1. Introduction
In the case of real-valued functions, bounds on a function
f are often determined from an inequality involving several derivatives of the function. As a simple example, a function
f twice continuously differentiable on an interval
can be considered and suppose that the differential operator
satisfies
for
. It is easily seen that such a function has the property that
for
.
In two articles in 1978 [
1] and 1981 [
2] the authors extended these ideas involving differential inequalities for the real-valued functions to complex-valued functions setting the basis for a new theory later named theory of differential subordinations or admissible functions theory which has developed rapidly in the following years since it provides means for proving a lot easier known results and for easily obtaining interesting, original ones.
The idea of introducing the notion of subordination originated in the observation that the differential inequality of the form (
1) does not have a direct analog for complex-valued functions since the real-valued function
cannot be replaced by a complex-valued function
, but the first inclusion from (
2) has a natural complex analog in the form
where
U denoting the unit disc of the complex plane and
.
If
satisfies this inclusion then, following the implication in relation (
2), the question is whether there exists a “smallest” set
such that
for the differential subordination theory, and a “smallest” set
for which
for the differential superordination theory, introduced in [
3].
The results presented in the original part of this paper will answer this last question.
Throughout the paper, we use the well-known classes:
denotes the class of analytic functions in the unit disc of the complex plane;
For
n a positive integer and
, let
denotes the class of convex functions in the unit disc;
.
The theory of differential superordination is based on the following definitions:
Definition 1. ([
3])
Let f and F be members of . The function f is said to be subordinate to F, or F is said to be superordinate to f, if there exists a function w, analytic in U, with and and such that . In such a case we write or . If F is univalent, then if and only if and . Definition 2. ([
3])
Let and let h be analytic in U. If p and are univalent in U and satisfy the (second-order) differential superordinationthen p is called a solution of the differential superordination. 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 (5). A subordinant that satisfies for all subordinants q of (5) is said to be the best subordinant of (5). Note that the best subordinant is unique up to a rotation of U. Definition 3. ([
3])
We denote by Q the set of functions f that are analytic and injective on whereand are such that , for . The subclass of Q for which is denoted by . Another notion used for obtaining the original results is that of subordination chains as it can be found in [
4]:
Definition 4. ([
4])
A function , , is a subordination chain if is analytic and univalent in U for all , is contiunously differentiable on for all and when . The applications of the hypergeometric functions in the field of complex analysis became obvious and worthy of investigation after L. de Branges used them in proving the famous Bieberbach’s Conjecture [
5]. One of the first papers to study the implications of using hypergeometric functions in the theory of univalent functions is the paper published by Miller and Mocanu in 1990 [
6]. In their paper, Gaussian and confluent (Kummer) hypergeometric functions are studied and conditions for their univalence are stated which inspired further study.
The confluent (Kummer) hypergeometric function of the first kind has the following definition:
Definition 5. ([
7], p. 5)
Let a and c be complex numbers with and considerThis function is called confluent (Kummer) hypergeometric function, is analytic in and satisfies Kummer’s differential equation: If we letthen (4) can be written in the form In paper [
6] the authors have determined conditions on
,
for the function
to be univalent in
U. In paper [
8], the theory of differential subordinations was used in order to obtain new conditions for the univalence of this function considering
,
.
Not much study was done using differential superordination theory on confluent hypergeometric function. Some superordination results were obtained for the generalized hypergeometric function as it can be seen in papers [
9,
10,
11] some superordination results were stated in the form of sandwich-type theorems.
Considering the results obtained in paper [
8] and motivated by the interesting outcomes published implementing the theory of differential superordination, means of this theory are used in this paper in order to investigate confluent (Kummer) hypergeometric function taking
,
. The results obtained here are connected to those from paper [
8] through some corollaries and a sandwich-type theorem.
A lemma is required in order to prove the original results contained in the next section.
Lemma 1. ([
7])
Let and let be analytic in U with and . If q is not subordinate to p, then there exists points and and an for which and(i) ;
(ii) and
(iii) .
2. Results
The original results contained in this paragraph answer the question generated by the applications of the inequalities in the complex plane written in the form of relation (
4).
Among the novelties that the results presented in this paper bring is the use of the theory of differential superordination in order to study the confluent hypergeometric function and obtain interesting superordinations for which it is the best subordinant. Another aspect of novelty is the geometrical interpretation of the obtained superodinations in terms of set inclusions which give sufficient conditions for the confluent hypergeometric function to be a function with positive real part.
The first superordination result is contained in the following theorem. Its proof is very simple since Lemma 1 presented in the Introduction is applied, also known as Miller–Mocanu lemma, a strong result which has been used in obtaining many interesting outcomes both in the theory of differential subordination and in the theory of differential superordination. Theorem 1 is very important as it facilitates the proof of the original results in the next theorems.
Theorem 1. Let , , , defined by (6), and h be analytic in U. Suppose that the differential equationhas a solution . If and is univalent in U, thenimpliesand is the best subordinant. Proof. Using Definition 1, from (
9) we have
since
is univalent in
U.
