Abstract
Keywords:
differential subordination; convex function; best dominant; differential operator; generalized Sălăgean operator; Ruscheweyh derivative1 Introduction
Denote by U the unit disc of the complex plane, and the space of holomorphic functions in U.
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 (secondorder) 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 (AlOboudi [1])
For , and , the operator is defined by ,
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 ,
Definition 1.3[4]
Let , . Denote by the operator given by ,
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].
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
Lemma 1.2 (Miller and Mocanu [12])
Letgbe a convex function inU, and let, for, whereandnis a positive integer.
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
then
and this result is sharp.
Proof By using the properties of operator , we have
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
then
where. The functionqis convex, and it is the best dominant.
Proof Let
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
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
holds, then
and this result is sharp.
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
then
where. The functionqis convex, and it is the best dominant.
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
holds, then
This result is sharp.
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
then
where. The functionqis convex, and it is the best dominant.
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
holds, then
This result is sharp.
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
then
where. The functionqis convex, and it is the best dominant.
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
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 , .
Using Theorem 2.9, we obtain
induce
Theorem 2.11Letgbe a convex function such that, and lethbe the function, .
If, , and the differential subordination
holds, then
This result is sharp.
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
then
where. The functionqis convex, and it is the best dominant.
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
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 , .
Using Theorem 2.12, we obtain
induce
Theorem 2.14Letgbe a convex function such that, and lethbe the function, .
If, , , and the differential subordination
holds, then
This result is sharp.
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
then
where. The functionqis convex, and it is the best dominant.
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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