For the univalent function given by (2.38), One now finds the image of the unit disk .
Let , where and are real. One has Elimination of yields Therefore, one concludes that which properly contains the half plane .
Corollary 2.13. Let and . If satisfies in and where then
Proof. If we let , then and (2.43) can be expressed as Hence, by taking ,โ,โ and in Theorem 2.5, we have . So, .
Setting and in Corollary 2.13, we get the following corollay.
Corollary 2.14. Let and . If satisfies in and where then
Remark 2.15. For the function given by (2.48), we have which properly contains the half plane , where
By putting and in Corollary 2.13, we get the following result of Tuneski [7].
Corollary 2.16. If and then
Remark 2.17. By putting ,โ, and in Corollary 2.13, we get the result obtained by Singh [8], which refines the result of Silverman [9].
Corollary 2.18. Let and be convex univalent in with and satisfy (2.18).
Let and
If
then
and is the best dominant.
Proof. By taking in Theorem 2.5, we have the above corollary.
Corollary 2.19. Let and be convex univalent in with and satisfy (i)If satisfies then (ii)If satisfies then and is the best dominant.
Proof. Proof of the first part follows from Corollary 2.18, by taking ,โ, and .
The proof of the second part follows from Corollary 2.18, by taking ,โ, and .
By taking where is a positive integer and in the first part of Corollary 2.19, we get the following result of Ponnusamy [10].
Corollary 2.20. Let , then for a positive integer , one has that implies and is the best dominant.
Remark 2.21. By taking and in Corollary 2.19 and and we get the result of Ponnusamy and Juneja [11].
By taking ,โ, ,โ, and in Theorem 2.5, we get the following result obtained by Owa and Obradoviฤ [12].
Corollary 2.22. Let and then and is the best dominant.
We remark here that is univalent if and only if .
Remark 2.23. For a special case when ,โ where and ,โ, and in Theorem 2.6, we have the result obtained by Srivastava and Lashin [13].
Corollary 2.24. If satisfies then and is the best dominant.
Proof. By taking and ,โ and in Theorem 2.5, we get the previous corollary.