# On Janowski Type Harmonic Meromorphic Functions with respect to Symmetric Point

Authors: Muhammad Ghaffar Khan; Bakhtiar Ahmad; Maslina Darus; Wali Khan Mashwani; Shahid Khan
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we define a new class of Sakaguchi type-meromorphic harmonic functions in the Janowski domain that are starlike with respect to symmetric point. Furthermore, we investigate some important geometric properties like sufficiency criteria, distortion bound, extreme point theorem, convex combination, and weighted means.


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## Body

## 1. Introduction and Definitions

One of the contemporary developments in Mathematics is the solicitations of harmonic analysis in other fields. Like various other fields, it has immensely influenced and nurtured the branch of geometric function theory. Jahangiri et al. [1] defined and studied a subclass of harmonic and univalent functions. Another example of such work would be an article of Porwal and Dixit [2], who used a certain convolution operator involving hypergeometric functions to define a class of univalent functions. As a consequence, many mathematicians generalized many ideas of this field and various important results with the help of some operators; the work of Porwal et al. [3], Porwal et al. [4], and Porwal and Dixit [5] are worth mentioning here. Recently, some subclasses of harmonic functions were investigated by Arif et al. [6] and Khan et al. [7]. To start with, we give preliminaries which will be useful in understanding the concepts of this research.A real-valued functionux,y is said to be harmonic in a domain D⊂ℂ if it has a continuous second partial derivative and satisfy the Laplace’s equation
(1)∂2u∂x2+∂2u∂y2=0.A continuous complex-valued functionf=u+iv is said to be harmonic in a complex domain U if both its real and imaginary parts are real harmonic in U. In any simply connected domain U⊂ℂ, one can write f=h+g¯, where h and g are analytic in U. The class of such functions is denoted by H. The condition h′z>g′z is necessary and sufficient for f to be locally univalent and sense preserving in U, see [8]. There are different papers on univalent harmonic functions defined in unit disc D=z:z<1, for details, see [9–14]. For z∈D∗=D\0, in the punctured open unit disc and let MH denote the class of functions
(2)fz=hz+gz¯=1z+∑n=1∞anzn+∑n=1∞bnzn¯,which are harmonic in D∗ where h is analytic in D∗ and has a simple pole at the origin with residue 1, while g is analytic in D. The class MH was studied in [15–17]. Furthermore, denoted by MH¯, a subclass of MH, consisting a functions of the form
(3)fz=hz+gz¯=1z+∑n=1∞anzn−∑n=1∞bnzn¯,which are harmonic univalent in punctured unit disc D∗.For functionsf∈MH given by (2) and F∈MH given by
(4)Fz=Hz+Gz¯=1z+∑n=1∞Anzn+∑n=1∞Bnzn¯,we recall the Hadamard product (or convolution) of f and F by
(5)f∗Fz=1z+∑n=1∞anAnzn+∑n=1∞bnBnzn¯z∈D∗.In terms of the Hadamard product (or convolution), we chooseF as a fixed function in H such that f∗Fz exists for any f∈H, and for various choices of F, we get different linear operators which have been studied in the recent past.Recently, Khan et al. [18] introduced and studied a class of meromorphic starlike functions with respect to symmetric point in circular domain i.e.,
(6)−2zf′zfz−f−z≺1+Az1+Bz.Motivated from the above discussion on harmonic functions and class of meromorphic starlike functions with respect to symmetric point, we introduced the class of meromorphic harmonic univalent functions as:Let−1≤B<A≤1. Then, the function f∈MH is in the class MH∗∗A,B if it satisfies the condition
(7)−2DHfzfz−f−z≺1+Az1+Bz,z∈D,where the symbol "≺" represent well-known subordination and
(8)DHfz=zh′z−zg′z¯,or equivalently(9)DHfz+fz−f−z/2BDHfz+Afz−f−z/2<1,z∈D.Furthermore, we denoteMH¯∗∗A,B subclass of MH∗∗A,B consisting of harmonic meromorphic functions f=h+g¯ of the form (3).


## 2. Main Results

Theorem 1.
Letf=h+g¯ be of the form (2) and satisfies the condition
(10)∑n=1∞αnan+βnbn≤1,with
(11)αn=1+Bn+1+A1−−1n/2A−Bandβn=1+Bn−1+A1−−1n/2A−B,then f is harmonic univalent sense-preserving in D∗ and f∈MH∗∗A,B.Proof.
For0<z1≤z2<1, we obtain
(12)fz1−fz2hz1−hz2≥1−gz1−gz2hz1−hz2=1−z1z2∑n=1∞bnz1n−z2nz1−z2−z1z2∑n=1∞anz1n−z2n>1−∑n=1∞nbn1−∑n=1∞nan>1−∑n=1∞βnbn1−∑n=1∞αnan≥0,where we have used (10) and this shows that the function is univalent.Now to showfz is sense-preserving harmonic mapping in D∗, consider
(13)h′z≥1z2−∑n=1∞nanzn−1≥1−∑n=1∞αnan≥∑n=1∞βnbn≥∑n=1∞nbn≥g′z.This shows that f is sense-preserving.Now, to show thatf∈MH∗∗A,B from (9), it is enough to show that
(14)DHfz+fz−f−z2−BDHfz+Afz−f−z2<0.For this, consider(15)DHfz+fz−f−z2−BDHfz+Afz−f−z2=∑n=1∞n+1−−1n2anzn+n+1−−1n2bnzn¯−A−Bz+∑n=1∞Bn+A1−−1n2anzn−Bn+A1−−1n2bnzn¯≤∑n=1∞n+1−−1n2anzn+n+1−−1n2bnzn−A−Bz∑n=1∞Bn+A1−−1n2anzn+Bn+A1−−1n2bnzn≤∑n=1∞1+Bn+1+A1−−1n2anzn+1+Bn+1+A1−−1n2bnzn−A−Bz=A−Bz∑n=1∞1+Bn+1+A1−−1n/2A−Banzn+1+1+Bn+1+A1−−1n/2A−Bbnzn+1−A−Bz<A−Bz∑n=1∞αnan+βnbn−1<0.Hence, complete the proof.Example 2.
The meromorphic univalent function(16)fz=1z+∑n=1∞xnαnzn+∑n=1∞ynβnz¯n,such that ∑n=1∞xn+yn=1, we have
(17)∑n=1∞αnan+βnbn=∑n=1∞xn+yn=1.Thus,f∈MH∗∗A,B and above coefficient bound given in (10) is sharp for this function.Theorem 3.
Letf=h+g¯∈MH¯ and of the form (3), then f∈MH¯∗∗A,Bif it satisfies the condition(18)∑n=1∞αnan+βnbn≤1,with
(19)αn=1+Bn+1+A1−−1n/2A−Bandβn=1+Bn−1+A1−−1n/2A−B.Proof.
The proof is similar to Theorem1, so omitted.Theorem 4.
Letf=h+g¯∈MH¯∗∗A,B and of the form (3), 0<z=r<1. Then,
(20)1r−A−B2+A+Br≤fz≤1r+A−B2+A+Br.Proof.
Consider(21)fz=1z+∑n=1∞anzn+bnzn¯≤1r+∑n=1∞an+bnrn≤1r+r∑n=1∞an+bn≤1r+A−B2+A+Br.Similarly, proceeding as above we get(22)fz=1z+∑n=1∞anzn+bnzn¯≥1r−r∑n=1∞an+bn=1r−A−B2+A+Br.Hence, this completes the result.Theorem 5.
Letf=h+g¯, and of the form (3) then f∈MH¯∗∗A,B if and only if
