On analytic multivalent functions associated with lemniscate of Bernoulli

: In this paper, we establish some su ﬃ cient conditions for analytic functions associated with lemniscate of Bernoulli. In particular, we determine conditions on α such that


Introduction and definitions
To understand in a clear way the notions used in our main results, we need to add here some basic literature of Geometric function theory.For this we start first with the notation A which denotes the class of holomorphic or analytic functions in the region D = {z ∈ C : |z| < 1} and if a function g ∈ A, then the relations g (0) = g (0) − 1 = 0 must hold.Also, all univalent functions will be in a subfamily S of A. Next we consider to define the idea of subordinations between analytic functions g 1 and g 2 , indicated by g 1 (z) ≺ g 2 (z) , as; the functions g 1 , g 2 ∈ A are connected by the relation of subordination, if there exists an analytic function w with the restrictions w (0) = 0 and |w (z)| < 1 such that g 1 (z) = g 2 (w(z)).Moreover, if the function g 2 ∈ S in D, then we obtain: In 1992, Ma and Minda [16] considered a holomorphic function ϕ normalized by the conditions ϕ(0) = 1 and ϕ (0) > 0 with Reϕ > 0 in D. The function ϕ maps the disc D onto region which is star-shaped about 1 and symmetric along the real axis.In particular, the function ϕ(z) = (1 + Az)/(1 + Bz), (−1 ≤ B < A ≤ 1) maps D onto the disc on the right-half plane with centre on the real axis and diameter end points 1−A 1−B and 1+A 1+B .This interesting familiar function is named as Janowski function [10].The image of the function ϕ(z) = √ 1 + z shows that the image domain is bounded by the right-half of the Bernoulli lemniscate given by |w 2 − 1| < 1, [25].The function ϕ(z) = 1 + 4 3 z + 2 3 z 2 maps D into the image set bounded by the cardioid given by (9x 2 + 9y 2 − 18x + 5) 2 − 16(9x 2 + 9y 2 − 6x + 1) = 0, [21] and further studied in [23].The function ϕ(z) = 1 + sin z was examined by Cho and his coauthors in [3] while ϕ(z) = e z is recently studied in [17] and [24].Further, by choosing particular ϕ, several subclasses of starlike functions have been studied.See the details in [2,4,5,11,12,14,19].
Recently, Ali et al. [1] have obtained sufficient conditions on α such that Similar implications have been studied by various authors, for example see the works of Halim and Omar [6], Haq et al [7], Kumar et al [13,15], Paprocki and Sokól [18], Raza et al [20] and Sharma et al [22].
In 1994, Hayman [8] studied multivalent (p-valent) functions which is a generalization of univalent functions and is defined as: an analytic function g in an arbitrary domain D ⊂ C is said to be p-valent, if for every complex number ω, the equation g(z) = ω has maximum p roots in D and for a complex number ω 0 the equation g(z) = ω 0 has exactly p roots in D. Let A p (p ∈ N = {1, 2, . ..}) denote the class of functions, say g ∈ A p , that are multivalent holomorphic in the unit disc D and which have the following series expansion: Using the idea of multivalent functions, we now introduce the class SL * p of multivalent starlike functions associated with lemniscate of Bernoulli and as given below: In this article, we determine conditions on α such that for each , for each j = 0, 1, 2, 3, are subordinated to Janowski functions implies g(z) These results are then utilized to show that g are in the class SL * p .

Lemma
Let w be analytic non-constant function in D with w (0) = 0.If then there exists a real number m (m ≥ 1) such that z 0 w (z 0 ) = mw (z 0 ) .This Lemma is known as Jack's Lemma and it has been proved in [9].

Theorem
Let g ∈ A p and satisfying

Proof
Let us define a function where the function p is analytic in D with p(0) = 1.Also consider Now to prove our result we will only require to prove that |w (z)| < 1. Logarithmically differentiating (2.3) and then using (2.2) , we get and so Now, we suppose that a point z 0 ∈ D occurs such that max Also by Lemma 1.1, a number m ≥ 1 exists with z 0 w (z 0 ) = mw (z 0 ).In addition, we also suppose that w (z 0 ) = e iθ for θ ∈ [−π, π] .Then we have which illustrates that the function φ (m) is increasing and hence φ (m) ≥ φ (1) for m ≥ 1, so . Now, by using (2.1) , we have p which contradicts the fact that p (z) ≺ 1+Az 1+Bz .Thus |w (z)| < 1 and so we get the desired result.Taking g (z) = z p+1 f (z) p f (z) in the last result, we obtain the following Corollary:

Corollary
Let f ∈ A p and satisfying with the condition on α is Then f ∈ SL * p .

Theorem
then g (z)

Proof
Let us choose a function p by in such a way that p is analytic in D with p(0) = 1.Also consider g (z) Using some simple calculations, we obtain and so .
Let a point z 0 ∈ D exists in such a way max Then, by virtue of Lemma 1.1, a number m ≥ 1 occurs such that z 0 w (z 0 ) = mw (z 0 ).In addition, we set w (z 0 ) = e iθ , so we have which contradicts (2.5) .Thus |w (z)| < 1 and so the desired proof is completed.Putting g (z) = z p+1 f (z) p f (z) in last Theorem, we get the following Corollary:

Corollary
If f ∈ A p and satisfying

Theorem
If g ∈ A p and satisfy the subordination relation with the condition on α

Proof
Let us define a function Then p is analytic in D with p(0) = 1.Also let us consider g (z) Using some simplification, we obtain and so .
Let us choose a point z 0 ∈ D such a way that max Then, by the consequences of Lemma 1.1, a number m ≥ 1 occurs such that z 0 w (z 0 ) = mw (z 0 ) and also put w (z 0 ) = e iθ ,for θ ∈ [−π, π] , we have which contradicts (2.7).Thus |w (z)| < 1 and so we get the required proof.
If we set g (z) = z p+1 f (z) p f (z) in last theorem, we easily have the following Corollary:

Theorem
If g ∈ A p satisfy the subordination .
Using some simple calculations, we obtain