On Certain Class of Bazilevič Functions Associated with the Lemniscate of Bernoulli

then gðζÞ is a solution of the differential subordination (4). The univalent function qðζÞ is called a dominant of the solutions of the differential subordination (4) if gðζÞ ≺ qðζÞ for all gðζÞ satisfying (4). A univalent dominant ~q that satisfies ~q ≺ q for all dominants of (4) is called the best dominant. Sokól and Stankiewicz [3] introduced the class SL∗ consisting of analytic functions f ∈A satisfying the following condition


Introduction
Let H ðUÞ be the class of analytic functions in the open unit disk and let A p denote the subclass of H ðUÞ consisting of functions of the form: We write A 1 = A. For f 1 , f 2 ∈ H ðUÞ, we say that f 1 ðζÞ is subordinate to f 2 ðζÞ, written symbolically, f 1 ≺ f 2 in U or f 1 ðζÞ ≺ f 2 ðζÞðζ ∈ UÞ, if there exists a Schwarz function ωðζÞ, which (by definition) is analytic in U with ωð0Þ = 0 and jωðζÞj < 1ðζ ∈ UÞ such that f 1 ðζÞ = f 2 ðωðζÞÞ ðζ ∈ UÞ. Further more, if the function f 2 ðζÞ is univalent in U, then we have the following equivalence (see [1,2]): Let ϕ : ℂ 2 × U ⟶ ℂ and hðζÞ be univalent in U. If gðζÞ is analytic in U and satisfies the first order differential subordination: then gðζÞ is a solution of the differential subordination (4). The univalent function qðζÞ is called a dominant of the solutions of the differential subordination (4) if gðζÞ ≺ qðζÞ for all gðζÞ satisfying (4). A univalent dominantq that satisfiesq ≺ q for all dominants of (4) is called the best dominant. Sokól and Stankiewicz [3] introduced the class SL * consisting of analytic functions f ∈ A satisfying the following condition which is equivalent to where the function maps U onto the domain O = fw ∈ ℂ : Rw > 0, jw 2 − 1j < 1g, and its boundary ∂O is the right-half of the lemniscate of Bernoulli ðx 2 + y 2 Þ 2 − 2ðx 2 − y 2 Þ = 0. Several geometric properties of SL * were investigated done by many authors in ( [4][5][6][7]). Now, we define a class B p ðλ, αÞ of Bazilevic functions associated with lemniscate of Bernoullia by using the principle of differential subordination as follows.
Definition 1. A function f ∈ A p is said to be the class B p ðλ, αÞ if it satisfies the following subordination condition: all the powers are principal values and throughout the paper unless otherwise mentioned the parameters λ, α, and p are constrained as λ ∈ ℂ, α > 0, p ∈ ℕ, and ζ ∈ U.
We note that In order to establish our main results, we need the following lemmas.
Lemma 2 [8]. Let the function h be analytic and convex (univalent) in U with hð0Þ = 1. Suppose also that the function gðζÞ given by is analytic in U. If then and qðζÞ is the best dominant. Lemma 3. [9]. For real or complex numbers a, b, cðc ≠ 0,−1, −2, ⋯Þ and ζ ∈ U, ð12Þ Lemma 4. [10]. Let F be analytic and convex in U.
Lemma 5 [11]. Let f ðζÞ = ∑ ∞ k=1 a k ζ k be analytic in U and Lemma 6 [12]. Let gðζÞ = 1 + ∑ ∞ k=1 c k ζ k ∈ P , i.e., let g be analytic in U and satisfy RfgðζÞg > 0 for ζ ∈ U, then the following sharp estimate holds The result is sharp for the functions given by Lemma 7. [12]. If gðζÞ = 1 + ∑ ∞ k=1 c k ζ k ∈ P , then when υ < 0 or ν > 1, the equality holds if and only if gðζÞ = ð1 + ζÞ/ð1 − ζÞ or one of its rotations. If 0 < ν < 1, then the equality holds if and only if gðζÞ = ð1 + ζ 2 Þ/ð1 − ζ 2 Þ or one of its rotations. If ν = 0, the equality holds if and only if or one of its rotations. If ν = 1, the equality holds if and only if g is the reciprocal of one of the functions such that equality holds in the case of ν = 0.

Journal of Function Spaces
Also, the above upper bound is sharp, and it can be improved as follows when 0 < ν < 1: In the present paper, we obtain subordination properties, inclusion relationship, convolution result, coefficients estimate, and Fekete-Szegö inequalities for the class B p ðλ, αÞ.

Main Results
We begin by presenting our first subordination property given by Theorem 8.
where the function QðζÞ given by is the best dominant.
Proof. Let f ∈ B p ðλ, αÞ and suppose that Then, the function gðζÞ is of the form (9), analytic in U, and gð0Þ = 1. By taking the derivatives in the both sides of (22), we get Since f ∈ B p ðλ, αÞ, we have Now, by using Lemma 2 for γ = pα/λ, we deduce that where we have made a change of variables followed by the use of identities in Lemma 3 with a = −1/2, b = pα/λn, and c = b + 1. This completes the proof of Theorem 8.
If the function f ∈ AðpÞ satisfies the subordination condition and F p,μ is the integral operator defined by (26), then where the function K given by is the best dominant of (28).
Proof. Let then g is analytic in U. Differentiating (31) with respect to ζ and using the identity (28) in the resulting relation, we get Employing the same technique that we used in the proof of Theorem 8, the remaining part of the theorem can be proved similarly.
Theorem 11. If f ∈ A p , then f ∈ B p ðλ, αÞ if and only if where Proof. For any function f ∈ A p , we can verify that First, in order to prove that (38) holds, we will write (8) by using the principle of subordination, that is, where wðζÞ is a Schwarz function, hence for all ζ ∈ U and θ ∈ 0, 2πÞ. From (40) and (41), the relation (43) may be written as which is equivalent to Journal of Function Spaces that is (38).
Reversely, suppose that f ∈ A p satisfy the condition (38). Like it was previously shown, the assumption (38) is equivalent to (41), that is, Denoting the relation (46) could be written as φðUÞ ∩ ψð∂UÞ = ∅. Therefore, the simply connected domain φðUÞ is included in a connected component of ℂ \ ψð∂UÞ. From this fact, using that φð0Þ = ψð0Þ = 1 together with the univalence of the function ψ, it follows that φðζÞ ≺ ψðζÞ, that is f ∈ B p ðλ, αÞ.
Theorem 13. If f given by (2) belongs to the class B p ðλ, αÞ, then The result is sharp.
Proof. If f ∈ B p ðλ, αÞ, then there is a Schwarz function ω in U such that where ϕðζÞ = ffiffiffiffiffiffiffiffiffi ffi 1 + ζ p . Define the function gðζÞ by Since ωðζÞ is a Schwarz function, we see that g ∈ P with gð0Þ = 1. Therefore, Now by substituting (54) in (52), we have Equating the coefficients of ζ and ζ 2 , we obtain Therefore, Our result now follows by an application of Lemma 6. The result is sharp for the functions This completes the proof of Theorem 13.
Corollary 14. If f given by (2) The result is sharp.
Corollary 15. If f given by (2) belongs to the class SL * p , then The result is sharp. Putting p = λ = 1 and α = 0 in Theorem 13, we obtain the following corollary.
Corollary 16. If f given by (2) (with p = 1) belongs to the class SL * , then The result is sharp. Applying Lemma 7 to (57) and (58), we obtain the following theorem.