1. Introduction
It is a known fact that the notion of an operator was used from the early stage of the study of complex-valued functions; many already known results can be proven easier with them, and new results are being obtained with them.
Many papers, such as [
1,
2], studied different operators defined by using the fractional integral of order
also used earlier by S. Owa [
3]. Despite that, we also refer to [
4,
5,
6] for theoretical and numerical analyses from real models described by classical PDEs and related operators. The results contained in the present paper were inspired by the outstanding results previously obtained using fractional integrals, and the study was done by applying them to a confluent hypergeometric function. The definition of a fractional integral can be seen in [
3] as follows:
Definition 1 ([
3])
. The fractional integral of order α is defined bywhere α is a positive real number, is an analytic function in a simply connected region of the z-plane containing the origin and the multiplicity of is removed by requiring to be real when In paper [
7] a new operator was introduced by using a fractional integral on the confluent (Kummer) hypergeometric function. The introduction of this operator was inspired by the studies done on this function having in view many aspects, from its combination with other functions, as can be seen in papers [
8,
9], to its univalence in paper [
10].
The confluent (Kummer) hypergeometric function of the first kind is defined in [
11] as follows:
Definition 2 ([
11])
. Let and considerThis function is called a confluent (Kummer) hypergeometric function, is analytic in and satisfies Kummer’s differential equation: Considering
the confluent (Kummer) hypergeometric function can be written as
The definition of the operator introduced in [
7] is the following:
where
.
The fractional integral of a confluent hypergeometric function can be written as
after a simple calculation. Evidently,
The original results which are shown in the next part of this paper were obtained by using this operator and differential subordination and superodination theories, synthesized in the monography [
12] published by Miller and Mocanu in 2000 and in paper [
13], respectively. The usual notion and definitions are considered.
is the unit disc of the complex plane, the class of analytic functions in U and with n a positive integer and .
Definition 3 ([
12])
. Let . The function f is said to be subordinate to F if there exists a Schwarz function analytic in U, with and , such that . In such a case, we write . If F is univalent, then if and only if and . Definition 4 ([
12])
. Let and let h be univalent in U. If p is analytic and satisfies the differential subordinationthen 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 (5). A dominant that satisfies for all dominants q of (5) is said to be the best dominant of (5). The notion related to differential superordinations was introduced in [
13].
Definition 5 ([
13])
. Let and let h be analytic in U. If p and are univalent in U and satisfy the differential superordinationthen p is called a solution of the differential superordination (6). An analytic function q is called a subordinant of the solutions of the differential superordination or more simply a subordinant, if for all p satisfying (6). A subordinant that satisfies for all subordinants q of (6) is said to be the best subordinant of (6). In the process of obtaining the original results from this paper, the following lemmas are needed:
Lemma 1 ([
12])
. Let the function q be univalent in the unit disc U and θ and ϕ be analytic in a domain D containing with when . Set and . Suppose that Q is star-like univalent in U and , .If p is analytic with , andthen and q is the best dominant. Lemma 2 ([
14]).
Let the function q be convex univalent in the open unit disc U and ν and ϕ be analytic in a domain D containing . Suppose that and is star-like univalent in U.If , with and is univalent in U andthen and q is the best subordinant. 2. Main Results
Continuing the work from [
7], we get:
Theorem 1. Let q be an analytic and univalent function in U with , ∀ and , where and a, Suppose that is star-like univalent in U. Considerwith , , and If the following subordination is satisfied by thenand q is the best dominant. Proof. By differentiating with respect to z, we get
and
By setting
and
evidently
is analytic in
∀
and
is analytic in
Considering
and
which reveals that
Q is a star-like univalent function in
U.
By differentiating, we obtain and
We deduce that .
