# Onq-ANALOGUE of Differential Subordination Associated with Lemniscate of Bernoulli

Authors: Mohsan Raza; Hira Naz; Sarfraz Nawaz Malik; Sahidul Islam
Journal: Journal of Mathematics
 (2021)

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## Abstract

This article comprises the study of differential subordination with analogue ofq-derivative. It includes the sufficient condition on γ for 1+γ∂zqhz/hnz to be subordinated by 1+Az/1+Bz, −1≤B<A≤1, and implies that hz≺1+z, where hz is the analytic function in the open unit disk. Moreover, certain sufficient conditions for q-starlikeness of analytic functions related with lemniscate of Bernoulli are determined.


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## Body

## 1. Introduction

Let a setA be considered as the class of analytic functions defined in open unit disk U=ς:ς∈ℂandς<1 under normalization conditions f0=0 and f′0=1, having(1)fς=ς+∑n=2∞anςn,ς∈U,as Taylor series. The class S comprises the normalized univalent functions, defined in U. The major subcategories of class S are C of convex functions and S∗ of starlike functions. The class P is another important class of analytic univalent functions whose co-domains are restricted to the right half plane and are used to determine the convexity and starlikeness of univalent functions. For more details, see [1, 2].Letf and g be two analytic functions in U. Then, f is subordinated by g, denoted as f≺g if f can be written in the form of composition of g and ϖ as fς=gϖς subject to the existence of analytic function ϖ which satisfies the condition that ϖ0=0 and ϖς<ς. Furthermore, if both f and g are univalent functions in U, then f≺g implies that f0=g0 and fU⊂gU.Subordination plays an important role in univalent function theory, and this concept was first introduced by Lindelöf, but Littlewood [3, 4] contributed remarkably to this field. Differential subordination is actually the generalized version of differential inequalities with real variables. Many researchers contributed in the work related to differential subordinations. Historical developments in the field of differential subordination are briefly described by Miller and Mocanu in [5].The advancement in the field of differential subordination starts with the usage of univalent functions. It was noticed in an article by Miller et al. [6] in 1974. Furthermore, many developments in this field have been achieved with the usage of differential subordinations in past fifty years. Differential inequality was a very well-known concept of real variables, and to study it in terms of complex variables, Miller and Mocanu [7] in 1981 were the first ones to introduce the idea of differential subordination. The contribution of Ruscheweyh and Singh [8] and Ruscheweyh and Wilken [9] is also of great importance in this field. Well-known Jack’s lemma [10] has brought the advancements in differential subordinations. Dziok [11] worked on some of the applications of Jack’s lemma. The research work carried out by Ma and Minda [12] in the function theory is worth to mention here, as they introduced the analytic function Φ, which satisfies the conditions of normalization Φ0=0 and Φ′0>1 having real part positive. The authors in [12] utilized the function Φ and introduced the subclass S∗Φ of starlike functions as follows:(2)S∗Φ=f∈A:ςf′ςfς≺Φς;ς∈U.The idea presented in [12] is very useful, and it helped many researchers for further studies in this direction. Ali et al. [13, 14] worked on differential subordination for sufficiency criteria of Janowski starlikeness and evaluated several differential subordinations such as 1+γςp′ς/pnς and found pς≺1+ς. Also, Ravichandran et al. [15] used this concept to find the sufficient conditions for starlikeness of Bernoulli’s lemniscate and Janowski functions. Sharma et al. [16] studied the differential subordinations to prove the starlikeness associated with cardioid domain and Halim et al. [17] introduced the concept for limacon domain.Jackson [18, 19] was the one who introduced the q-derivatives and q-integrals. After Jackson, Srivastava was amongst the pioneers to contribute in the q-calculus for its usage for analytic functions and their subclasses. Not only this, but he also applied q-hypergeometric function in the functions theory. All these contributions are comprised in his book (pp. 347 in [20]). Ismail et al. [21] contributed in the q-calculus for the study of starlike functions. Anastassiu and Gal [22, 23] also played their part in the development of complex variables with q-generalization. Purohit et al. [24] have used fractional q-calculus operators to apply subordination conditions on the class of non-Bazilevic functions. Sahoo and Agrawal [25] worked on starlike functions in q-calculus and extended the idea of q-starlikeness for particular subclasses of starlike functions. The involvement of q-derivative in the class S∗Φ gave the formation of following subclass Sq∗Φ of starlike functions which was introduced by Aouf and Seoudy [26].(3)Sq∗Φ=f∈A:ςDqfςfς≺Φς;ς∈U.The class described above has drawn the attention of many researchers. ReplacingΦς with different functions such as Janowski, lemniscate of Bernoulli, cardioid, and limacon, the researchers got the new directions to the study. Srivastava et al. [27] studied q-derivatives to find the relation between different classes of q-starlike functions related to Janowski function. Srivastava et al. [28] introduced the class of q-starlike functions by using general conic domains. They also obtained the bounds on Hankel and Toeplitz determinants for q-starlike functions and continued working par excellence. They produced unmatchable results that worked as great motivation for many researchers worldwide. To have an idea of their remarkable work, one can see [29–31], [20, 27, 28, 32–43]. Contributions of Haq et al. [44] and Zainab et al. [45] are also worth to mention. They studied q-analogue of differential subordinations for star-like functions related to limacon and cardioid domains, and Janowski functions. The q-derivative is the foundation of all this work in q-analogue, and it is defined as follows.Theq-derivative of a complex valued function f, defined in the domain U, is given as follows:(4)Dqfς=fς−fqς1−qς,ς≠0,f′0,ς=0,where 0<q<1. This implies the following:(5)limq⟶1−Dqfς=limq⟶1−fς−fqς1−qς=f′ς,on the assumption that the function f is differentiable in U. The q-derivative Dqf of an analytic function f has Taylor series of the form(6)Dqfς=∑n=0∞nqanςn−1,where(7)nq=1−qn1−q,n∈ℂ,∑k=0n−1qk,n∈ℕ.For more details aboutq-derivative and recent work on it, we refer the readers to [29–31], [20, 27, 28, 32–43]. In addition, the q-analogue of Jack’s lemma has played a vital role in this paper which states as follows.Lemma 1 (see [46]).
Letϖ be an analytic function in U with ϖ0=0. For maximum of ϖ on ς=1 at ς0=aeiθ, where θ∈−π,π and 0<q<1, then we have(8)ς0Dqϖς0=mϖς0,where m∈ℝ with m≥1.


## 2. Main Results

Theorem 1.
Assume that(9)γ≥A−B2+3−q1−B,−1<B<A≤1.Consider an analytic functionh on U with h0=1 which satisfies(10)1+γςDqhς≺1+Aς1+Bς,ς∈U.Also, suppose(11)1+γςDqhς=1+Aϖς1+Bϖς,ς∈U.Here,ϖ is an analytic function in U such that ϖ0=0. Then, we have(12)hς≺1+ς.Proof.
Suppose that(13)pς=1+γςDqhς,where p is analytic, and we have p0=1. Also, consider that(14)hς=1+ϖς.
Now, we prove thatϖς<1, where(15)ϖς=pς−1A−Bpς.
Using (13) and (14), we obtain(16)pς=1+γςDqϖς1+ϖς+1+ϖς−ςDqϖς1−q.
Also, we have(17)pς−1A−Bpς=γςDqϖςA−B1+ϖς+1+ϖς−ςDqϖς1−q−BγςDqϖς.
Consider a pointς0∈U such that(18)maxς≤ς0ϖς=ϖς0=1.
Now, by using Lemma1, we have ς0Dqϖς0=mϖς0,m≥1. Now, consider that ϖς0=eiθ,θ∈−π,π; then, for ς0∈U, we obtain(19)pς0−1A−Bpς0=γς0Dqϖς0A−B1+ϖς0+1+ϖς0−ς0Dqϖς01−q−Bγς0Dqϖς0,≥γmA−B1+eiθ+1+eiθ+meiθ1−q+Bγm,=γmA−B2+2+m1−q+Bγm.
Consider a new function(20)Ξm=γmA−B2+2+m1−q+Bγm.
Then,(21)Ξ′m=γA−B2+2+m1−q−γmA−B1−q/22+m1−qA−B2+2+m1−q+Bγm2>0.
Above expression represents that the functionΞ has increasing behavior, so we have its minimum value at m=1 and(22)Ξ1=γA−B2+3−q+Bγ.
So, we conclude that(23)pς0−1A−Bpς0≥γA−B2+3−q+Bγ.
Now, from (9), we have(24)pς0−1A−Bpς0≥1,
Since this result contradicts (10), therefore, ϖς<1, which completes the proof.
By takinghς=ςDqfς/fς, we deduce the following result.Corollary 1.
Letγ≥A−B2+3−q/1−B,−1<B<A≤1 and f∈A, satisfy the subordination(25)1+γςDqςDqfςfς≺1+Aς1+Bς.Then,f∈Sq∗1+ς.Theorem 2.
Assume that(26)γ≥2A−B2+3−q1−B,−1<B<A≤1.Consider an analytic functionh on U with h0=1 which satisfies(27)1+γςDqhςhς≺1+Aς1+Bς,ς∈U.Also, suppose(28)1+γςDqhςhς=1+Aϖς1+Bϖς,ς∈U,where ϖ is analytic function on U with ϖ0=0. Then,(29)hς≺1+ς.Proof.
We define a function(30)pς=1+γςDqhςhς,where p is analytic and p0=1. Now, consider that(31)hς=1+ϖς.
To obtain the result, we have to show thatϖς<1. Using (30) and (31), we obtain the result(32)pς=1+γςDqϖς1+ϖς1+ϖς+1+ϖς−ςDqϖς1−q.
Also, we have(33)pς−1A−Bpς=γςDqϖςA−B1+ϖς1+ϖς+1+ϖς−ςDqϖς1−q−BγςDqϖς.
Consider a pointς0∈U such that(34)maxς≤ς0ϖς=ϖς0=1.
Now, by using Lemma1, we have ς0Dqϖς0=mϖς0,m≥1. Now, consider that ϖς0=eiθ,θ∈−π,π; then, for ς0∈U, we obtain(35)pς0−1A−Bpς0=γmeiθA−B1+eiθ1+eiθ+1+eiθ−meiθ1−q−Bγmeiθ,≥γmA−B1+eiθ1+eiθ+1+eiθ+meiθ1−q+Bγm,=γmA−B22+2+m1−q+Bγm.
Consider(36)Ξ1m=γmA−B22+2+m1−q+Bγm.
Then,(37)Ξ1′m=γA−B22+2+m1−q−γm2A−B1−q/22+m1−qA−B22+2+m1−q+Bγm2>0.
Above expression represents that functionΞ1 has increasing behavior, so we have its minimum value at m=1 and(38)Ξ11=γA−B22+3−q+Bγ.
So, we conclude that(39)pς0−1A−Bpς0≥γA−B22+3−q+Bγ.
Now, from (26), we have(40)pς0−1A−Bpς0≥1,which contradicts (27), and hence, ϖς<1, which completes the proof.
