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# Differential subordinations using the Ruscheweyh derivative and the generalized Sălăgean operator

DOI: 10.1186/1687-1847-2013-252

Accepted: 6 August 2013

Published: 20 August 2013

## Abstract

In the present paper, we study the operator, using the Ruscheweyh derivative and the generalized Sălăgean operator , denote by , , , where is the class of normalized analytic functions. We obtain several differential subordinations regarding the operator .

MSC:30C45, 30A20, 34A40.

### Keywords

differential subordination convex function best dominant differential operator generalized Sălăgean operator Ruscheweyh derivative

## 1 Introduction

Denote by U the unit disc of the complex plane, and the space of holomorphic functions in U.

Let 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 an univalent function in U. If p is analytic in U and satisfies the (second-order) differential subordination
(1.1)

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 (Al-Oboudi [1])

For , and , the operator is defined by ,

Remark 1.1 If and , then , .

Remark 1.2 For , in the definition above, we obtain the Sălăgean differential operator [2].

Definition 1.2 (Ruscheweyh [3])

For , , the operator is defined by ,

Remark 1.3 If , , then , .

Definition 1.3 [4]

Let , . Denote by the operator given by ,

Remark 1.4 If , , then , .

This operator was studied also in [46] and [7].

Remark 1.5 For , , where and for , , where .

For , we obtain , which was studied in [811].

For , , where .

Lemma 1.1 (Hallenbeck and Ruscheweyh [[12], 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 [12])

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 , for .

If , , and satisfies the differential subordination
(2.1)
then

and this result is sharp.

Proof By using the properties of 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
(2.2)
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
(2.3)
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 [[13], Example 4.2.1, p.125].

Theorem 2.4 Let g be a convex function such that , and let h be the function , .

If , , and the differential subordination
(2.4)
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 , , and satisfies the differential subordination
(2.5)
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 , .

If , , and the differential subordination
(2.6)
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 an holomorphic function, which satisfies the inequality , , and .

If , , and satisfies the differential subordination
(2.7)
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
(2.8)
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
(2.9)
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
(2.10)
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
(2.11)
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
(2.12)
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
(2.13)
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
(2.14)
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
(2.15)
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. □

## Author’s contributions

The author drafted the manuscript, read and approved the final manuscript.

## Declarations

### Acknowledgements

The author thanks the referee for his/her valuable suggestions to improve the present article.

## Authors’ Affiliations

(1)
Department of Mathematics and Computer Science, University of Oradea

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