New Class of Close-to-Convex Harmonic Functions Defined by a Fourth-Order Differential Inequality

Department of Basic Science, College of Science and eoretical Studies, Saudi Electronic University, Riyadh 11673, Saudi Arabia Faculty of Computer Science, Arab Open University, Riyadh, Saudi Arabia Department of Mathematics, Riphah International University, Islamabad 44000, Pakistan Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad 22060, Pakistan Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600, Malaysia


Introduction and Definitions
Let H represent the class of all harmonic functions f s + v in open unit disk U z ∈ C: |z| < 1 { }, and all these harmonic functions are normalized by and f can be expressed as f s + v, where s is the analytic part and v is the coanalytic part of f in U, and also, these functions have a series of the form Harmonic functions f is locally univalent and sensepreserving in U, if it satis es a necessary su cient condition |s ′ (z)| > |v ′ (z)| [1,2]. If coanalytic part of f is zero, then class H of complex valued harmonic functions reduces to class A of normalized analytic functions.
Let S denote the family of analytic univalent and normalized functions in U and also S ⊂ S H which are de ned as Also, let class H 0 de ne as where class S H represent the class of functions f which are harmonic, univalent, and sense-preserving in open unit disk U.
We can see that class S 0 H is compact and normal, but class S H is only normal. Let S 0, * H , K 0 H , and C 0 H are the subclasses of S 0 H which map open unit disk U onto starlike, convex, and close-to-convex domains for harmonic functions, respectively. We can observe that In this class, they studied about close-to-convexity of harmonic functions. After that, Li and Ponnusamy [3,4] discussed univalency and convexity of the abovementioned class. A class W 0 H of harmonic functions f � s + v ∈ H 0 defined by Nagpal and Ravichandran in [5] and the functions in this class satisfy the condition Note that and members of W 0 H are fully starlike in U. Recently, Ghosh and Vasudevarao [6] for α ≥ 0 defined a new class W 0 H (α) for harmonic functions f � s + v∈ H 0 satisfying the condition Re s ′ (z) + αzs ″ (z) > v ′ (z) + αzs ″ (z) , for z ∈ U. (9) Rajbala and Prajapat [7] for δ ≥ 0, 0 ≤ λ < 1, defined a new class W 0 H (δ, λ) of harmonic functions which satisfy the following inequality: For this class, authors used Gaussian hypergeometric function and created harmonic polynomials for the class W 0 H (δ, λ).
By taking the inspiration from the abovementioned work, we define new class of harmonic functions in U which satisfy the fourth-order differential inequality.
In this section, we prove that all the members of the class are close-to-convex. We will derive coefficient bounds, growth estimates, and sufficient coefficient condition for the class R 0 H (λ, δ, c). Furthermore, we investigate example of harmonic polynomial belonging to R 0 H (λ, δ, c).
be analytic in U, such that en, for n ∈ R, n ≥ j ≥ 1, such that or Re en, and hence, Proof. Let L ∈ R(λ, δ, c), and we have en, Let φ be an analytic function in U with Journal of Mathematics We have to show that |φ(z)| < 1, ∀z ∈ U. en, Since φ is analytic in U, such that en, by using Lemma 2, we may write For all z 0 ∈ U, we get cos θ) , Journal of Mathematics which opposes the hypothesis. Hence, there is no z 0 ∈ U, such that Hence, |φ(z)| < 1 for all z ∈ U. So, we get Proof. According to Lemma 3, we derive that L μ � s + μv ∈ R(λ, δ, c) are close-to-convex in U, ∀μ(|μ| � 1). erefore, in the light of eorem 1 and Lemma 1, we get L μ ∈ R 0 , for j ≥ 2.
(36) e equality is satisfied for Proof. δ, c). Applying the series of v(z), we get Taking r ⟶ 1−, we get required result.