Sufficient conditions for univalence obtained by using Briot-Bouquet differential subordination

In this paper, we define the operator I : A → A, I [f ] (z) = (1− λ)S [f ] (z) + λL [f ] (z), z ∈ U , differential-integral operator, where S is Sălăgean differential operator and L is Libera integral operator. By using the operator I the class of univalent functions denoted by M (m,β, λ), 0 ≤ β < 1, 0 ≤ λ ≤ 1, m ∈ N is defined and several differential subordinations are studied. Even if the use of linear operators and introduction of new classes of functions where subordinations are studied is a well-known process, the results are new and could be of interest for young researchers because of the new approach derived from mixing a differential operator and an integral one. By using this differential–integral operator, we have obtained new sufficient conditions for the functions from some classes to be univalent. For the newly introduced class of functions, we show that is it a class of convex functions and we prove some inclusion relations depending on the parameters of the class. Also, we show that this class has as subclass the class of functions with bounded rotation, a class studied earlier by many authors cited in the paper. Using the method of the subordination chains, some differential subordinations in their special Briot-Bouquet form are obtained regarding the differential–integral operator introduced in the paper. The best dominant of the Briot-Bouquet differential subordination is also given. As a consequence, sufficient conditions for univalence are stated in two criteria. An example is also illustrated, showing how the operator is used in obtaining Briot–Bouquet differential subordinations and the best dominant.

Next we give the main classes of univalent functions used in the paper. Let U denote the unit disk of the complex plane: U = {z ∈ C : |z| < 1}.
Let H (U ) be the space of holomorphic functions in U , and let A n = f ∈ H (U ) : f (z) = z + a n+1 z n+1 + ..., z ∈ U , Mathematics and Statistics 8 (2): 126-136, 2020 127 be the class of holomorphic and univalent functions in the open unit disk U , with conditions f (0) = 0, f (0) = 1, that is the holomorphic and univalent functions with the following power series development f (z) = z + a 2 z 2 + a 3 z 3 + ..., z ∈ U.
For a ∈ C and n ∈ N * , denote by H [a, n] = f ∈ H (U ) : f (z) = a + a n z n + a n+1 z n+1 + ..., z ∈ U , the class of normalized convex functions in U and the class of normalized starlikeness functions in U . We now remind the definition of the subordination as it is given in the most important monograph related to the differential subordination method or the admissible function method which contains its fundamental notions, published in 2000 ( [10]). [10, p. 4]). If f and g are analytic functions in U , then f is subordinate to g, written f ≺ g, or

Definition 1. (Subordination
Definition 2. (Second-order differential subordination [10, p. 7]). Let ψ : C 3 × U → C and let h be univalent in U . If p is analytic and satisfies the (second-order) differential subordination the 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 p ≺ q for all p satisfies (i). A dominant q that satisfies q ≺ q for all dominants q of (i) is said to be the best dominant of (i). (Note that the best dominant is unique up to a rotation of U ).
If we require the more restrictive condition p ∈ H [a, n], then p will be called an (a, n)-solution, q an (a, n)-dominant and q the best (a, n)-dominant.
Definition 3. (Briot-Bouquet differential subordination [10, p. 80]). Let β, γ ∈ C, β = 0, and let h be a univalent function in U , with h (0) = a, and let p ∈ H [a, n], satisfy This first-order differential subordination is called the Briot-Bouquet differemtial subordination. The name derives from the fact that a differential equation of the form is called a differential equation of Briot-Bouquet type [5, p. 403].
The original results shown in the present paper are obtained using the well-known Sȃlȃgean differential operator which was introduced in 1983 ( [18]) and is widely used by researchers in the field of Geometric Function Theory.
Definition 4. ( [18]). For f ∈ A, m ∈ N = {0, 1, 2, ...} the differential operator S m : A → A, is defined by (1.1) In the same paper ( [13]), the following result related to the operator in Definition 6 is stated: We next give a definition and an important lemma from the admissible function method: Definition 8. ([10, Definition 2.2b, p. 21]). We denote by Q the set of functions q that are analytic and injective on U \E (q), where and are such that q (ζ) = 0 for ζ ∈ ∂U \E (q). The set E (q) is called exception set.
be analytic in U , with p (z) = a and n ≥ 1. If p is not subordinate to q, then there exist points τ 0 = r 0 e iθ0 ∈ U and ζ 0 ∈ ∂U \E (q), and an p ≥ n ≥ 1 for which p (U r 0 ) ⊂ q (U ) , In this paper, the original results are obtained by using the method of the subordination chains. The definition of the subordination chain and some important lemmas related to this method now follow.

