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

Authors: K. A. Selvakumaran; P. Rajaguru; S. D. Purohit; D. L. Suthar
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we determine the radius ofλ-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.


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## Body

## 1. Introduction

Letr>0 be a real number and A be the class of analytic functions defined in the disk Ur=w∈ℂ:w<r and satisfy the normalization conditions f0=f′0−1=0. Let an, where an∈ℂ,∀n≥2 be a sequence with
(1)1limn⟶+∞supan1/n=rf≥0,whererf means the radius of convergence of the series w+∑n=2∞anwn=fw∈A. If limn⟶+∞supan1/n=0, then rf=+∞.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 functionf∈A is said to be in λ−UCVδ, the class of λ-uniformly Convex of order δ [3], iff
(2)Re1+wf"wf′w>λwf"wf′w+δ,λ≥0,δ∈0,1∀w∈Ur.A functionf∈A is said to be in λ−STδ, the class of λ-starlike function of order δ [10], iff
(3)Rewf′wfw>λwf′wfw−1+δ,λ≥0,δ∈0,1∀w∈Ur.Geometrically, the conditions (2) and (3) mean that for f∈λ−UCVδ and f∈λ−STδ, the images of Ur under the functions 1+wf″w/f′w and wf′w/fw are in the conic domain Ωλδ contained in the right half plane for which 1∈Ωλδ and ∂Ωλδ is the curve defined by the equation
(4)∂Ωλδ=ω=u+iv:u−δ2=λ2u−12+v2,λ≥0.Moreover,Ωλδ is an elliptic region for λ>1, parabolic for λ=1, and hyperbolic for 0<λ<1, and finally, Ω00 is the whole right half plane.The radius ofλ-uniform convexity of order δ denoted by rucfλδ and radius of λ-starlikeness of order δ denoted by rstfλδ are defined by
(5)rucfλδ=supr∈0,rf:Re1+wf″wf′w>λwf″wf′w+δ,∀w∈Ur,(6)rstfλδ=supr∈0,rf:Rewf′wfw>λwf′wfw−1+δ,∀w∈Ur,whereλ≥0,δ∈0,1.By specializing the parameters, we observerucf00=rfc, radius of convexity, rucf0δ=rfcδ, radius of convexity of order δ, rstf0δ=rf∗δ, radius of starlikeness of order δ and rstf00=rf∗, radius of starlikeness.Letα∈ℝ and α∈0,1. A function f∈A is said to be in Mαδ, the class of α-convex functions (Mocanu functions) of order δ [14, 16] iff
(7)Re1−αwf′wfw+α1+wf″wf′w>δ,w∈Ur,δ∈0,1.The radius ofα-convexity (Mocanu functions) of order δ denoted by rcfαδ is defined by, for 0≤δ<1,
(8)rcfαδ=supr∈0,rf:Re1−αwf′wfw+α1+wf″wf′w>δ,w∈Ur.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
(9)Bw=wehw∏n=1∞1−wcnexp∑k=1qn1kwcnk,is an entire function for a proper choice ofqn≤n with zeros cn and no other zeros, where hw is an entire function with h0=0, cn≠0∀n, qn≥0 are certain nonnegative integers, and for each n in which qn=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 Bw, and as a special case, the authors determined the radius of starlikeness of
(10)1Γw=wewγ∏n=1∞1+wne−w/n.Later in [17], Szasz obtained the radius of convexity for Bw.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 Bw 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.Lemma 1 (see [17]).
Ifa,b∈ℝ and a>b>0, then
(11)a+wb+w2≤a−wb−w2,forw<b,w∈U=U1.


## 2. Main Results

Theorem 2.
