1. Introduction
Hypergeometric functions began to be studied more intensely in terms of their relation to the field of complex analysis after proving useful in demonstrating Bieberbach’s conjecture in 1985 [
1]. Before that, the interest on hypergeometric functions came from their applications in different branches of mathematics such as representation theory, algebraic geometry and Hodge theory, number theory, mirror symmetry. The Gaussian hypergeometric function presented high interest, with studies concerning its applications in complex analysis being done as soon as this relation between domains was considered. One of the first papers which contained new results due to this study appeared in 1990 [
2] when Miller and Mocanu considered Gaussian and confluent (Kummer) hypergeometric functions and stated some conditions for their univalence. Gaussian hypergeometric function was investigated in terms of close-to-convexity properties in [
3], and some results related to its univalence were obtained before 1990 [
4], but it has already been established that they were different from the ones contained in [
2].More recently, results on the univalence conditions for the Gaussian hypergeometric function were given in [
5,
6,
7], but in this paper the new results related to the univalence of this function are stated using the well-known theory of differential subordination introduced in two papers in 1978 [
8] and 1981 [
9] by Miller and Mocanu, further developed in the coming years by many authors and synthesized in the work published in 2000 [
10] by the authors who introduced the notion.
Miller and Mocanu’s paper published in 1990 [
2] first served as inspiration for obtaining some new results related to the confluent (Kummer) hypergeometric function stated by first using notions concerning differential subordination theory [
11] and then notions of differential superordiantion theory [
12], with the relation between those results and the original results of Miller and Mocanu being emphasized. The study done to obtain the results contained in this paper follows the same pattern as in paper [
11], but now it takes the Gaussian hypergeometric function into consideration.
All the classical definitions of notions and established notations for the special classes of univalent functions are used throughout the paper.
Let and let denote the class of analytic functions in the unit disc. The class is seen as the subclass of analytic functions having the particular form with and . Another important subclass of which the study refers to consists of functions written as , this class being denoted by , with . All functions from class having the property are called starlike and their class is denoted by , while functions from the same class with the property are called convex and their class is denoted by .
Definitions for the relation of subordination, solution of the differential subordination, dominant and best dominant for a differential subordination are well-known and can be seen in [
11] (Definition 2 and Definition 3) and in Miller and Mocanu’s monograph, [
10] (p.4) and they are used just as they are given there.
The proofs of the original theorems from the next part of this paper require the use of a lemma which can be found in [
10] (Th. 3.4h, p.132).
Lemma 1. [10] Let be univalent in and let and be analytic in a domain containing , with , when Set and suppose that either
- (i)
in convex, or
- (ii)
is starlike.
In addition, assume that
- (iii)
.
Ifis analytic in U, withandthenandis the best dominant. The notion of differential subordination was introduced after observing certain inequalities and inclusions of sets valid in real analysis and adapting them for the case of complex-valued functions. A short history of the emergence of the notion can be read in the introduction of paper [
12]. In the same cited paper, some geometrical interpretations of superordination results written as inclusions of sets are shown. Such a perspective on using set inclusions to express a new outcome was approached in another paper published in 2020 [
13], which proves that this direction is of interest at this time and generates an interesting outcome. This technique is applied for the original results obtained in this paper for differential subordinations.
The definition of the Gaussian hypergeometric function is reminded in [
10] and presented below:
Definition 1. [10] Let a, b,c,The function:is called a Gaussian hypergeometric function. A property easily obtained for this function given in [
2] is:
Next to the mirror symmetry formula for Gauss hypergeometric function, it is known that this function has the symmetric property remaining unaltered if the numerator parameters and are interchange while, keeping the denominator parameter fixed.
The original results obtained in relation to this function and which have led to stating two criteria for its univalence can be seen in the following theorems and corollaries.
2. Main Results
Using Lemma 1 mentioned in the introduction, the first stated theorem proves a subordination result which, for certain specific functions used, gives criteria for the univalence of the Gaussian hypergeometric function. The criteria are presented in two corollaries that follow the theorem, and the geometrical interpretation of the results in the corollaries using the inclusion of sets can also be seen.
Theorem 1. Consider a convex functionhaving the propertyand letwhenknowing thatfor
LetandTake: Letwith the conditionbe met when.
