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Some differential subordinations using Ruscheweyh derivative and Sălăgean operator
Advances in Difference Equations volume 2013, Article number: 150 (2013)
Abstract
In the present paper we study the operator defined by using the Ruscheweyh derivative and the Sălăgean operator , denoted by , , , where is the class of normalized analytic functions with . We obtain several differential subordinations regarding the operator .
MSC:30C45, 30A20, 34A40.
1 Introduction
Denote by U the unit disc of the complex plane, , and by the space of holomorphic functions in U. Let with and for and . Denote by the class of normalized convex functions in U.
If f and g are analytic functions in U, we say that f is subordinate to g, written , if there is a function w analytic in U, with , , for all , such that for all . If g is univalent, then if and only if and .
Let and let h be a univalent function in U. If p is analytic in U and satisfies the (second-order) differential subordination
then p is called a solution of the differential subordination. The univalent function q is called a dominant of the solutions of the differential subordination, or more simply a dominant, if for all p satisfying (1.1).
A dominant that satisfies for all dominants q of (1.1) is said to be the best dominant of (1.1). The best dominant is unique up to a rotation of U.
Definition 1.1 (Sălăgean [1])
For , , the operator is defined by ,
Remark 1.1 If , , then , .
Definition 1.2 (Ruscheweyh [2])
For , , the operator is defined by ,
Remark 1.2 If , , then , .
Definition 1.3 ([3])
Let , . Denote by the operator given by ,
Remark 1.3 If , , then , .
This operator was studied also in [3–5].
Lemma 1.1 (Hallenbeck and Ruscheweyh [[6], Th. 3.1.6, p.71])
Let h be a convex function with , and let be a complex number with . If and
then
where , .
Lemma 1.2 (Miller and Mocanu [6])
Let g be a convex function in U and let , for , where and n is a positive integer.
If , , is holomorphic in U and
then
and this result is sharp.
2 Main results
Theorem 2.1 Let g be a convex function, and let h be the function , .
If , , and satisfies the differential subordination
then
and this result is sharp.
Proof By using the properties of the operator , we have
Consider , .
We deduce that .
Differentiating, we obtain , .
Then (2.1) becomes
By using Lemma 1.2, we have
□
Theorem 2.2 Let h be a holomorphic function which satisfies the inequality , , and .
If , , and satisfies the differential subordination
then
where . The function q is convex and it is the best dominant.
Proof Let
for , .
Differentiating, we obtain , , and (2.2) becomes
Using Lemma 1.1, we have
and q is the best dominant. □
Corollary 2.3 Let be a convex function in U, where .
If , , and satisfies the differential subordination
then
where q is given by , . The function q is convex and it is the best dominant.
Proof Following the same steps as in the proof of Theorem 2.2 and considering , the differential subordination (2.3) becomes
By using Lemma 1.1 for , we have , i.e.,
□
Remark 2.1 For , , , we obtain the same example as in [[7], Example 2.2.1, p.26].
Theorem 2.4 Let g be a convex function such that and let h be the function , , where .
If , , and the differential subordination
holds, then
and this result is sharp.
Proof For , , we have , .
Consider and we obtain .
Relation (2.4) becomes
By using Lemma 1.2, we have
□
Theorem 2.5 Let h be a holomorphic function which satisfies the inequality , , and .
If , is a complex number with , , and satisfies the differential subordination
then
where . The function q is convex and it is the best dominant.
Proof Let , , . Differentiating, we obtain , , and (2.5) becomes
Using Lemma 1.1, we have
and q is the best dominant. □
Theorem 2.6 Let g be a convex function such that and let h be the function , , where .
If , , and the differential subordination
holds, then
This result is sharp.
Proof Let . We deduce that .
Differentiating, we obtain , .
Using the notation in (2.6), the differential subordination becomes
By using Lemma 1.2, we have
and this result is sharp. □
Theorem 2.7 Let h be a holomorphic function which satisfies the inequality , , and .
If , is a complex number with , , and satisfies the differential subordination
then
where . The function q is convex and it is the best dominant.
Proof Let , , .
Differentiating, we obtain , , and (2.7) becomes
Using Lemma 1.1, we have
and q is the best dominant. □
Theorem 2.8 Let g be a convex function such that and let h be the function , .
If , , and the differential subordination
holds, then
This result is sharp.
Proof Let . We deduce that .
Differentiating, we obtain , .
Using the notation in (2.8), the differential subordination becomes
By using Lemma 1.2, we have
and this result is sharp. □
Theorem 2.9 Let h be a holomorphic function which satisfies the inequality , , and .
If , , and satisfies the differential subordination
then
where . The function q is convex and it is the best dominant.
Proof Let , , .
Differentiating, we obtain , , and (2.9) becomes
Using Lemma 1.1, we have
and q is the best dominant. □
Corollary 2.10 Let be a convex function in U, where .
If , , and satisfies the differential subordination
then
where q is given by , . The function q is convex and it is the best dominant.
Proof Following the same steps as in the proof of Theorem 2.9 and considering , the differential subordination (2.10) becomes
By using Lemma 1.1 for , we have , i.e.,
□
Example 2.1 Let be a convex function in U with and .
Let , . For , , we obtain .
Then ,
We have .
Using Theorem 2.9, we obtain
induce
Theorem 2.11 Let g be a convex function such that and let h be the function , .
If , , and the differential subordination
holds, then
This result is sharp.
Proof Let . We deduce that .
Differentiating, we obtain , .
Using the notation in (2.11), the differential subordination becomes
By using Lemma 1.2, we have
and this result is sharp. □
Theorem 2.12 Let h be a holomorphic function which satisfies the inequality , , and .
If , , and satisfies the differential subordination
then
where . The function q is convex and it is the best dominant.
Proof Let , , .
Differentiating, we obtain , , and (2.12) becomes
Using Lemma 1.1, we have
and q is the best dominant. □
Corollary 2.13 Let be a convex function in U, where .
If , , and satisfies the differential subordination
then
where q is given by , . The function q is convex and it is the best dominant.
Proof Following the same steps as in the proof of Theorem 2.12 and considering , the differential subordination (2.13) becomes
By using Lemma 1.1 for , we have , i.e.,
□
Example 2.2 Let be a convex function in U with and .
Let , . For , , we obtain , .
Then ,
We have .
Using Theorem 2.12, we obtain
induce
Theorem 2.14 Let g be a convex function such that and let h be the function , .
If , , , and the differential subordination
holds, then
This result is sharp.
Proof Let . We deduce that .
Differentiating, we obtain , .
Using the notation in (2.14), the differential subordination becomes
By using Lemma 1.2, we have
and this result is sharp. □
Theorem 2.15 Let h be a holomorphic function which satisfies the inequality , , and .
If , , , and satisfies the differential subordination
then
where . The function q is convex and it is the best dominant.
Proof Let , , .
Differentiating, we obtain , , and (2.15) becomes
Using Lemma 1.1, we have
and q is the best dominant. □
References
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Alb Lupaş A: On special differential subordinations using Sălăgean and Ruscheweyh operators. Math. Inequal. Appl. 2009, 12(4):781–790.
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Acknowledgements
Dedicated to Professor Hari M Srivastava.
The author thanks the referee for his/her valuable suggestions to improve the present article.
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Alb Lupaş, A. Some differential subordinations using Ruscheweyh derivative and Sălăgean operator. Adv Differ Equ 2013, 150 (2013). https://doi.org/10.1186/1687-1847-2013-150
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DOI: https://doi.org/10.1186/1687-1847-2013-150