Letcnn∈ℕ/0 be a sequence with cn=cneiϕ∈ℂ,cn≥1 for n∈ℕ/0, r0=infcn:n∈ℕ/0, and let hw be an analytic function in Ur0 with weiϕh′weiϕ∈ℝ and weiϕh′weiϕ≤Rwh′w, for w∈Ur0. If the function μ:0,r0⟶ℝ defined by μr=reiϕh′reiϕ is decreasing with respect to r and Bw is of the form (9) with qn∈ℕ/0 for n∈ℕ/0, then the radius of λ-starlikeness of order δ of the function Bw is rstBλδ, the absolute value of the root of the equation 1+λwB′w−λ+δBw=0 having the smallest modulus and argument ϕ.Proof.
By logarithmic differentiation, (9) becomes
(12)wB′wBw=1+wh′w−∑n=1∞cnwqn+11−cnw.
Forw∈U and k,n∈ℕ,
(13)Rwn1−wk≤wn1−wk=wn1−wk≤wn1−wk.
Sincew/cn≤1, (12) along with (13) implies
(14)RwB′wBw≥1+Rwh′w−∑n=1∞w/cnqn+11−w/cn≥1+weiϕh′weiϕ−∑n=1∞w/cnqn+11−w/cn=weiϕB′weiϕBweiϕ.Also,(15)wB′wBw−1≤wh′w+∑n=1∞w/cnqn+11−w/cn≤−weiϕh′weiϕ+∑n=1∞w/cnqn+11−w/cn=1−weiϕB′weiϕBweiϕ.From (14) and (15), we have
(16)RwB′wBw−λwB′wBw−1−δ≥1+λweiϕB′weiϕBweiϕ−λ+δ,δ∈0,1,λ≥0.By the virtue of minimum principle for harmonic functions,(17)infw<rRwB′wBw−λwB′wBw−1−δ=1+λreiϕB′reiϕBreiϕ−λ+δ,r∈0,ro.We observe that the functionφ:0,r0⟶ℝ defined by
(18)φr=1+λreiϕB′reiϕBreiϕ−λ+δis strictly decreasing; also,limr⟶0φr=1−δ>0 and limr⟶r0φr=−∞.Hence, the equation1+λeiϕrB′reiϕ−λ+δBeiϕr=0 has a unique root in 0,r0, and this root is rstBλδ.Remark 3.
λ≥0 in Theorem 2 means that, if B∈λ−STδ, then the image of Ur under the function wB′w/Bw is in conic domain Ωλδ contained in the right half plane for which 1∈Ωλδ and ∂Ωλδ is the curve defined by equation (4).In the following remarks, we deduce the radius of some special classes by specializing the parameters in Theorem2.Remark 4.
Takingλ≥0,δ=0 in Theorem 2, we get rstBλ, the radius of λ-starlikeness of the function Bw. rstBλ is the absolute value of the root of the equation 1+λwB′w−λBw=0 having the smallest modulus and argument ϕ.Remark 5.
Lettingλ=0,0≤δ<1 in Theorem 2, we get rB∗δ, the radius of starlikeness of order δ of the function Bw. rB∗δ is the absolute value of the root of the equation wB′w−δBw=0 having the smallest modulus and argument ϕ.In the following, we obtain the radiusrucBλδ of λ-uniform convexity of order δ for Bw.Theorem 6.
Letcnn∈ℕ/0 be a sequence with cn=cneiϕ∈ℂ,cn≥1 for n∈ℕ/0, r0=infcn:n∈ℕ/0, and let hw be an analytic function in Ur0 with weiϕh′weiϕ∈ℝ, weiϕh′weiϕ≤Rwh′w, and w2h″w≤−w2ei2ϕh″weiϕ for w∈Ur0. If the function μ:0,r0⟶ℝ defined by μr=reiϕh′reiϕ is decreasing, the function ϑr:0,r0⟶ℝ defined by ϑr=−r2e2iϕh″reiϕ is increasing with respect to r and Bw is of the form (9) with qn∈ℕ/0 for n∈ℕ/0; then, λ-uniform convexity of order δ of the function Bw is rucBλδ, the absolute value of the root of the equation 1+λwB″w+1−2λ−δB′w=0 having the smallest modulus and argument ϕ.Proof.
