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

Loriana Andrei

Author Affiliations

Department of Mathematics and Computer Science, University of Oradea, 1 Universitatii street, Oradea, 410087, Romania

Advances in Difference Equations 2013, 2013:252  doi:10.1186/1687-1847-2013-252

 Received: 13 June 2013 Accepted: 6 August 2013 Published: 20 August 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### 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 [4-6] and [7].

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

For , we obtain , which was studied in [8-11].

For , , where .

Lemma 1.1 (Hallenbeck and Ruscheweyh [[12], Th. 3.1.6, p.71])

Lethbe a convex function with, and letbe a complex number with. Ifand

then

where, .

Lemma 1.2 (Miller and Mocanu [12])

Letgbe a convex function inU, and let, for, whereandnis a positive integer.

If, , is holomorphic inUand

then

and this result is sharp.

### 2 Main results

Theorem 2.1Letgbe a convex function, , and lethbe 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 2.2Lethbe a holomorphic function, which satisfies the inequality, , and.

If, , and satisfies the differential subordination

(2.2)

then

where. The functionqis 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 2.3Letbe a convex function inU, where.

If, , and satisfies the differential subordination

(2.3)

then

whereqis given by, . The functionqis 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 2.4Letgbe a convex function such that, and lethbe 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 2.5Lethbe a holomorphic function, which satisfies the inequality, , and.

If, , and satisfies the differential subordination

(2.5)

then

where. The functionqis 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 2.6Letgbe a convex function such that, and lethbe 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 2.7Lethbe an holomorphic function, which satisfies the inequality, , and.

If, , and satisfies the differential subordination

(2.7)

then

where. The functionqis 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 2.8Letgbe a convex function such that, and lethbe 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 2.9Lethbe a holomorphic function, which satisfies the inequality, , and.

If, , and satisfies the differential subordination

(2.9)

then

where. The functionqis 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 2.10Letbe a convex function inU, where.

If, , and satisfies the differential subordination

(2.10)

then

whereqis given by, . The functionqis 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 2.11Letgbe a convex function such that, and lethbe 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 2.12Lethbe a holomorphic function, which satisfies the inequality, , and.

If, , and satisfies the differential subordination

(2.12)

then

where. The functionqis 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 2.13Letbe a convex function inU, where.

If, , and satisfies the differential subordination

(2.13)

then

whereqis given by, . The functionqis 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 2.14Letgbe a convex function such that, and lethbe 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 2.15Lethbe a holomorphic function, which satisfies the inequality, , and.

If, , , and satisfies the differential subordination

(2.15)

then

where. The functionqis 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. □

### Competing interests

The author declares that she has no competing interests.

### Author’s contributions

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

### Acknowledgements

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

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