# Sufficiency Criteria forq-Starlike Functions Associated with Cardioid

Authors: Saira Zainab; Ayesha Shakeel; Muhammad Imran; Nazeer Muhammad; Hira Naz; Sarfraz Nawaz Malik; Muhammad Arif
Journal: Journal of Function Spaces
 (2021)

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## Abstract

This article deals with theq-differential subordinations for starlike functions associated with the lemniscate of Bernoulli and cardioid domain. The primary goal of this work is to find the conditions on γ for 1+γz∂qhz/hnz≺1+z, where hz is analytic function and is subordinated by the function which is producing cardioid domain as its image domain while mapping the open unit disk. Along with this, certain sufficient conditions for q-starlikeness of analytic functions are determined.


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## Body

## 1. Introduction

Consider the classA of analytic functions defined in open unit disk F with normalization condition f0=0 and f′0=1 which provides the Taylor series expansion of the form(1)fz=z+∑n=2∞anzn,z∈F.The classS consists of functions from A which are univalent functions in F, and the class P contains the analytic functions whose codomains are bounded by the open right half plane. For more details, see [1, 2].The concept of differential subordination plays a vital role in the study of geometric properties of analytic functions. It was first introduced by Lindelof, but Littlewood [3] did the remarkable work in this field. Many researchers contributed in the study of differential subordinations. History and the development of works in the field related to differential subordination are briefly described and included in the book by Miller and Mocanu [4]. The major development in the field of differential subordination started in 1974 by Miller et al. [5].An analytic functionf is considered to be subordinated by analytic function g, denoted as f≺g, if there exists another analytic function w with the property that w0=0 and wz<z such that fz=gwz. Moreover, in case of univalent functions in F, we can have(2)f≺g⇔f0=g0,fF⊂gF.Recently, many mathematicians have used this concept of differential subordinations to prove many helpful results. Familiar Jack’s lemma [6] has produced several advancements for the generalization of differential subordinations and found many applications in this field. The work of Ma and Minda [7] in this field is not negligible as they studied the function Φ which is analytic, and condition of normalization given for prescribed function is defined as Φ0=1 and Φ′0>0 with a positive real part. With the help of the function Φ, they introduced the following subclasses for starlike and convex functions.(3)S∗Φ=f∈A:zf′zfz≺Φz;z∈F,CΦ=f∈A:1+zf′′zf′z≺Φz;z∈F.These subclasses helped many researchers for further studies in the field of differential subordination. Ali et al. [8] used the concept of differential subordination to prove analytic functions to be Janowski starlike. Ali et al. [9] also evaluated several differential subordinations: 1+γzp′z/pnz and found the γ for pz≺1+z. Raina and Sokol [10] used subordinations for coefficient estimation of starlike functions. Similar kinds of works have also been done by Sharma et al. [11] by using starlikeness for cardioid function, and Yunus et al. [12] studied for limacon.Quantum calculus is the new branch of mathematics and is equally important for its applications both in physics and in mathematics as well. Jackson [13, 14] presented the functions of q-derivatives and q-integrals and highlighted their definitions for the first time. He also holds the credit for the systematic initiation of q-calculus. Ismail et al. [15] were the pioneers to contribute in the application of q-calculus in geometric function theory. The new form of the subclass of starlike functions S∗Φ with the involvement of q-derivative was introduced by Seoudy and Aouf [16]. By choosing different image domains instead of Φz, so many attractive subclasses of starlike functions are obtained. Mahmood et al. [17] have dealt with the class of q-starlike functions by relating them with conic domains. The most recent work related to q-starlikeness of functions is done by Srivastava et al. [18]. The contributions of Haq et al. [19] are remarkable. They proved differential subordinations with q-analogue for cardioid and limacon domain with the involvement of Janowski function and found the sufficient conditions for q-starlike functions. The q version of Jack’s lemma which is the soul of our work was given by Çetinkaya and Polatoglu [20]. These recent efforts of mathematicians discussed above motivated us and provide strength to contribute in the field of differential subordinations with the involvement of its q-analogue, which is the main idea of this article. The foundation of all this work in q-analogue is the q-derivative which is defined below.Theq-derivative of a complex-valued function f, defined in the domain F, is given as follows:(4)Dqfz=fz−fqz1−qz,z≠0,f′0,z=0,where 0<q<1. This implies the following:(5)limq⟶1−Dqfz=limq⟶1−fz−fqz1−qz=f′z,provided the function f is differentiable in domain F. The function Dqf has Maclaurin’s series representation(6)Dqfz=∑n=0∞nqanzn−1,where(7)nq=1−qn1−q,n∈ℂ,∑k=0n−1qk=1+q+q2+⋯+qn−1,n∈ℕ.For more details aboutq-derivatives and recent work on it, we refer the reader to [21–25].Definition 1.
The functionfz∈A is said to be in the class Sq,c∗, if(8)z∂qfzfz≺1+43z+23z2,z∈F.Lemma 2 (q-Jack’s lemma, [20]).
Consider an analytic functionw in F with w0=0. For a maximum value of w on the circle z=1 at z0=aeiθ, where θ∈−π,π, and 0<q<1, then, we have(9)z0∂qwz0=mwz0.Here, m is real and m≥1.By using the above lemma, we have proved our main results.