From (9),
(19)1+wB″wB′w=2+wh′w−∑n=1∞w/cnqn+11−w/cn−1−w2h″w+∑n=1∞qnw/cnqn+1−qn−1w/cnqn+2/1−w/cn21+wh′w−∑n=1∞w/cnqn+1/1−w/cn(20)=1+wB′wBw−1−w2h″w+∑n=1∞qnw/cnqn+1−qn−1w/cnqn+2/1−w/cn2wB′w/Bw.Using (14) and the inequality of Lemma 1, we have
(21)R1−w2h″w+∑n=1∞qnw/cnqn+1−qn−1w/cnqn+2/1−w/cn2wB′w/Bw≤1−w2h″w+∑n=1∞qnw/cnqn+1−qn−1w/cnqn+2/1−w/cn2wB′w/Bw≤1+w2h″w+∑n=1∞w/cnqn+1qn−qn−1w/cn/1−w/cn2RwB′w/Bw≤1−w2ei2ϕh″weiϕ+∑n=1∞qnw/cnqn+1−qn−1w/cnqn+2/1−w/cn2weiϕB′weiϕ/Bweiϕ.From (14), (20), and (21),
(22)R1+wB″w/B′w≥1+RwB′w/Bw−R1−w2h″w+∑n=1∞qnw/cnqn+1−qn−1w/cnqn+2/1−w/cn2wB′w/Bw≥1+weiϕB′weiϕBweiϕ−1−w2ei2ϕh″weiϕ+∑n=1∞qnw/cnqn+1−qn−1w/cnqn+2/1−w/cn2weiϕB′weiϕ/Bweiϕ=1+weiϕB″weiϕB′weiϕ.Also, we have(23)wB″wB′w≤wB′wBw−1−w2h″w+∑n=1∞qnw/cnqn+1−qn−1w/cnqn+2/1−w/cn21+wh′w−∑n=1∞w/cnqn+1/1−w/cn≤wB′wBw+1−w2h″w+∑n=1∞qnw/cnqn+1−qn−1w/cnqn+2/1−w/cn21+wh′w−∑n=1∞w/cnqn+1/1−w/cn≤2−weiϕB′weiϕBweiϕ+1−w2ei2ϕh″weiϕ+∑n=1∞qnw/cnqn+1−qn−1w/cnqn+2/1−w/cn21+weiϕh′weiϕ−∑n=1∞w/cnqn+1/1−w/cn≤2−weiϕB′weiϕBweiϕ+1−w2ei2ϕh″weiϕ+∑n=1∞qnw/cnqn+1−qn−1w/cnqn+2/1−w/cn2weiϕB′weiϕ/Bweiϕ=2−weiϕB″weiϕB′weiϕ.From (22) and (23),
(24)R1+wB″wB′w−λwB″wB′w−δ≥1+weiϕB″weiϕB′weiϕ−λ2−weiϕB″weiϕB′weiϕ−δ=1+λweiϕB″weiϕB′weiϕ+1−δ−2λ,δ∈0,1,0≤λ<1−δ2.By the virtue of minimum principle for harmonic functions,(25)infw<rR1+wB″wB′w−λwB″wB′w−δ=1+λreiϕB″reiϕB′reiϕ+1−δ−2λ,where r∈0,r0. The function ψ:0,r0⟶ℝ, defined by ψr=1+λreiϕB″reiϕ/B′reiϕ+1−δ−2λ is strictly decreasing; also, observe that limr⟶0ψr=1−δ−2λ>0, limr⟶r0ψr=−∞. Thus, it follows that the equation 1+λeiϕrB″eiϕr+1−2λ−δB′eiϕr=0 has a unique root situated in 0,r0, and this root is rucBλδ.Remark 7.