By using (
11), we obtain
By using (
9), we have
By applying Lemma 1, we get , ∀ which means , ∀ and the function q is the best dominant. □
Corollary 1. Let a, and relation (7) holds for , with . Ifwith , which is defined in (8), thenand is the best dominant. Proof. We get the corollary considering , in Theorem 1. □
Corollary 2. Let , a, , Assume that (7) holds for , . Ifwith is defined in (8), thenand is the best dominant. Proof. Put , in Theorem 1 to obtain the corollary. □
Theorem 2. Let q be an analytic and univalent function in U with ∀ such that is a star-like univalent function in U and If and is an univalent function in U, where is defined by (8) and a, , , thenimpliesand function q is the best subordinant. Proof. Considering
and
it is easy to show that
is analytic in
,
,
and
is also analytic in
Since , it yields
, where
From (
11) and (
13) we get
Applying Lemma 2, we obtain
; therefore,
and the best subordinant is function
q. □
Corollary 3. Consider and a, , Assume that (12) holds for , . If andwhere is defined in (8), thenand the best subordinant is . Proof. When , consider in Theorem 2 and obtain the corollary. □
Corollary 4. Let a, , Assume that (12) holds for , . If andwhere is defined in (8), thenand the best subordinant is Proof. Put in Theorem 2 , when . □
The sandwich theorem is obtained combining Theorems 1 and 2.
Theorem 3. (Sandwich-type result) Consider and analytic and univalent functions in with and , ∀, such that and are star-like univalent. Assume that satisfies relation (7) and satisfies relation (12). If and is defined by (8) and is univalent in and a, , , thenfor , impliesand and are, respectively, the best subordinant and the best dominant. For , , where , we have the following corollary.
Corollary 5. Let a, and Assume that (7) and (12) hold for , and , . If andwith defined by (8), thenhence and are the best subordinant and the best dominant. For , , where , we have the following corollary.
Corollary 6. Let a, Assume that (7) and (12) hold for , and , . If andwhere is defined in (8), thenhence and are the best subordinant and the best dominant, respectively. By changing the functions and in Theorem 1, we get:
Theorem 4. Consider a, the convex and univalent function q in U with and ∀. Suppose thatwhere , β, and If the following subordination is satisfied by thenand the best dominant is the function q. Proof. Define
with
, an analytic function in
U. Differentiating with respect to
z we get
and
Let
be analytic in
with
and
analytic in
Consider
star-like univalent in
U and
We obtain .
From (
19), we get
By using (
17), we get
Lemma 1 gives , ∀ so we obtain , ∀ and the best dominant is function q. □
Corollary 7. Let a, and , fulfilling the relation (15). Ifwith defined by (16), thenand the best dominant is . Proof. In Theorem 4 consider , with □
Corollary 8. Let a, β , and fulfilling the relation (15). Ifwith defined by (16), thenand is the best dominant. Proof. In Theorem 4 put , with □
By changing the functions and in Theorem 3 to be the same as in Theorem 4, we get:
Theorem 5. Let a, and q be a convex and univalent function in U with . Suppose that If and defined in (16) is univalent in U, thenimpliesand the best subordinant is the function q. Proof. Define the analytic function
and
Set
to be analytic in
with
, ∀
and
analytic in
Since
, from (
20) we get
, with
Relation (
21) gives the differential superordination
and by applying Lemma 2, we obtain
, which means
and the best subordinant is the function
q. □
Corollary 9. Assume that (20) holds for , with a, . If andwhere is defined in (16), , thenand the best subordinant is . Proof. In Theorem 5 consider , with . □
Corollary 10. Assume that (20) holds for , a, If andwhere is defined in (16), and thenand the best subordinant is Proof. In Theorem 5 consider , with □
The sandwich theorem is obtained combining Theorem 4 and Theorem 5.
Theorem 6. (Sandwich-type result) Consider and convex and univalent functions in U with and , ∀. Assume that relation (15) is satisfied by and relation (20) is satisfied by . If and is univalent in U defined by (16), a, , thenimpliesand the best subordinant is and the best dominant is . Letting , , where , in Theorem 6 we get
Corollary 11. Assume that (15) and (20) hold for and , and , a, , If andwhere is defined in (16), and thenhence, the best subordinant is and the best dominant is . By setting and , where , in Theorem 6 we obtain
Corollary 12. Assume that (15) and (20) hold for and , and a, , If andwhere is defined by (16), thenhence, the best subordinant is and the best dominant is .