## 2. Main Results

Theorem 3.
Assume that(10)γ≥32+121−q,and we define an analytic function h on F with h0=1 which satisfies(11)1+γz∂qhz≺1+z.
In addition, we suppose that(12)1+γz∂qhz=1+wz,where w is analytic in F with w0=0. Then,(13)hz≺1+43z+23z2.Proof.
Consider the function(14)pz=1+γz∂qhz,which is analytic in F with the condition p0=1 and the function(15)hz=1+43wz+23w2z,where w is an analytic function in F with w0=0. To prove the result, it would be sufficient to show that wz≤1 for(16)wz=p2z−1.From (14) and (15), we deduce the following:(17)pz=1+γ3z∂qwz41+wz−21−qz∂qwz,and with this, one can have(18)wz=p2z−1=γ3z∂qwz41+wz−21−qz∂qwz2+2γ3z∂qwz41+wz−21−qz∂qwz.This implies that(19)p2z−1=γ3z∂qwz41+wz−21−qz∂qwz2+2γ3z∂qwz41+wz−21−qz∂qwz=2+γ3z∂qwz41+wz−21−qz∂qwz×γ3z∂qwz41+wz−21−qz∂qwz.Now, considering the existence of a pointz0∈F such that(20)maxz≤zwz=wz0=1.Now, we use theq-Jack’s lemma which implies that there exist a number m≥1 such that z0∂qwz0=mwz0. This, with the consideration that wz0=eiθ,θ∈−π,π for z0∈F, we have(21)p2z0−1=2+γ3z0∂qwz041+wz0−21−qz0∂qwz0.γ3z0∂qwz041+wz0−21−qz0∂qwz0=2+γ3meiθ41+eiθ−2m1−qeiθ.γ3meiθ41+eiθ−2m1−qeiθ=4+49γ2m24+2−m1−q2+163γm1+γm32−m1−qcosθ+83γm2−m1−qcos2θ.mγ316+4−2m1−q2+84−2m1−qcosθ.The function(22)Gθ=4+49γ2m24+2−m1−q2+163γm1+γm32−m1−qcosθ+83γm2−m1−qcos2θ.mγ316+4−2m1−q2+84−2m1−qcosθis clearly an even function. So, in order to find the maximum value of G, we will consider the interval 0,π. Thus,(23)G′θ=−8γm9+4γ2m2+γ2m22−m1−q2+12γmcosθ+4γ2m22−m1−qcosθ+6γm2−m1−qcos2θ4−2m1−qsinθ916+4−2m1−q2+84−2m1−qcosθ+19γm16+4−2m1−q2+84−2m1−qcosθ−12γmsinθ−4γ2m22−m1−qsinθ−12γm2−m1−qsin2θ9+4γ2m2+γ2m22−m1−q2+12γmcosθ+4γ2m22−m1−qcosθ+6γm2−m1−qcos2θgives G′θ=0 for θ=0 and π. Also, we can see that G′′π>0 for 1<m<2.5, which results that Gθ≥Gπ. Now, consider the function(24)Θm=4+49γ2m24+2−m1−q2−163γm1+γm32−m1−q+83γm2−m1−qmγ316+4−2m1−q2−84−2m1−q=49γ2m41−q2−43γm21−q.So we have(25)Θ′m=163γ2m31−q2−83γm1−q>0.Thus,Θm is an increasing function which gives a minimum value for m=1. Then, we have(26)p2z0−1≥49γ21−q2−43γ1−q.From (10), we conclude that(27)p2z0−1≥1,but this result contradicts (11). Hence, wz<1 and this leads us to the desired result.By takinghz=z∂qfz/fz, the above result reduces to the following.Corollary 4.
Letγ≥32+1/21−q and f∈A satisfy the subordination(28)1+γz∂qz∂qfzfz≺1+z.