Suppose that functions satisfy the conditions from Lemma 1, in .
Assume
. By Lemma 1 there exists points
and
and
that satisfy the conditions:
From
,
,
, we have
Since this contradicts (
10) we must have
Since
satisfies the Equation (
8), the conclusion is that
q is the best subordinant. □
Remark 1. From this theorem we see that the problem of finding the best subordinant of (9) essentially reduces to showing that the differential Equation (8) has a univalent solution. The conclusion of the theorem can be written in the symmetric form
or
which implies
We can simplify Theorem 1 for the case of first order differential superordination. The simplified form contained in Theorem 2 is used in the proof of the next theorems, hence it is also a key result of this paper, next to Theorem 1.
Theorem 2. Let , , , defined by (6), and h be analytic in U. If and is univalent in U, thenimpliesand is the best subordinant. The first result which appeals to Theorem 1 for its proof is the next theorem which also generates two interesting corollaries for particular uses of function p involved giving sufficient conditions for the confluent hypergeometric function to be a function with positive real part.
Theorem 3. Let , , , defined by (6) and let h be defined by If , is univalent in U, andthenand is the best subordinant. Proof. Let
, where
For
,
,
, we have
For
,
,
, we get
Using (
18) and (
19), superordination (
16) becomes
By using Theorem 1, we have
and
is the best subordinant. □
Remark 2. Using , a convex function in U, from Theorem 3, we obtain the following corollary.
Corollary 1. Let , , , defined by (6), let and let h be defined by If , , a convex function in U, andis univalent in U and satisfiesortheni.e.,orand is the best subordinant. Proof. From relation (
12) we have
Since
is a convex function in
U, the differential superordination (
22) is equivalent to
or
and
is the best subordinant. □
Remark 3. (a) The conclusion of this corollary has been previously obtained using different methods by Miller and Mocanu in [6] for a and c real numbers satisfying the conditions: (i) and , or
(ii) and for .
(b) The result in Corollary 1 was proved in [8] by using the theory of differential subordination. Remark 4. For convex in U, , , from Theorem 3 we obtain the following corollary which gives an even more precise information on the confluent hypergeometric function’s real part estimate showing how it may change by imposing other conditions on function p used.
Corollary 2. Let , , , defined by (6), let and let h be defined by If , , a convex function in U, with andis univalent in U and satisfiestheni.e.,orand is the best subordinant. Proof. From relation (
12) we have
Since
is a convex function in
U, and
, the differential superordination (
23) is equivalent to
and
is the best subordinant. □
Remark 5. The result in Corollary 2 was proved in [8] using the theory of differential subordination. In the proof of the next result, Theorem 2 is invoked since first order differential superodinations for the confluent hypergeometric function are involved. The order has decreased since the method of subordination chains is applied being a well-known fact that implementing its techniques together with the theory of differential superordination has lead to easier proofs of known results and obtaining new and interesting others.
Theorem 4. Let , , , defined by (6) and let , If , , , is a subordination chain, and is univalent in U, thenimpliesand is the best subordinant. Proof. Since
is a subordination chain, from Definition 4 we have
For
becomes
Using (
27) in (
24), we get
Using (
28), the differential superordination (
26) becomes
for all
and
.
The differential superordination (
29) is equivalent to
By using Theorem 2, we have
Since
is the univalent solution of the Equation (
24), the conclusion is that
is the best subordinant of the differential superordination (
25). □
The next theorem provides means to connect the original results of this paper with the ones previously obtained in paper [
8] through a sandwich-type theorem. Theorem 2 is implemented in the proof of the theorem since it contains results related to first order differential superordinations written for the confluent hypergeometric function in order the facilitate the connection with the results in paper [
8].
Theorem 5. Let h be convex in U, with , let with and consider the function , , given by (6). Let and suppose that is univalent in U. If the differential equationhas a univalent solution that satisfies andthenimplies The function is the best subordinant.
Proof. We will use Theorem 2 to prove this result.
Fir
,
, the relation (
34) becomes
For
,
and since
is a solution of Equation (
31), we have
Using (
35) and (
36) in (
33), we get
For
,
,
, the relation (
34) becomes
Using (
39) in (
38), we have
Using (
32) in (
40), we obtain
Using Theorem 2, the differential superordination (
33) implies
Since
satisfies the differential Equation (
31), we have that
is the best subordinant. □
If in Theorem 1 from paper [
8] we take
and combine the result with Theorem 5, we obtain the sandwich-type theorem that follows, a form of theorems obtained when the dual theories of differential subordination and differential superordination are both used for the study on the same function, underlining the similarities and the differences between the outcomes.
Theorem 6. Let , , , holomorphic in U, given by (6), let h be a convex function in U, let with and If p and v are analytic in U, with , is analytic in U, is univalent in U, and relationis satisfied, then The functions p and v are the best subordinant and the best dominant, respectively.
In order to conclude the study, an example is provided which shows how following the theoretical statements, interesting applications emerge.
Example 1. For , , , , Let the function h be defined as Let , and Using Corollary 1, we have
If or , thenor