# Close-to-Convexity of Convolutions of Classes of Harmonic Functions

Authors: Raj Kumar Garg; Jay M. Jahangiri
Journal: International Journal of Mathematics and Mathematical Sciences
 (2018)

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## Abstract

Forj=1,2 and for positive integers m and n, we consider classes of harmonic functions fj=hj+gj¯, where g1(z)=znh1(z) and g2′(z)=znh2′(z) or g1′(z)=znh1′(z) and g2′(z)=zmh2′(z), and we prove that their convolution f1⁎f2=h1⁎h2+g1⁎g2¯ is locally one-to-one, sense-preserving, and close-to-convex harmonic in z<1.


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## Body

## 1. Introduction

LetA denote the class of functions that are analytic in the open unit disk E≔{z:z<1} and let A′ be the subclass of A consisting of functions h with the normalization h(0)=h′(0)-1=0. Consider the family of complex-valued harmonic functions f=u+iv, where u and v are real harmonic in E. Such functions can be expressed as f=h+g¯, where h∈A and g∈A. By Lewy’s Theorem (see [1, 2] or [3]), a necessary and sufficient condition for the harmonic function f=h+g¯ to be locally one-to-one and sense-preserving in E is that its Jacobian Jf=|h′|2-|g′|2 should be positive or equivalently if and only if h′≠0 in E and the second complex dilatation ω of f satisfies ω=|g′/h′|<1 in E. In the sequel, without loss of generality, we consider those locally one-to-one and sense-preserving harmonic functions f=h+g¯ that are normalized by f(0)=h(0)=0 and fz(0)=1 and have the representation(1)fz=hz+gz¯=z+∑n=2∞anzn+∑n=1∞bnzn¯;z∈E.The Hadamard product or convolution of two power series h1(z)=∑n=1∞anzn and h2(z)=∑n=1∞cnzn is given by h1(z)∗h2(z)=(h1∗h2)(z)=∑n=1∞ancnzn. Similarly, the convolution of two harmonic functions f1=h1+g1¯ and f2=h2+g2¯ is given by f1∗f2=h1∗h2+g1∗g2¯.A simply connected proper subdomainD of the complex domain C is said to be convex if the linear segment joining any two points of D lies entirely in D and is said to be close-to-convex if its complement in C is the union of closed half-lines with pairwise disjoint interiors. Consequently, a univalent analytic or harmonic function f:E→C is said to be convex or close-to-convex in E if f(E) is convex or close-to-convex there. For -1/2≤α<1, a function h∈A′ is said to be in the class K(α) if Re[1+zh′′(z)/h′(z)]>α, z∈E. It can easily be verified that K(β)⊂K(α) if -1/2≤α<β<1. If h∈(α) for 0≤α<1 in E, then h is said to be convex of order α in E (e.g., see [3] or [4]). A function h∈K(0) is simply called a convex function in E.Ruscheweyh and Sheil-Small [5] proved that the Hadamard product or convolution of two analytic convex functions is also convex analytic and that the convolution of an analytic convex function and an analytic close-to-convex function is close-to-convex analytic in the unit disk E. Ironically, these results can not be extended to the harmonic case, since the convolution of harmonic functions, unlike the analytic case, proved to be very challenging.Recently, Ahuja and Jahangiri [6] proved the following theorem.Theorem 1.
Let the functionsh1∈A′ and h2∈A′ be in the class K(α) in E. If either g1=zh1, g2′=zh2′, α≥-1/2 or g1′=znh1′, g2′=znh2′, α≥0, then F=h1∗h2+g1∗g2¯ is locally one-to-one, sense-preserving, and close-to-convex harmonic in E.The following question is asked in [6].Question 2.
Is Theorem1 true for g1=znh1 and g2′=znh2′ if n>1?In Theorem3, we address Question 2. Moreover, in Theorem 4, we allow variations in the powers of z for the dilatations of harmonic functions. Also note that the techniques presented here to prove our theorems are different from those used in [6].Theorem 3.
Let the functionsh1∈A′ and h2∈A′ be so that h1∗h2 is convex in E. Set g1(z)=znh1(z) and g2′(z)=znh2′(z), where n∈N≔{1,2,3,…}. Then the convolution function F(z)=h1(z)∗h2(z)+g1(z)∗g2(z)¯ is locally one-to-one, sense-preserving, and close-to-convex harmonic in E.Theorem 4.
Let the functionsh1∈A′ and h2∈A′ be convex of order 1/2 in E. Set g1′(z)=znh1′(z), n∈N≔{1,2,3,…}, and g2′(z)=zmh2′(z), m∈N. Then the convolution function F(z)=h1(z)∗h2(z)+g1(z)∗g2(z)¯ is locally one-to-one, sense-preserving, and close-to-convex harmonic in E.In the following example, we demonstrate a case of close-to-convexity of convolutions of two harmonic functions.Example 5.
Consider(2)f1z=h1z+g1z¯=z1-z+z1-z+log⁡1-z¯,f2z=h2z+g2z¯=arctan⁡z+12log⁡1+z2¯.For j=1,2, it is easy to verify that gj′(z)=zhj′(z) and(3)Fz=Hz+Gz¯=f1z∗f2z=h1z∗h2z+g1z∗g2z¯=arctan⁡z+12log⁡1+z2-12∫0z1tlog⁡1+t2¯dt=arctan⁡z+12∑n=1∞-1n-12n-12n2z2n¯.Also,(4)G′zH′z=z-1+z2log⁡1+z22z=z21-z22+z46-z612+z820+⋯≤z21+z22+z46+z612+z820+⋯<121+12+16+112+120+⋯<122=1.Therefore, f1∗f2 is locally one-to-one, sense-preserving, and close-to-convex harmonic in E.
The images ofE under f1, f2, and f1∗f2 are shown in Figures 1, 2, and 3, respectively.Figure 1
Image ofE under f1.Figure 2
Image ofE under f2.Figure 3
Image ofE under f1∗f2.