Then,fz∈Sq,c∗.Theorem 5.
Assume that(29)γ≥2+121−q,and we define an analytic function h on F with h0=1 which satisfies(30)1+γz∂qhzhz≺1+z.In addition, we suppose that(31)1+γz∂qhzhz=1+wz,where w is analytic in F with w0=0. Then,(32)hz≺1+43z+23z2.Proof.
Consider the function(33)pz=1+γz∂qhzhz,which is analytic in F with the condition p0=1 and the function(34)hz=1+43wz+23w2z,where w is an analytic function in F with w0=0. Using (33) and (34), we obtain(35)pz=1+γ/3z∂qwz41+wz−21−qz∂qwz1+4/3wz+2/3w2z.Proving the fact thatwz≤1 will be sufficient to prove our assertion. For this, consider(36)p2z−1=23+4wz+2w2z+γz∂qwz41+wz−21−qz∂qwz3+4wz+2w2z·γz∂qwz41+wz−21−qz∂qwz3+4wz+2w2z.Considering the existence of a pointz0∈F such that(37)maxz≤zwz=wz0=1,we can make use of q-Jack’s lemma which implies that there exists a number m≥1 such that z0∂qwz0=mwz0. Now, consider that wz0=eiθ,θ∈−π,π, then for z0∈F, we have(38)p2z0−1=23+4eiθ+2e2iθ+γmeiθ41+eiθ−2m1−qeiθ3+4eiθ+2e2iθ×γmeiθ41+eiθ−2m1−qeiθ3+4eiθ+2e2iθ.Now, one can easily see that the function(39)Gθ=Ψ129+40cosθ+29cos2θ·mγ16+4−2m1−q2+84−2m1−qcosθ29+40cosθ+29cos2θwith(40)Ψ1=68+48γm+8γm2+4γ2m4+32γ2m2−16γ2m3−8γm2q+16γ2m3q−8γ2m4q+4γ2m4q2+160cosθ+48γm2qcos2θ+32cosθγm2q+16cosθγ2m3q+96cos2θ−48γm2cos2θ+96γmcos2θ+144cosθγm−16cosθγ2m3+32cosθγ2m2−32cosθγm2is clearly an even function. So, in order to find the maximum value of G, we will consider the interval 0,π. Now, we have(41)G′θ=0for θ=0 and π. Also, we can see that G′′π>0 for m≥1, thus we conclude that Gθ≥Gπ. So we have the function(42)Θm=4−8γm2+8γm2q+4γ2m4−82m4q+4γ2m4q2·mγ2m1−q=2γm21−q−22γm21−q.This gives(43)Θ′m=16γ2m31−q2−8γm1−q>0.Thus,Θm is an increasing function which gives a minimum value for m=1. Then, we have(44)p2z0−1≥4γ21−q2−4γ1−q.From (29), we conclude that(45)p2z0−1≥1,but this result contradicts (30). Hence, wz<1 which provides the required result.By takinghz=z∂qfz/fz, the above result reduces to the following.Corollary 6.
Letγ≥2+1/21−q and f∈A satisfy the subordination(46)1+γzfzz∂qfz∂qz∂qfzfz≺1+z.
Then,fz∈Sq,c∗.Theorem 7.
Assume that(47)γ≥2+12.31−q,and we define an analytic function h on F with h0=1 which satisfies(48)1+γz∂qhzh2z≺1+z.In addition, we suppose that(49)1+γz∂qhzh2z=1+wz,where w is analytic in F with w0=0. Then,(50)hz≺1+43z+23z2.Proof.