(23)fz=∑n=0∞γnhn+δngn,where
(24)h0z=1z,hnz=1z+A−B1+Bn+1+A1−−1n/2zn,n=1,2,⋯,g0z=1z,gnz=1z−A−B1+Bn+1+A1−−1n/2zn¯,n=1,2,⋯,where 1≥γn≥0,1≥δn≥0, and ∑n=0∞γn+δn=1.Proof.
Let(25)fz=γ0h0+δ0g0+∑n=1∞γnhn+δngn=γ0+δ01z+∑n=1∞γn1z+A−B1+Bn+1+A1−−1n/2zn+∑n=1∞δn1z−A−B1+Bn+1+A1−−1n/2zn¯,=∑n=1∞γn+δn1z+∑n=1∞γnA−B1+Bn+1+A1−−1n/2zn−∑n=1∞δnA−B1+Bn+1+A1−−1n/2zn¯.Thus,(26)∑n=1∞αnA−B1+Bn+1+A1−−1n/2γn+βnA−B1+Bn+1+A1−−1n/2δn=∑n=1∞γn+δn=1−γ0−δ0≤1,hence, f∈MH¯∗∗A,B. Conversely, let f∈MH¯∗∗A,B. Set
(27)γn=1+Bn+1+A1−−1n/2A−Ban,0≤γn≤1,δn=1+Bn+1+A1−−1n/2A−Bbn,0≤δn≤1,γ0=1−∑n=1∞γn−∑n=1∞δn.Therefore,f can be written as
(28)fz=1z+∑n=1∞anzn−∑n=1∞bnzn¯=1z+∑n=1∞A−B1+Bn+1+A1−−1n/2γnanzn−∑n=1∞A−B1+Bn+1+A1−−1n/2δnbnzn¯=1z+∑n=1∞A−B1+Bn+1+A1−−1n/2γnanzn−∑n=1∞A−B1+Bn+1+A1−−1n/2δnbnzn¯=γ0+δ01z+∑n=1∞γn1z+A−B1+Bn+1+A1−−1n/2zn+∑n=1∞δn1z−A−B1+Bn+1+A1−−1n/2zn¯=∑n=1∞γnhn+δngn,hencerequired.Theorem 6.
The classMH¯∗∗A,B is closed under convex combination.Proof.
Fork∈ℕ, let fk∈MH¯∗∗A,B, be of the form
(29)fkz=1z+∑n=1∞ak,nzn−bk,nzn¯.Then form (10), we get
(30)∑n=1∞αnak,n+βnbk,n≤1.For∑k=1∞δk=1,0≤δk≤1, the convex combination of fk is
(31)∑k=1∞δkfkz=1z+∑n=1∞∑k=1∞δkak,nzn−∑n=1∞∑k=1∞δkbk,nzn¯.Using (10), we have
(32)∑n=1∞αn∑k=1∞δkak,n+βn∑k=1∞δkbk,n=∑k=1∞δk∑n=1∞αnak,n+βnbk,n≤∑k=1∞δk=1,thus prove our desired results.Theorem 7.
Letfk∈MH¯∗∗A,B, for k=1,2 be of the form (29), then, their weighted Fi mean is also in the class MH¯∗∗A,B, where Fi is defined below
(33)Fiz=1−if1z+1+if2z2.Proof.
From (33), one may easily write
(34)Fiz=1z+1−if1z+1+if2z2=1z+∑k=1∞1−jan,1+1+jan,22zn−1−jbn,1+1+jbn,22zn¯.To show thatFi∈MH¯∗∗A,B, it is enough to show that
(35)∑k=1∞1−jan,1+1+jan,22αn+1−jbn,1+1+jbn,22βn≤1.Now consider(36)∑k=1∞1−jan,1+1+jan,22αn+1−jbn,1+1+jbn,22βn=∑k=1∞1−jan,1αn+1−jbn,1βn2+1+jan,2αn+1+jbn,2βn2=1−j2∑k=1∞an,1αn+bn,1βn+1+j2∑k=1∞an,2αn+bn,2βn≤1−j2+1+j2=1.Hence,Fi∈MH¯∗∗A,B.



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