Certain Aspects of Univalent Function with Negative Coefficients Defined by Bessel Function

Abstract

Key words: In recent years, applications of Bessel functions have been effectively used in the modelling of chemical engineering processes and theory of univalent functions.In this paper, we study a new class of analytic and univalent functions with negative coefficients in the open unit disk defined by Modified Hadamard product withBessel function. We obtain coefficient bounds and exterior points for this new class.

Higher-Order Derivatives of Differential Subordination of Multivalent Functions

“…The theory about harmonic as well as analytic [1] univalent functions for (bi or just multi-types) [2][3][4][5][6] constitutes a few about the most significant ideas associated with complex analysis. Thus, a few unique elements are described within this theory to establish novel interesting certain groups or just subclasses [7][8][9] for special functions associated to multiple operators [10][11][12][13][14] that could have maximized as well as maximized a number real problem via a certain functional relative that results via the theory of conventional functions by way of a few characteristics for complex functions [15,16].…”

On certain subclass of univalent functions with finitely many fixed coefficients defined by Bessel function

“…using the generalised Bessel function ω u,b,c (z) is studied by many researchers [2,3]. Ramachandran et al [6] obtained the following series representation for the function ϕ u,b,c (z) given by (1.5)…”

“…For convenience, ϕ u,b,c (z) is replaced by ϕ κ,c (z). Ramachandran et al [6] introduced a operator B c κ : S →S which is defined by the convolution…”

“…. Motivated by the work of Ramachandran et al [6], here we consider and study the subclass BSD(α, λ , c) with fixed finitely many coefficients defined by modified Hadamard product with Bessel function. We begin with the definition of the class BSD(α, λ , c).…”

On Bohr Radius of Certain Analytic Functions with Negative Coefficients

“…For our ease we write ϕ(u, c, d)(z) = ϕ(κ, d). The operator B (d,κ) : A −→ A is defined by B (d,κ) (g(z)) = ϕ(κ, d) * g(z), ∀g(z) ∈ A.Ramachandran et al[18] introduced a class U B(γ, η, β, d) of analytic functions with negative coefficients, using the normalized form of generalized Bessel functions of first kind defined as:Let d > 1, 0 ≤ γ < 1, β ≥ 0, 0 ≤ η < 1 and z ∈ ∇, then a function g(z) ∈ τ is said to be in the class U B(γ, η, β, d), if and only if [ zG (z) G (z) ] > β| zG (z) G (z) − 1| + η, where G(z) = (1 − γ)B (d,κ) (g(z)) + γ(B (d,κ) (g(z))) .(4. 17)Now by using some results of[18] and the above technique of finding the Bohr radius we can show that the class U B(γ, η, β, d) satisfies Bohr's phenomenon for the following Bohr radiusr = (η + β + 2(2 − η)ς) β + 2(2 − η)ς + (β + 2(2 − η)ς) 2 + 4(1 − η)(η + β + 2(2 − η)ς − 1) ,(4.18) with ς = (1 + γ)(| −d 4κ |).…”