Let us define the function(51)pz=1+γz∂qhzh2z,which is analytic in F with the condition p0=1 and the function(52)hz=1+43wz+23w2z,where w is an analytic function in F with w0=0. Using (51) and (52), we get(53)pz=1+γ/3z∂qwz41+wz−21−qz∂qwz1+4/3wz+2/3w2z2.To prove the assertion, it would be enough to show thatwz≤1. Therefore,(54)p2z−1=2+γ/3z∂qwz41+wz−21−qz∂qwz1+4/3wz+2/3w2z2·γ/3z∂qwz41+wz−21−qz∂qwz1+4/3wz+2/3w2z2,which after using (9) gives(55)p2z0−1=23+4wz0+2w2z02+3γz0∂qwz041+wz0−21−qz0∂qwz03+4wz0+2w2z02×3γz0∂qwz041+wz0−21−qz0∂qwz03+4wz0+2w2z02.Now, we consider the function(56)Gθ=Ψ229+40cosθ+12cos2θ·3γm16+4−2m1−q2−84−2m1−qcosθ29+40cosθ+12cos2θ,where(57)Ψ2=1156+1104γm−360γm2−960γm2cosθ+360γm2q+144γ2m3q−72γ2m4q+36γ2m4q2+288γ2m2−144γ2m3+36γ2m4+9664cos2θ+768γmcos3θ+2784γmcos2θ−624γm2cosθ+3120γmcosθ+288γ2m2cosθ−144γ2m3cosθ+2304cos4θ+7680cos3θ+5440cosθ+624γm2qcos2θ+960γm2qcosθ+144γ2m3qcosθ.As we see thatGθ is an even function, so G′θ=0 at θ=0, π and also we see that G′′π>0 for m≥1. Thus, we conclude that Gθ≥Gπ and we get a new function(58)Θm=4−24γm2+24γm2q−72γ2m4q+36γ2m4q2+36γ2m4·3γm2m1−q,and we have(59)Θ′m=72γ2m31−q2+126γm21−q−2γm1−q>0.SoΘm is an increasing function, and it has its minimum value at m=1. Then, we have(60)p2z0−1≥6γ1−q−26γ1−q.Using (47), we get(61)p2z0−1≥1,but this result contradicts (48). Hence, wz<1 which proves the required result.By takinghz=z∂qfz/fz, the above result reduces to the following.Corollary 8.
Letγ≥2+1/2.31−q and f∈A satisfy the subordination(62)1+γzfzz∂qfz2∂qz∂qfzfz≺1+z.
Then,fz∈Sq,c∗.Theorem 9.
Assume that(63)γ≥2+12.321−q,and we define an analytic function h on F with h0=1 which satisfies(64)1+γz∂qhzh3z≺1+z.In addition, we suppose that(65)1+γz∂qhzh3z=1+wz,where w is analytic in F with w0=0. Then,(66)hz≺1+43z+23z2.Proof.
Let us define the function(67)pz=1+γz∂qhzh3z,which is analytic in F with the condition p0=1 and the function(68)hz=1+43wz+23w2z,where w is an analytic function in F with w0=0. Using (67) and (68), we obtain(69)pz=1+γ/3z∂qwz41+wz−21−qz∂qwz1+4/3wz+2/3w2z3.To prove the result, we have to show thatwz≤1. Therefore,(70)p2z−1=2+γ/3z∂qwz41+wz−21−qz∂qwz1+4/3wz+2/3w2z3·γ/3z∂qwz41+wz−21−qz∂qwz1+4/3wz+2/3w2z3.Hence, by applying (9), we obtain(71)p2z0−1=23+4wz0+2w2z03+9γz0∂qwz041+wz0−21−qz0∂qwz03+4wz0+2w2z03×9γz0∂qwz041+wz0−21−qz0∂qwz03+4wz0+2w2z03.Now, consider