By takinghς=ςDqfς/fς,we deduce the following result.Corollary 2.
Letγ≥2A−B2+3−q/1−B,−1<B<A≤1 and f∈A, which satisfies the subordination(41)1+γςfςςDqfςDqςDqfςfς≺1+Aς1+Bς.Then,f∈Sq∗1+ς.Theorem 3.
Assume that(42)γ≥2A−B2+3−q1−B,−1<B<A≤1.Consider an analytic functionh on U with h0=1 which satisfies(43)1+γςDqhςh2ς≺1+Aς1+Bς,ς∈U.Also, suppose(44)1+γςDqhςh2ς=1+Aϖς1+Bϖς,ς∈U,where ϖ is analytic function on U with ϖ0=0. Then,(45)hς≺1+ς.Proof.
We define a function(46)pς=1+γςDqhςh2ς,where p is analytic, and we have p0=1. Now, consider that(47)hς=1+ϖς.
To obtain the result, we have to show thatϖς<1. Using (46) and (47), we obtain the result(48)pς=1+γςDqϖς1+ϖς1+ϖς+1+ϖς−ςDqϖς1−q.
Also, we have(49)pς−1A−Bpς=γςDqϖςA−B1+ϖς1+ϖς+1+ϖς−ςDqϖς1−q−BγςDqϖς.
Consider a pointς0∈U such that(50)maxς≤ς0ϖς=ϖς0=1.
Now, by using Lemma1, we have ς0Dqϖς0=mϖς0,m≥1. Now, consider that ϖς0=eiθ,θ∈−π,π; then, for ς0∈U, we obtain(51)pς0−1A−Bpς0=γmeiθA−B1+eiθ1+eiθ+1+eiθ−meiθ1−q−Bγmeiθ,≥γmA−B1+eiθ1+eiθ+1+eiθ+meiθ1−q+Bγm,=γmA−B22+2+m1−q+Bγm.
Consider a function(52)Ξ2m=γmA−B22+2+m1−q+Bγm.
Then,(53)Ξ2′m=γA−B22+2+m1−q−γm2A−B1−q/22+m1−qA−B22+2+m1−q+Bγm2>0.
Here,Ξ2 is clearly an increasing function, so we have its minimum value at m=1 and(54)Ξ21=γ2A−B2+3−q+Bγ.
So, we conclude that(55)pς0−1A−Bpς0≥γ2A−B2+3−q+Bγ.
Now, from (42), we have(56)pς0−1A−Bpς0≥1,which contradict (43), and hence, ϖς<1, which completes the proof.
By takinghς=ςDqfς/fς, we deduce the following result.Corollary 3.
Letγ≥2A−B2+3−q/1−B,−1<B<A≤1 and f∈A, satisfies the subordination(57)1+γςfςςDqfς2DqςDqfςfς≺1+Aς1+Bς.Then,f∈Sq∗1+ς.Theorem 4.
Assume that(58)γ≥22A−B2+3−q1−B,−1<B<A≤1.Consider an analytic functionh on U with h0=1 which satisfies(59)1+γςDqhςh3ς≺1+Aς1+Bς,ς∈U.Also, suppose(60)1+γςDqhςh3ς=1+Aϖς1+Bϖς,ς∈U,where ϖ is analytic function on U with ϖ0=0. Then,(61)hς≺1+ς.Proof.
We define a function(62)pς=1+γςDqhςh3ς,where p is analytic, and we have p0=1. Now, consider that(63)hς=1+ϖς.
To obtain the result, we have to show thatϖς<1. Using (62) and (63), we obtain(64)pς=1+γςDqϖς1+ϖς3/21+ϖς+1+ϖς−ςDqϖς1−q.
Also, we have(65)pς−1A−Bpς=γςDqϖςA−B1+ϖς3/21+ϖς+1+ϖς−ςDqϖς1−q−BγςDqϖς.
Consider a pointς0∈U such that(66)maxς≤ς0ϖς=ϖς0=1.
Now, by using Lemma1, we have ς0Dqϖς0=mϖς0,m≥1. Now, consider that ϖς0=eiθ,θ∈−π,π; then, for ς0∈U, we obtain(67)pς0−1A−Bpς0=γmeiθA−B1+eiθ3/21+eiθ+1+eiθ−meiθ1−q−Bγmeiθ,≥γmA−B1+eiθ3/21+eiθ+1+eiθ+meiθ1−q+Bγm,=γmA−B23/22+2+m1−q+Bγm.
Consider a function(68)Ξ3m=γm23/2A−B2+2+m1−q+Bγm.
Then,(69)Ξ3′m=γA−B23/22+2+m1−q−γm23/2A−B1−q/22+m1−qA−B23/22+2+m1−q+Bγm2>0.