If the subordination:is satisfied, thenwhenwritten as the sets inclusionandis the best dominant. Proof.
The proof uses Lemma 1, and hence the conditions from the lemma must be met. To prove the compliance of the first condition from the lemma, the functions
,
and
are defined as:
Putting in (5), the following equalities are obtained:
and
Using (3), it results in:
and:
Putting in (5), the following equalities are obtained:
and
Using (3) again, it results in:
Using (8) and (9), the differential subordination (4) becomes:
To apply Lemma 1, the starlikeness of function
needs to be proved. In the hypothesis of the theorem, function
was taken as convex. Thus, it is known that:
Differentiating (6) and making a simple calculus, we obtain:
Using (11) in (12), we have:
which is equivalent to
being starlike. It is now proved that the first condition needed in order to apply Lemma 1 is satisfied.
Differentiating (7) and after a simple calculus, using (11) and the hypothesis of the theorem, we get:
which means that the second condition from Lemma 1 is also satisfied.
Using Lemma 1, the differential subordination (10) implies:
when
Since is a convex function which satisfies the equation it is the best dominant of the proved differential subordination. □
Remark 1. If in Theorem 1, the function:is considered since it is a convex function, and ifgiven by (1), then the following univalence criterion for the Gaussian hypergeometric function is obtained. Corollary 1. Consider the convex function:and letwhenknowing thatfor Letgiven by (1) withwhen.
If the subordination:is satisfied, then:whenwritten as the sets inclusion:i.e.,written as the sets inclusion:Hence,is univalent in Proof.
Next, we differentiate the equality and we obtain:
Then, the differential subordination (15) becomes:
Using relation (14) from the proof of Theorem 1 results in:
and:
Using (17) in (16), we have:
Using the convexity of function
, the differential subordination (18) is equivalent to:
written as the sets inclusion:
concluding that the function
is univalent. □
Remark 2. In Theorem 1, take the functions: The following univalence criterion for the Gaussian hypergeometric function can be written:
Corollary 2. Consider the convex function:and letwhenknowing thatfor If the function:is analytic inand satisfies the differential subordination: Then:i.e.,written as the sets inclusion:Hence, the functionis convex in Proof.
Differentiating the equality and using the expressions for
and
in (19), we get:
Using relation (14) obtained in the proof of Theorem 1 results in:
By replacing
with its expression written above, we obtain:
and function
is the best dominant.
Using the convexity of function
, the above differential subordination is equivalent to:
written as the sets inclusion:
concluding that function
is convex in
.
This corollary is a key result as it gives the connection between the original outcome of this paper and the findings of Miller and Mocanu in their paper [
2] which have inspired this study, and it is also used in the proof of the next two original theorems. □
Remark 3. The outcome proved in Corollary 2 has been previously obtained in [2] having consideredwith the propertieswhere The next theorem gives results concerning the starlikeness of the derivative of the Gaussian hypergeometric function. The proof of this theorem is short because it uses Corollary 2.
Theorem 2. Letgiven by (1). Then,is starlike.
Proof.
Differentiating and doing some simple calculus, the following equality is obtained:
Corollary 2 states that function
is a convex function in
hence:
which means that function
is starlike, i.e.,
is starlike. □
The next theorem states a starlikeness result for the Gaussian hypergeometric function and also invokes Corollary 2 in the proof. The theorem makes another connection with the results from [
2].
Theorem 3. Letgiven by (1). Then,is starlike in
Proof.
From (2), it is known that:
which is equivalent to:
Define the function
as:
having
.
Differentiating
we obtain:
Using Corollary 2, it is known that function
is convex in
, which gives:
meaning that the function
is starlike in
However, the function was defined as and hence the proof is concluded. □
Remark 4. The starlikeness result of this theorem was previously given in [
2]
taking with the properties where The study presented is concluded with two examples illustrating the way to apply the theoretical findings presented in the paper. The first example refers to this last proved result and the second is related to Corollary 2.
Example 1. Take
Define the functionWe next prove that Theorem 3 can be applied for this function, meaning that g(z) is a starlike function. We conclude that functionis starlike and: Example 2. TakeThen,and: Differentiating, we get:and: Using Corollary 2, we obtain:
Then:written as the sets inclusion:hence,is a convex function. Indeed:which shows thatis convex inwritten as the sets inclusion: