# On Quantum Differential Subordination Related with Certain Family of Analytic Functions

Authors: Afis Saliu; Khalida Inayat Noor; Saqib Hussain; Maslina Darus
Journal: Journal of Mathematics
 (2020)

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## Abstract

Recently, there is a rapid increase of research in the area of Quantum calculus (known asq-calculus) due to its widespread applications in many areas of study, such as geometric functions theory. To this end, using the concept of q-conic domains of Janowski type as well as q- calculus, new subclasses of analytic functions are introduced. This family of functions extends the notion of α-convex and quasi-convex functions. Furthermore, a coefficient inequality, sufficiency criteria, and covering results for these novel classes are derived. Besides, some remarkable consequences of our investigation are highlighted.


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## Body

## 1. Introduction

Recently, there is a rapid increase in the area of Quantum calculus (known asq-calculus) due to its widespread applications in many areas of study such as geometry functions theory (GFT), combinatorial analysis, Lie theory, mechanical engineering, cosmology, and statistics. The concept of q-integral was first introduced and studied by Jackson et al. [6] at the beginning of the twentieth century.The development of the concept ofq-calculus in GFT had its history from the work of Ismail et al. [5], where the notion of q-starlike functions was extensively studied. As such, many subclasses of univalent functions correlated with q-calculus have been on increase (see [1, 14, 17, 20, 23–25, 27, 28]). In recent times, various family of q- extension of starlike functions, which are connected to Janowski functions in the open unit disc U, were initiated and examined from many different viewpoints and perspective (see [17, 20, 28]).In an attempt to generalize the notion of uniformly closed-to-convex functions considered by Goodman [3], Kanas and Wisniowska [8–10, 13] and Kanas and Srivastava [12] introduced the conic domain Ωmm≥0 and studied the classes m−UCV and m−UST of m-uniformly convex and starlike functions. Furthermore, Noor and Malik [22], using the concept of Janowski class, extended the domain Ωm to Ωmγ,λ,−1≤λ<γ≤1. In the latest article by Mahmood et al. [17], the importance of q-calculus was used to improve the Noor–Malik conic domains to Ωq,mγ,λ. Using this domain, they examined the coefficient inequalities associated with the class m−USTqγ,λ of q-uniformly starlike functions. Afterward, the same coefficient problems were also explored for the classes m−UCVqγ,λ,m−UKVqγ,λ,m−UC∗Vqγ,λ of m-uniformly q-convex, close-to-convex and quasi-convex functions by Naeem et al. [20].Motivated by these recent articles [15, 17, 20, 28], our aim is to introduce the novel classes m−UMqα,γ,λ and m−UQqα,γ,λ consisting of m-uniformly q-alpha convex and quasi-convex functions. We study the coefficient inequalities associated with these classes and some other related properties. Some relevant consequences of our results which were studied in previous work show the significance of our investigation.