Above expression represents that functionΞ3 has increasing behavior, so we have its minimum value at m=1 and(70)Ξ31=γ23/2A−B2+3−q+Bγ.
So, we conclude that(71)pς0−1A−Bpς0≥γ23/2A−B2+3−q+Bγ.
Now, from (58), we have(72)pς0−1A−Bpς0≥1,which contradicts (59), and hence, ϖς<1, which completes the proof.
By takinghς=ςDqfς/fς, we deduce the following result.Corollary 5.
Letγ≥22A−B2+3−q/1−B,−1<B<A≤1 and f∈A, satisfy the subordination(73)1+γςfςςDqfς3DqςDqfςfς≺1+Aς1+Bς.Then,f∈Sq∗1+ς.Theorem 6.
Assume that(74)γ≥2n/2A−B2+3−q1−B,−1<B<A≤1.Consider an analytic functionh on U with h0=1 which satisfies(75)1+γςDqhςhnς≺1+Aς1+Bς,ς∈U.Also, suppose(76)1+γςDqhςhnς=1+Aϖς1+Bϖς,ς∈U,where ϖ is analytic function on U with ϖ0=0. Then,(77)hς≺1+ς.Proof.
We define a function(78)pς=1+γςDqhςhnς,where p is analytic, and we have p0=1. Now, consider that(79)hς=1+ϖς.
To obtain the result, we have to show thatϖς<1. Using (78) and (79), we obtain(80)pς=1+γςDqϖς1+ϖςn/21+ϖς+1+ϖς−ςDqϖς1−q.
Also, we have(81)pς−1A−Bpς=γςDqϖς/1+ϖςn/21+ϖς+1+ϖς−ςDqϖς1−qA−B1+γςDqϖς/1+ϖςn/21+ϖς+1+ϖς−ςDqϖς1−q=γςDqϖςA−B1+ϖςn/21+ϖς+1+ϖς−ςDqϖς1−q−BγςDqϖς.
Consider a pointς0∈U such that(82)maxς≤ς0ϖς=ϖς0=1.
Now, by using Lemma1, we have ς0Dqϖς0=mϖς0,m≥1. Now, consider that ϖς0=eiθ,θ∈−π,π; then, for ς0∈U, we obtain(83)pς0−1A−Bpς0=γς0Dqϖς0A−B1+ϖς0n/21+ϖς0+1+ϖς0−ς0Dqϖς01−q−Bγς0Dqϖς0,=γmeiθA−B1+eiθn/21+eiθ+1+eiθ−meiθ1−q−Bγmeiθ,≥γmA−B1+eiθn/21+eiθ+1+eiθ−meiθ1−q−Bγm,=γmA−B1+eiθn/21+eiθ+1+eiθ−meiθ1−q+Bγm,≥γmA−B1+eiθn/21+eiθ+1+eiθ+meiθ1−q+Bγm,=γmA−B2n/22+2+m1−q+Bγm.
Consider(84)Ξ4m=γm2n/2A−B2+2+m1−q+Bγm.
Then,(85)Ξ4′m=γA−B2n/22+2+m1−q−γm2n/2A−B1−q/22+m1−qA−B2n/22+2+m1−q+Bγm2>0.
Above expression represents that functionΞ4 has increasing behavior, so we have its minimum value at m=1 and(86)Ξ41=γ2n/2A−B2+3−q+Bγ.
So, we conclude that(87)pς0−1A−Bpς0≥γ2n/2A−B2+3−q+Bγ.
Now, from (74), we have(88)pς0−1A−Bpς0≥1,which contradict (75), and hence, ϖς<1, which completes the proof.
By takinghς=ςDqfς/fς, we deduce the following result.Corollary 7.
Letγ≥2n/2A−B2+3−q/1−B,−1<B<A≤1 and f∈A, satisfy the subordination(89)1+γςfςςDqfςnDqςDqfςfς≺1+Aς1+Bς.Then,f∈Sq∗1+ς.


## 3. Conclusion

In this article, we have investigated theq-differential subordination by using q-version of well-known Jack’s Lemma. We have found the condition on γ such that 1+γςDqhς/hnς≺1+Aς/1+Bς implies that hς≺1+ς. These results have been utilized to find sufficient conditions for star-like functions related to lemniscate of Bernoulli. This method can further be applied to find sufficient conditions for star-like functions of Ma–Minda type.



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