subordination chain if and only if
Re Lemma 12. C ([10, Theorem 3.44, p. 132]). Let q be univalent in U and let θ and φ be analytic in a domain D containing q (U ), and q is the best dominant.
The function q is convex and is the best (a, n)-dominant.
A paper containing subordination results related to a class of univalent functions obtained by the use of an operator introduced by using a differential operator and an integral one has been published recently ( [17]) and inspired the results from this paper.

Main Results
Firstly, we define a new differential-integral operator.
Next, we define a new class of univalent functions by using the newly introduced operator.
we let M (m, β, λ) stand for the class of functions f ∈ A, which satisfy the inequality

3)
where the differential-integral operator I m [f ] is given by (2.1).
called the class of functions with bounded rotation. This class of functions was studied by Alexander [1] and he proved that R ⊂ S, Krzyz [4] and Mocanu [11] have proved that R ⊂ S * . A more systematic study of the class R was done by Mac Gregor [6]. Proof. Let the functions where α kj = a kj (1 − λ) k m + λ 2 m (k+1) m , be in the class M (m, β, λ). It is sufficient to show that the function (2.4)
We let f ∈ M (m, β, λ). From Definition 16, we have (2.10) Using (2.2) in (2.10), we have Using (2.11) in (2.9), we have Re (p (z) + zp (z)) > β, z ∈ U. (2.12) Relation (2.12) can be written as a subordination of the form Using Lemma 13 for γ = 1, n = 1, we have The function q is convex and is the best dominant. Since q is a convex function and p (z) ≺ q (z), z ∈ U , we have Usin(2.10) in (2.13), we have (2.14) Using (2.2), we have Since q is convex function, we have Theorem 21. Let q be univalent in U , and let θ and φ be analytic functions in a domain D containing q (U ), with θ : and suppose that we have (i) Q is starlike; If p is analytic in U with p (0) = q (0), p (U ) ⊂ D, then and q is the best dominant.
Proof. From (i) we know that the function Q is starlike and from (ii) we know that the function h is close-to-convex. Let the function: This function is analytic in U for all t ≥ 0 and is continuously differentiable on [0, ∞) for z ∈ U. Differentiating (2.18) with respect to z we obtain For z = 0 we have Differentiating (2.18) with respect to t we obtain We deduced Hence a 1 (t) = 0, lim t→∞ a 1 (t) = ∞ and Re z ∂L(z,t) ∂z ∂L(z,t) ∂t > 0, for z ∈ U and t ≥ 0. Using Lemma 11, L (z, t) is a subordination chain which by Definition 10 implies (2.20) Using (2.20) and Definition 10 we get Let the function ψ : C 2 × U → C, and θ and φ be analytic functions ψ (r, s) = θ (r) + sφ (r) , θ (r) = r, φ (r) = 1 r .
(2.24) Using (2.2) and (2.24), we have Differentiating (2.24) and after short calculation, we obtain In order to prove that (2.16) or (2.26) implies p is subordinate to function q, we applying Lemma 9. For that we assume that the functions p, q and h satisfy the conditions in Lemma 9 in the unit disk U .
Since q is the solution of the univalent equation we have that q is the best dominant.
From Theorem 21 we deduce the following sufficient conditions for univalent function.
From Definition 16 we have f ∈ M (m, β, λ) and f is an univalent function.
From Definition 16 we have f ∈ M (0, 0, λ) = R and f is an univalent function.