## 2. Materials and Methods

Now, we give some useful preliminaries which are necessary for our study.LetA be the class of normalized analytic functions fς in U=ς:ς<1 with(1)fς=ς+a2ς2+a3ς3+⋯.LetS,CV,ST,QV, and KV be the subclasses of A consisting functions that are univalent, convex, starlike, quasi-convex, and close-to-convex functions, respectively. The function fς of form (1) is subordinate to the analytic function gς (written as fς≺gς) of the form(2)gς=ς+b2ς2+b3ς3+⋯,if there exists a Schwarz function wς in U such that(3)fς=gwς,ς∈U.Let−1≤λ<γ≤1. Then, class Pγ,λ  (see [7]) of function pς satisfies the subordination condition(4)pς≺1+γς1+λς,ς∈U,or equivalently,(5)pς=1+γhς+1−γ1+λhς+1−λ,ς∈U,where h∈P (class of functions with positive real part). For γ=1−2β,0≤β<1, and λ=−1, the class Pγ,λ reduces to the class Pβ, the class of functions whose real part is greater than β.The conic domainsΩmγ,λm≥0 of Janowski type introduced by Noor and Malik [22] are defined as follows:(6)Ωmγ,λ=u+iv:λ2−1u2+v2−2γλ−1u+γ2−12>m2−2λ+1u2+v2+2γ+λ+2u−2γ+12+4γ−λ2v2.Geometrical interpretation ofΩmγ,λ and its effect on Ωm were also demonstrated in [22]. The class m−Pγ,λ represents the class of all functions that maps U onto Ωmγ,λ. Equivalently, a function pς belongs to m−Pγ,λ if and only if(7)pς≺γ+1pmς−γ−1λ+1pmς−λ−1,m≥0,−1≤λ<γ≤1,where pmς has its definition in [10, 11] and given by(8)pmς=1+ς1−ς,m=0,1+2π2log1+ς1−ς2,m=1,1+21−m2sinh22πarccosmarctanς,0<m<1,1+11−m2sinπ2Rt∫0uς/t11−x21−tx2dx+1m2−1,m>1,where uς=ς−t/1−tς,t∈0,1,ς∈U and t is chosen such that m=coshπR′t/4Rt, Rt is Legendre’s complete elliptic integral of the first kind, and R′t is the complementary integral of RtDefinition 1 (see [2]).
Letq∈0,1. Then, the q-number nq is given as(9)nq=1−qn1−q,n∈ℂ,∑ι=0n−1qι=1+q+q2+⋯+qn−1,n∈ℕ,n,asq⟶1−,and the q-derivative of a complex valued function fς in U is given by(10)Dqfς=fqς−fςq−1ς,ς≠0,f′0,ς=0,f′ς,asq⟶1−.
From the above explanation, it is easy to see that, forfς given by (1),(11)Dqfς=1+∑ι=2∞ιqaιςι.Definition 2 (see [28]).
An analytic functionpς in U belongs to Pqγ,λ if and only if the condition(12)pς=O1γ+O3hς−γ−1O1O1λ+O3hς−λ−1O1,is satisfied, where O1=1+q,O3=3−q and h∈P.Definition 3 (see [17]).
An analytic functionpς in U belongs to m−Pqγ,λ if and only if the subordination condition(13)pς≺O1γ+O3pmς−γ−1O1O1λ+O3pmς−λ−1O1,is satisfied, where pmς is given by (8). Equivalently, p∈m−Pqγ,λ if and only if pς conformally maps U onto the domain Ωq,mγ,λ defined by(14)Ωq,mγ,λ=pς=u+iv:ReΦ>mΦ−1,where(15)Φ=λ−1O1pς−γ−1O1λO1+O3pς−γO1+O3.
Equivalently,(16)Ωq,mγ,λ=u+iv:O12λ−1λO1+O3u2+v2−λ−1γO1+O3+γ−1λO1+O3u+γ−1γO1+O32>4m2−λO1+O3u2+v2+γ+λO1+2O3u−γO1+O32+γ−λ2O12v2,m≥0,0<q<1.
We note that(i)
m−Pqγ,λ⊂Pβ, where β is given by(17)β=4m+1−γO14m+1−λO1.(ii)
Asq⟶1−, the class m−Pqγ,λ becomes the class m−Pγ,λ and Ωq,mγ,λ≡Ωmγ,λ [22].(iii)
Whenq⟶1− and γ=1,λ=−1, the class m−Pqγ,λ reduces to the class Ppm and Ωq,mγ,λ≡Ωm [10].Definition 4 (see [20]).
Letgς of form (2) be in A and α≥0,m∈0,∞. Then, g∈m−UCVqγ,λ if and only if(18)DqςDqgςDqgς≺O1γ+O3pmς−γ−1O1O1λ+O3pmς−λ−1O1,where pmς is given by (8).
Inspired by the above recent mentioned work, we announce the following novel classes of analytic functions.Definition 5.
Letfς∈A,α≥0,m∈0,∞. Then, f∈m−UMqα,γ,λ if and only if(19)Reλ−1O1Jqα,f;ς−γ−1O1λO1+O3Jqα,f;ς−γO1+O3>mλ−1O1Jqα,f;ς−γ−1O1λO1+O3Jqα,f;ς−γO1+O3−1.
Equivalently,f∈m−UMqα,γ,λ if and only if(20)Jqα,f;ς=1−αςDqfςfς+αDqςDqfςDqfς≺O1γ+O3pmς−γ−1O1O1λ+O3pmς−λ−1O1.Definition 6.
Letfς∈A,α≥0,m∈0,∞. Then, f∈m−UQqα,γ,λ if and only if there exists an analytic function gς∈m−UCVqγ,λ such that(21)Reλ−1O1Jqα,f;ς−γ−1O1λO1+O3Jqα,f;ς−γO1+O3>mλ−1O1Jqα,f;ς−γ−1O1λO1+O3Jqα,f;ς−γO1+O3−1.
Equivalently,f∈m−UQqα,γ,λ if and only if(22)Jqα,f;ς=1−αDqfςDqgς+αDqςDqfςDqgς≺O1γ+O3pmς−γ−1O1O1λ+O3pmς−λ−1O1.
We note the following special cases:(i)
Whenq⟶1−, the classes m−UMqα,γ,λ and m−UQqα,γ,λ reduce to the classes m−UMα,γ,λ [21] and m−UQα,γ,λ [18], respectively.(ii)
Whenq⟶1− and α=0, the classes m−UMqα,γ,λ and m−UQqα,γ,λ cut down to the classes m−USTγ,λ [22] and m−UKVγ,λ [16].(iii)
Whenq⟶1− and α=1, the classes m−UMqα,γ,λ and m−UQqα,γ,λ scale down to the classes m−UCVγ,λ [22] and m−UC∗Vγ,λ [16].(iv)
Whenα=0 and α=1, the classes m−UMqα,γ,λ and m−UQqα,γ,λ diminish, respectively, to those classes of functions considered in [17, 20].(v)
Whenα=0=m in Definition 2, the class m−UMqα,γ,λ becomes the class q-starlike functions STqγ,λ of Janowski type recently explored by Srivastava et al. [28].
To effectively establish our findings, the following set of lemmas is required.


