Theorem 2.1 Let g be a convex function, and let h be the function , .
If , , and satisfies the differential subordination
(2.1)
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
(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 [[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
(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 , is a complex number with , , 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 , , where .
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 a holomorphic function which satisfies the inequality , , and .
If , is a complex number with , , 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. □