Properties of Certain Multivalent Analytic Functions Associated with the Lemniscate of Bernoulli

: Using differential subordination, we consider conditions of β so that some multivalent analytic functions are subordinate to ( 1 + z ) γ (0 < γ ≤ 1). Notably, these results are applied to derive sufﬁcient conditions for f ∈ A to satisfy the condition (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) zf (cid:48) ( z ) f ( z ) (cid:17) 2 − 1 (cid:12)(cid:12)(cid:12)(cid:12) < 1. Several previous results are extended.

For the two functions f and g analytic in D, the function f is said to be subordinate to g, written as f (z) ≺ g(z) (z ∈ D) , if there exists a function w analytic in D with w(0) = 0 and |w(z)| < 1, such that f (z) = g(w(z)). Notably, if g is univalent in D, then f (z) ≺ g(z) is equivalent to f (0) = g(0) and f (D) ⊂ g(D).
In [1] Sokól and Stankiewicz defined and studied the class From (2), one can see that a function f ∈ SL if z f (z)/ f (z) lies in the region bounded by the right-half of the lemniscate of Bernoulli, given by |w 2 − 1| < 1. All functions in SL are univalent starlike functions. Several authors ( [2][3][4][5]) considered differential subordination for functions belonging to the class SL.
Recently, many scholars introduced and investigated various subclasses of multivalent analytic functions (see, e.g., [3][4][5][6][7][8][9][10][11][12][13][14][15] and the references cited therein). Some properties, such as distortion bounds, inclusion relations and coefficient estimates, were considered. In [16], Seoudy and Shammaky introduced a class of multivalently Bazilevič functions involving the Lemniscate of Bernoulli and obtained subordination properties, inclusion relationship, convolution result, coefficients estimate, and Fekete-Szegǒ problems for this class. In [14], Xu and Liu investigated some geometric properties of multivalent analytic functions associated with the lemniscate of Bernoulli and obtained a radius of starlikeness of the order ρ. In [2], Ali, Cho, Ravichandran and Kumar considered conditions on β so that 1 + βzp (z) subordinate to √ 1 + z. Furthermore, Srivastava [8] carried out a systematic investigation of various analytic function classes associated with operators of q-calculus and fractional q-calculus. In this paper, we will consider conditions of β so that some multivalent analytic functions are subordinate to (1 + z) γ (0 < γ ≤ 1), and derive several sufficient conditions of multivalent analytic functions associated with the lemniscate of Bernoulli. Some previous results are extended. In order to prove our results, the following lemmas will be recalled.

Lemma 1 ([17]
). Let q be univalent in D, and let ϕ be analytic in a domain containing q(D). Also let , and q is the most dominant.

Lemma 2 ([17]
). Let q be univalent in the unit disk D, and let θ and ϕ be analytic in a domain , and q is the best dominant.
Next, we need only to prove q(z) ≺ 1 + βzq (z). Consider the function h by For z = e it , t ∈ [−π, π], we have The minimum of |q −1 (h(e it ))| is obtained at t = 0. Thus Thus h(D) ⊃ q(D). It follows that q(z) ≺ h(z), and the conclusion (4) is proved. Now, we define the function φ by then φ is analytic in D and φ(0) = 1. By a simple calculation, we have The proof of the theorem is completed.
then f ∈ SL or z f (z)/ f (z) lies in the region bounded by the right-half of the lemniscate of Bernoulli. The lower bound β 0 is sharp.
Define the function φ by then, φ is analytic in D and φ(0) = 1. A simple calculation shows that From (6)- (8), we obtain Now, we complete the proof of Theorem 2.

Corollary 2.
Let β 0 = 4( √ 2 − 1) ≈ 1.65 and f ∈ A with f (z) f (z) = 0 when z = 0. If f satisfies the subordination then f ∈ SL or z f (z)/ f (z) lies in the region bounded by the right-half of the lemniscate of Bernoulli. The lower bound β 0 is sharp.
The lower bound β 0 is sharp.
Proof. We first prove the following conclusion: where φ is analytic in D with φ(0) = 1, β ≥ β 0 and the lower bound β 0 is the best possible.
Let q(z) = (1 + z) γ with q(0) = 1. Then, q is a convex function in D. Define the function Q by
Thus we complete the proof of Theorem 4.