Certain Geometric Properties of the Canonical Weierstrass Product of an Entire Function Associated with Conic Domains

In this paper, we determine the radius of λ -uniform convexity, λ -starlikeness, and α -convexity of order δ for the Weierstrass canonical product of an entire function as a root having smallest modulus and argument ϕ of a functional equation. As special cases, we also determine the radius of λ -uniform convexity, λ -starlikeness, and α -convexity of order δ for the entire function 1/ Γ .


Introduction
Let r > 0 be a real number and A be the class of analytic functions defined in the disk UðrÞ = fw ∈ ℂ : jwj < rg and satisfy the normalization conditions f ð0Þ = f ′ð0Þ − 1 = 0: Let ða n Þ, where a n ∈ ℂ, ∀n ≥ 2 be a sequence with 1 lim n⟶+∞ sup a n j j 1/n = r f ≥ 0, ð1Þ where r f means the radius of convergence of the series w + ∑ ∞ n=2 a n w n = f ðwÞ ∈ A. If lim n⟶+∞ sup ja n j 1/n = 0, then r f = +∞: In 1999, Kanas and Wisniowska [9] (also refer Goodman [7,8], Rønning [15], and Ma and Minda [12]) proposed the idea of λ-uniform convexity denoted by λ − UCV : A function f ∈ A is said to be in λ − UCV ðδÞ, the class of λ-uniformly Convex of order δ [3], iff A function f ∈ A is said to be in λ − ST ðδÞ, the class of λ-starlike function of order δ [10], iff Geometrically, the conditions (2) and (3) mean that for f ∈ λ − UCV ðδÞ and f ∈ λ − ST ðδÞ, the images of UðrÞ under the functions 1 + wf ″ðwÞ/f ′ðwÞ and wf ′ðwÞ/f ðwÞ are in the conic domain Ω δ λ contained in the right half plane for which 1 ∈ Ω δ λ and ∂Ω δ λ is the curve defined by the equation Moreover, Ω δ λ is an elliptic region for λ > 1, parabolic for λ = 1, and hyperbolic for 0 < λ < 1, and finally, Ω 0 0 is the whole right half plane.
Let α ∈ ℝ and α ∈ ½0, 1Þ. A function f ∈ A is said to be in M α ðδÞ, the class of α-convex functions (Mocanu functions) of order δ [14,16] iff The radius of α-convexity (Mocanu functions) of order δ denoted by r α cð f Þ ðδÞ is defined by, for 0 ≤ δ < 1, Addressing radius problems for some special functions is a new direction in the geometric function theory. For recent studies on radius problems, we refer to [2,4,6,11].
By the Weierstrass factorization theorem [18], the function is an entire function for a proper choice of q n ≤ n with zeros c n and no other zeros, where hðwÞ is an entire function with hð0Þ = 0, c n ≠ 0∀n, q n ≥ 0 are certain nonnegative integers, and for each n in which q n = 0, the value of exponential factor becomes 1.
The product (9) is called the canonical Weierstrass product [1]. In Theorem 3 of [13] Merkes et al. determined the radius of starlikeness of the canonical Weierstrass product BðwÞ, and as a special case, the authors determined the radius of starlikeness of Later in [17], Szasz obtained the radius of convexity for BðwÞ.
Motivated by the results of Szász [17] and Merkes et al. [13], we determine the radius of λ-uniformly convexity, λ -starlikeness, and α-convexity of order δ for the function BðwÞ given by (9). Consequently, we also determine the radius of λ-uniform convexity, λ-starlikeness, and α-convexity of order δ for the function 1/Γ in this paper. In order to prove the main result, we require the following lemma.
By specializing the parameters in Theorem 6, we have Remark 8. Substituting δ = 0 and λ ∈ ½0, 1/2Þ in Theorem 6, we get the radius r λ ucðBÞ of λ -uniform convexity given by the absolute value of the root of the equation ð1 + λÞwB″ðwÞ + ð1 − 2λÞB ′ ðwÞ = 0 having the smallest modulus and argument ϕ.