Asδ∈0,1 and 0≤λ<1−δ/2≤1/2,we have λ∈0,1/2, which means that if B∈λ−UCVδ, then the image of Ur under the function contained in the right half plane for which 1+wB″w/B′w is in hyperbolic domain Ωλδ contained in the right half plane for which 1∈Ωλδ and ∂Ωλδ is the curve defined by equation (4).By specializing the parameters in Theorem6, we haveRemark 8.
Substitutingδ=0 and λ∈0,1/2 in Theorem 6, we get the radius rucBλ of λ-uniform convexity given by the absolute value of the root of the equation 1+λwB″w+1−2λB′w=0 having the smallest modulus and argument ϕ.Remark 9 (see [17]).
Takingλ=0,0≤δ<1 in Theorem 6, we get the radius rBcδ of convexity of order δ given by the absolute value of the root of the equation wB″w+1−δB′w=0, having the smallest modulus and argument ϕ.Theorem 10.
Letcnn∈ℕ/0 be a sequence with cn=cneiϕ∈ℂ,cn≥1 for n∈ℕ/0,r0=infcn:n∈ℕ/0, and let hw be an analytic function in Ur0 with weiϕh′weiϕ∈ℝ, weiϕh′weiϕ≤Rwh′w, and w2h″w≤−w2ei2ϕh″weiϕ for w∈Ur0. If the function μ:0,r0⟶ℝ defined by μr=reiϕh′reiϕ is decreasing, the function ϑr:0,r0⟶ℝ defined by ϑr=−r2e2iϕh″reiϕ is increasing with respect to r, and Bw is of the form (9) with qn∈ℕ/0 for n∈ℕ/0 and α∈0,1; then, the radius of α-convexity of order δ of the function Bw is the smallest positive root of the equation 1−αwB′w/Bw+α1+wB″w/B′w=δ having the smallest modulus and argument ϕ.Proof.
Consider(26)RMα,Bw=R1−αwB′wBw+α1+wB″wB′w=1+wh′w−∑n=1∞w/cnqn+11−w/cn+α1−1−w2h″w+∑n=1∞qnw/cnqn+1−qn−1w/cnqn+2/1−w/cn21+wh′w−∑n=1∞w/cnqn+1/1−w/cn≥1+weiϕh′weiϕ−∑n=1∞w/cnqn+11−w/cn+α1−1−w2ei2ϕh″weiϕ+∑n=1∞qnw/cnqn+1−qn−1w/cnqn+2/1−w/cn21+weiϕh′weiϕ−∑n=1∞w/cnqn+1/1−w/cn≥1−αweiϕB′weiϕBweiϕ+α1+weiϕB″weiϕB′weiϕ=Mα,eiϕBw,for everyw<r0, and the equality holds for w=weiϕ. By the virtue of minimum principle for harmonic functions,
(27)infw<rRMα,Bw=Mα,reiϕ,r∈0,r0,Also,Mα,reiϕ is strictly decreasing; also, limr⟶0Mα,reiϕ=1>0 and limr⟶r0Mα,reiϕ=−∞. Hence, the equation 1−αreiϕB′reiϕ/Breiϕ+α1+reiϕB″reiϕ/B′reiϕ=δ has a unique root in 0,r0, and this root is rcBαδ.Remark 11 (see [17]).
Takingα=0 in Theorem 10, we get the radius rB∗δ of starlikeness of order δ, given by the absolute value of the root of the equation wB′w−δBw=0, having the smallest modulus and argument ϕ.Remark 12 (see [17]).
Takingα=1, in Theorem 6, we get the radius rBcδ of convexity of order δ, given by the absolute value of the root of the equation wB″w+1−δB′w=0, having the smallest modulus and argument ϕ.In the following remark, we discuss the radius ofλ-starlikeness, λ-uniform convexity, and α-convexity of order δ for the function 1/Γ.Remark 13.