## 2. Preliminary Lemmas and Proofs

To prove our theorems, we shall need the following four lemmas. Lemmas6 and 9 are according to Clunie and Sheil-Small [2], Lemma 7 is a well-established result by Robinson [7], and Lemma 8 is a celebrated result by Ruscheweyh and Sheil-Small [5].Lemma 6.
Letg and h be analytic in E so that |g′(0)|<|h′(0)|. If h+ϵg is close-to-convex analytic in E for each ϵ;(|ϵ|=1), then f=h+g¯ is close-to-convex harmonic in E.Lemma 7.
IfP(z) and Q(z) are analytic in E, Q′z≤P′z for all z∈E, Q(0)=P(0)=0, and if P(z) maps E onto a region which is starlike with respect to origin, then Qz≤Pz for all z∈E.A functionϕ analytic in E is convex of order α;0≤α<1 in E if and only if zϕ′(z) is starlike of order α;0≤α<1 in E (e.g., see Duren [4]).Lemma 8.
Letϕ and ψ be analytic and starlike of order 1/2 in E. Then for each function F analytic in E, the convolution (ϕ∗Fψ)/(ϕ∗ψ) takes only values in the convex hull of F(E).Lemma 9.
Ifh and g are analytic in E, h is convex in E and if f=h+g¯ is locally univalent in E, then the function f=h+g¯ is locally one-to-one, sense-preserving, and close-to-convex harmonic in E.Proof of Theorem3.
For the convolution function(5)Fz=h1z∗h2z+g1z∗g2z¯=Hz+Gz¯we note that G′0=0<H′0. To satisfy the condition of Lemma 6, we will show that H(z)+ϵG(z) is close-to-convex analytic in E for each ϵ;(|ϵ|=1). To do so, it suffices to show that there exists a function ϕ analytic and convex in E so that(6)ReHz+ϵGz′ϕ′z>0;z<1.We observe that(7)Hz+ϵGz′=H′z+ϵG′z=h1z∗h2z′+ϵg1z∗g2z′=1zh1z∗h2′z+ϵ1zg1z∗g2′z=1zh1z∗h2′z+ϵ1zznh1z∗znh2′z=1zh1z∗h2′z+ϵzn1zh1z∗h2′z=1+ϵzn1zh1z∗h2′z. Now letting ϕ(z)=h1(z)∗h2(z) yields(8)ReHz+ϵGz′ϕ′z=Re1+ϵzn1/zh1z∗h2′zϕ′z=Re1+ϵzn1/zh1z∗h2′z1/zh1z∗h2′z=Re1+ϵzn>0.Thus by Lemma 6, F(z)=h1(z)∗h2(z)+g1(z)∗g2(z)¯ is locally one-to-one, sense-preserving, and close-to-convex harmonic in E.Proof of Theorem4.
We need to show that the convolution function(9)Fz=h1z∗h2z+g1z∗g2z¯=Hz+Gz¯is locally univalent and sense-preserving in E.
Without loss of generality, we consider only the two cases ofm=n and m>n.
Case 1. If m=n, then we have to show that H′z>G′z in E. We observe that(10)h1′z∗h2′z>znh1′z∗h2′z=g1′z∗g2′z.Inequality (10) is equivalent to(11)h1z∗zh2′z′>g1z∗zg2′z′or(12)z1zh1z∗h2′z′>z1zg1z∗g2′z′or(13)zh1z∗h2z′′>zg1z∗g2z′′.Since h1(z)∗h2(z) is convex in E, z[h1(z)∗h2(z)]′ is starlike in E. Therefore, by Lemma 7, inequality (13) yields(14)zh1z∗h2z′>zg1z∗g2z′or(15)h1z∗h2z′>g1z∗g2z′or(16)H′z>G′z.
Case 2. If m>n, then we have(17)h1′z∗h2′z>znh1′z∗h2′z=znh1′z∗znh2′z=g1′z∗zn-mg2′z=g1′z∗g2′z×g1′z∗zn-mg2′zg1′z∗g2′z.We note that(18)g1′z∗zn-mg2′zg1′z∗g2′z=znh1′z∗zn-mzmh2′zznh1′z∗zmh2′z=znh1′z∗znh2′zznh1′z∗zmh2′z=zn-1zh1′z∗zh2′zzn-1zh1′z∗zm-nzh2′z=zh1′z∗zh2′zzh1′z∗zm-nzh2′z.Letting ϕ(z)=zh1′(z), ψ(z)=zh2′(z), and F(z)=zm-n in Lemma 8 yields(19)zh1′z∗zh2′zzh1′z∗zm-nzh2′z>1.Therefore(20)h1′z∗h2′z>g1′z∗g2′z×g1′z∗zn-mg2′zg1′z∗g2′z>g1′z∗g2′z.This is exactly inequality (10). So a similar argument following inequality (10) will lead to the conclusion that(21)H′z>G′z.Therefore, for either of cases m=n or m>n,(22)Fz=h1z∗h2z+g1z∗g2z¯=Hz+Gz¯is locally univalent and sense-preserving in E. Thus, by Lemma 9,(23)Fz=h1z∗h2z+g1z∗g2z¯is locally one-to-one, sense-preserving, and close-to-convex harmonic in E.



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*Source: 3808513-2018-05-02.xml*