the function(72)Gθ=Ψ329+40cosθ+12cos2θ3/2·9γm16+4−2m1−q2+84−2m1−qcosθ29+40cosθ+12cos2θ3/2,where(73)Ψ3=19652+18288γm2qcos2θ+13824γm2qcos3θ+4608cos4θγm2q+12384cosθγm2q+1296cosθγ2m3q+18432cos5θγm−720γm−3384γm2+138720cosθ+41184γmcos2θ+80640γmcos3θ+64512cos4θγm+3384γm2q−18288γm2cos2θ+2592cosθγ2m2+5904cosθγm−1296cosθγ2m3−12384cosθγm2−13824γm2cos3θ−4608cos4θγm2+1296γ2m3q−648γ2m4q+324γ2m4q2+409632cos2θ+647680cos3θ+578304cos4θ+276480cos5θ+55296cos6θ+2592γ2m2−1296γ2m3+324γ2m4.The above function is clearly an even function. So in order to find its maximum value, we will consider the interval0,π. Now, we have G′θ=0 for θ=0 and π. Clearly, G′′π>0, and hence, we have obtained the minimum value of G at θ=π, and thus, we conclude that Gθ≥Gπ. So now consider the function(74)Θm=4−72γm2+72γm2q−648γ2m4q+324γ2m4q2+324γ2m4·9γm16+4−2m1−q2−84−2m1−q,which gives(75)Θ′m=648γ2m21−q3>0.Thus,Θm is an increasing function. So for m=1, it gives a minimum value. Then, we have(76)p2z0−1≥18γ1−q−218γ1−q.Using (63), we get(77)p2z0−1≥1,but this result contradicts (64). Hence wz<1 which proves the required result.By takinghz=z∂qfz/fz, the above result reduces to the following.Corollary 10.
Letγ≥2+1/2.321−q and f∈A satisfy the subordination(78)1+γzfzz∂qfz3∂qz∂qfzfz≺1+z.
Then,fz∈Sq,c∗.Theorem 11.
Assume that(79)γ≥2+12.3n−11−q,and we define an analytic function h on F with h0=1 which satisfies(80)1+γz∂qhzhnz≺1+z.In addition, we suppose that(81)1+γz∂qhzhnz=1+wz,where w is analytic in F with w0=0. Then,(82)hz≺1+43z+23z2.We omit the proof of this result as it can be done by using a similar technique as applied in the above results.By takinghz=z∂qfz/fz, the above result reduces to the following.Corollary 12.
Letγ≥2+1/2.3n−11−q and f∈A satisfy the subordination(83)1+γzfzz∂qfzn∂qz∂qfzfz≺1+z.
Then,fz∈Sq,c∗.


## 3. Conclusion

In this article, we have worked onq-differential subordinations associated with lemniscate of Bernoulli and defined sufficient conditions for q-starlikeness related to cardioid domain. We have also determined the conditions on γ to prove the starlikeness of prescribed function such as(84)1+γz∂qhzhzn≺1+zforn=0,1,2,3,then(85)hz≺1+43z+23z2.We can use these results to study the sufficiency criteria of other analytic functions.



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