Lethw=γw where γ is the Euler-Mascheroni constant [5], and let qn=1,cn=−n,n∈ℕ, and ϕ=0. Then,
(28)Bw=1Γw=weγw∏n=1∞1+wne−w/n.We now havewh′w=γw, w2h″w=γw2, and it is easy to verify Rwh′w≥γw,w∈U with equality iff w∈ℝ and w2h″w=γw2, w∈U. The conditions of Theorem 2, Theorem 6, and Theorem 10 are satisfied.By Theorem2, the radius rst1/Γλδ of λ-starlikeness of order δ of the function 1/Γw is the modulus of the biggest negative root of the equation wΓ′w/Γw+λ+δ/1+λ=0. Numerical approach gives rst1/Γ00=0.504083, rst1/Γ1/20=0.416321, rst1/Γ01/2=0.358071, rst1/Γ11/4=0.30431, and rst1/Γ21/2=0.180823.By Theorem6, the radius ruc1/Γλδ of λ-uniform convexity of order δ of the function 1/Γw is the modulus of the biggest negative root of the equation
(29)wΓ″wΓ′w−2wΓ′wΓw+1−2λ−δ1+λ=0.Numerical approach givesruc1/Γ00=0.266701, ruc1/Γ01/2=0.190771, ruc1/Γ1/30=0.125966,ruc1/Γ1/41/4=0.108467, and ruc1/Γ1/51/3=0.116513.By Theorem10, the radius rc1/Γαδ of α-convexity of order δ for the function 1/Γw is the modulus of the biggest negative root of the equation
(30)α1+wΓ″wΓ′w+1+αwΓ′wΓw=δ.Numerical approach givesrc1/Γ00=0.504083, rc1/Γ10=0.266701, rc1/Γ1/31/2=0.258289, rc1/Γ1/21/3=0.269676, and rc1/Γ1/21/2=0.234978.In the following remark, we give an example which shows that the Theorems2, 6, and 10 work even if wehw is not starlike. That is, the example given in the following remark shows that the hypotheses of Theorems 2, 6, and 10 are free from the hypothesis of Theorem 3 in [13], proved by Merkes et al.Remark 14.
Lethw=w2/w2−1 with ϕ=0, and let qn=1,cn=n. Clearly, wehw is not starlike. Then, we have wh′w=−2w2/w2−12 and w2h″w=2w2+6w4/w2−13. Also, Rwh′w≥−2w2/w2−12 and w2h″w≥2w2+6w4/1−w23, w∈U with equality iff w∈ℝ.By Theorem2, the radius rstBλδ of λ-starlikeness of order δ of the function
(31)B1w=wew2/w2−1∏n=1∞1−wnew/nis the smallest positive root of the equation 1−2r2/1−r22−∑n=1∞r2/nn−r−λ+δ/1+λ=0. Numerical approach gives rstB100=rB1∗=0.426948, rstB101/2=rB1∗1/2=0.325887, rstB111/2=0.241843, rstB121/2=0.201282, and rstB11/21/2=0.274465.By Theorem6, the radius rucB1λδ of λ-uniform convexity of order δ is the smallest positive root of the equation
(32)1−2r21−r22−∑n=1∞r2nn−r−1+2r2+6r4/1−r23+∑n=1∞r2/n−r21−2r2/1−r22−∑n=1∞r2/nn−r+1−δ−2λ1+λ=0.Numerical approach givesrucB100=rB1c=0.242015, rucB101/2=rB1c1/2=0.187093, rucB11/61/2=0.108455, and rucB11/41/4=0.126439.By Theorem10, the radius rcB1αδ of α-convexity of order δ is the smallest positive root of the equation
(33)1−2r21−r22−∑n=1∞r2nn−r+α1−1+2r2+6r4/1−r23+∑n=1∞r2/n−r21−2r2/1−r22−∑n=1∞r2/nn−r=δ.Numerical approach givesrcB100=rB1∗=0.426948, rcB110=rB1c=0.242015, rcB101/2=rB1∗1/2=0.325887, rcB11/2=rB1c1/2=0.187093, rcB11/21/2=0.222952, rcB11/41/4=0.139653, and rcB11/41/2=0.254242.



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