A new inner approach for differential subordinations

In this paper we introduce and examine the differential subordination of the form \[ p(z)+zp'(z)\varphi(p(z),zp'(z))\prec h(z),\quad z\in\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}, \] where $h$ is a convex univalent function with $0\in h(\mathbb {D}).$ The proof of the main result is based on the original lemma for convex univalent functions and offers a new approach in the theory. In particular, the above differential subordination leads to generalizations of the well-known Briot-Bouquet differential subordination. Appropriate applications among others related to the differential subordination of harmonic mean are demonstrated. Related problems concerning differential equations are indicated.


Introduction
Given r > 0, let D r := {z ∈ C : |z| < r}.Let D := D 1 and T := {z ∈ C : |z| = 1}.For D being a domain in C let H(D) be the family of all analytic functions f : D → C and H := H(D).Let A be the subclass of H of f normalized by f (0) = 0 = f (0) − 1, and S be the subclass of A of univalent functions.
A function f ∈ H is said to be subordinate to a function g ∈ H if there exists ω ∈ H such that ω(0) = 0, ω(D) ⊂ D and f = g • ω in D. We write then f ≺ g.If g is univalent, then (e.g., [3,Vol. I,p. 85]) (1.1) Given ψ : C 2 → C, let H[ψ] be the subset of H of all p such that a function D z → ψ(p(z), zp (z)) is well-defined and analytic.

A. Lecko
Let ψ : C 2 → C and h ∈ H be univalent.We say that a function p ∈ H [ψ] satisfies the first-order differential subordination and is called its solution if ψ (p(z), zp (z)) ≺ h(z), z ∈ D. (1.2) If q ∈ H is a univalent function such that p ≺ q for all p satisfying (1.2), then q is called a dominant of (1.2).A dominant q is called the best dominant if q ≺ q for all dominants q of (1.2).Finding those q for which the subordination (1.2) yields p ≺ q, in particular, p ≺ q, is the basis in the theory of differential subordinations.Further details and references can be found in the book of Miller and Mocanu [8].
The classical example of (1.2) is related to the arithmetic mean and has been studied by many authors (see e.g., [8, pp. 120-145]).Given ϕ ∈ H(D) and α ∈ [0, 1], consider written equivalently as which is known as the Briot-Bouquet differential subordination.Let ϕ : C 2 → C, p ∈ H[ϕ] and h ∈ H be a convex univalent function which means that h maps univalently D onto a convex domain h(D).In this paper, we propose a study of the differential subordination of the form (1.5) The case when 0 ∈ ∂h(D) has been studied in [5].Let us remark that the case where 0 is a boundary point of h(D) requires different methods of proofs than those when the origin is the interior point of h(D).The differential subordination (1.5) is a special case of (1.2), but it offers interesting applications.In particular, it generalizes the Briot-Bouquet differential subordination (1.4).In addition, we prove in a new way some recent results regarding the differential subordination related to the harmonic mean.The problem of the best dominant in the case where h is a linear function is also discussed.The proof of the main result is based on the original lemma 2.1 on convex univalent functions.Therefore, the proof of Theorem 2.4 is strictly analytical in nature, while until now in the proofs of analogous propositions, analytical arguments have been used in conjunction with geometric considerations (cf.[8]).By applying lemma 2.1, a series of theorems from the monographs [8]) underlying the theory of the differential subordinations can be proved again by using a purely analytical argumentation.
Let Q be the subclass of S c of all convex functions analytic on D with h (ζ) = 0 at every ζ ∈ T.
We will now prove the lemma that will be used in the proof of the main theorem.This results is geometrically obvious.
Proof.Since h r for every r ∈ (0, 1), is analytic on D and convex in D, from (2.1) it follows that for z ∈ D and ζ ∈ T, where u := rζ ∈ D and v := rz ∈ D. Hence and by the fact that h which shows (2.3).
We need also the following lemma which is a special case of lemma 2.2d [8, p. 22].

Lemma 2.2. Let h be an analytic univalent function in D, p ∈ H be a nonconstant function with p
and for some m 1.
The theorem below follows directly from the Lindelöf Principle (e.g., [3, Vol.I, p. 86]).However, it will be useful in proving the main theorem.

2.2.
We now prove the main theorem of this paper.In the proof we apply lemma 2.1 and Theorem 2.3.Therefore the argumentation is purely analytical without using a geometrical property based on the behaviour of the normal vector to the boundary curve ∂h(D), standardly used in the theory (cf.[8]).In further discussion we present new type of the differential subordination generalizing the well known Briot-Bouquet differential subordination.The significance of Theorem 2.4 is emphasized also in the presented applications.
Theorem 2.4.Let h be a convex function and ϕ : C 2 → C be such that for each m 1 a function is well-defined and analytic satisfying the condition Re ϕ(h(z), mzh (z)) 0, z ∈ D. (2.10) with p(0) = h(0), and ) Proof.Note first that if Re ϕ(h(z 0 ), mz 0 h (z 0 )) = 0 for a certain z 0 ∈ D, then by the minimum principle for harmonic function Re ϕ(h(z), mzh (z)) = 0 for all z ∈ D and hence ϕ(h(z), mzh (z)) = ia for some a ∈ R and all z ∈ D.
In Theorem 2.4 instead of ϕ we can put a function φ : D → C such that a function D z → φ(h(z)) is well-defined and analytic satisfying the condition Re φ(h(z)) 0 for z ∈ D. Then we obtain the result due to Miller and Mocanu [7] (see also [8,Theorem 3.4a, p. 120]).
In the following theorem the assumption (2.19) is based on the idea of [1] (see also [8, pp. 124-125]), where in the proof of the main result Löwner chains were used.Our argumentation is analogous to that in the proof of Theorem 2.4. (

3.1.
We now discuss special cases of Theorem 2.4.
Thus the function has an analytic extension on D by setting φ(0) + βk at zero, i.e., p ∈ H [ϕ]. At the end note that Thus, the assumptions of Theorem 2.4 are satisfied, which ends the proof of the corollary.
For β = 0 the above theorem reduces to Corollary 2.5, with the additional assumption that h(0) = 0.In fact, this assumption is not required in Corollary 2.5.
then p ≺ h.

3.2.
By selecting h ∈ S c we can get a number of new results.It is natural to take into account the following convex functions keeping the origin fixed: for M > 0,

The best dominant
To find the best dominant of (2.12) is an interesting problem to study related to the theory of the differential equations.By applying Theorem 2.3e of [8] we can expect that the best dominant q of (1.5) should be a univalent solution of the differential equation if such a solution exists.Here we restrict our interest to the differential subordination (3.6) with h(z) := Mz, z ∈ D, where M > 0. For this purpose, we will find a univalent solution of the differential equation The following theorem provides a solution to this problem.
and q is the best dominant of (4.2).

A. Lecko
Proof.We apply the technique of power series to find a univalent solution of (4.1) of the form Since q is assumed to be univalent, we see that From (4.1) we equivalently have Hence using (4.3) we have Comparing coefficients we obtain and in general, for n = 2k − 1, k 2, which in view of (4.8) yield a 2k−1 = 0. Thus we proved that a n = 0 for all n 2. In this way by (4.3) and (4.8) we see that is a unique univalent solution of (4.1).From Theorem 2.3e of [8] it follows that q is the best dominant of (4.2) which completes the proof of the lemma.
For α = 1, β = 1 and M = 1, the above result reduces to the well known special case of the first order Euler differential subordination (see [8, pp. 334-340]).and q is the best dominant.