## 3. A Set of lemmas

Lemma 1 (see [4]).
Letm≥0 and pmς given by (8) be of the form pmς=1+Q1ς+Q1ς2+⋯. Then,(23)Q1=8π2,m=1,2T21−m2,0≤m<1,π24K2tt2−11+tt,m>1,(24)Q2=23Q1,m=1,2+T23Q1,0≤m<1,4K2tt2+6t+1−π224K2t1+ttQ1,m>1,where T=2cos−1m/π and t∈0,1 is chosen such that m=coshπK′t/Kt, where Kt is Legendre’s complete elliptic integral of the first kind.Lemma 2 (see [17]).
Ifpς=1+c1ς+c2ς2+⋯∈P, then, for any real number δ,(25)c2−δc12≤−4δ+2,δ≤0,2,0≤δ≤1,4δ+2,δ≥1,when δ<0 or δ>1, the equality holds if and only if pς=1+ς/1−ς or one of its rotations. If 0<δ<1, then the equality holds if and only if pς=1+ς2/1−ς2 or one of its rotations. If δ=0, then the equality holds if and only if(26)pς=1+ν21+ς1−ς+1−ν21−ς1+ς,0≤ν≤1,or one of its rotations. If δ=1, then the equality holds if and only if pς is the reciprocal of one of the functions such that equality holds for the case δ=0. Although the above upper bound is sharp, when 0<δ<1 it can be improved as follows:(27)c2−δc12+c12≤2,0<δ≤12,c2−δc12+1−δc12≤2,12<δ≤1.Lemma 3 (see [20]).
Letgς be of form (2) and g∈m−UCVqγ,λ. Then,(28)bn≤1nq∏ι=0n−2Q1O1γ−λ−4qιq4qι+1q,where Q1 is defined by (23).


## 4. Results and Discussion

We now turn our attention to the main results of this article.


### 4.1. Sufficient Conditions

Theorem 1.
A functionfς of form (1) belongs to the class m−UMqα,γ,λ if it satisfies the condition(29)ϕnα,m,q,γ,λ<λ−γO1,where(30)ϕnα,m,q,γ,λ=∑n=2∞4m+1nq−11+αnq−1+λO1+O32nq+αnq−12−γO1+O3nq+1|an+∑n=2∞∑ι=2n−14m+1ιq−1+αn+1−ιq−ιq+|λO1+O3ιq+αn+1−ιq−ιq−γO1+O3|n+1−ιqaιan+1−ι.Proof.
Suppose condition (29) holds. Then, we need to prove that(31)mλ−1O1Jqα,f;ς−γ−1O1λO1+O3Jqα,f;ς−γO1+O3−1−Reλ−1O1Jqα,f;ς−γ−1O1λO1+O3Jqα,f;ς−γO1+O3−1<1.
Therefore,(32)mλ−1O1Jqα,f;ς−γ−1O1λO1+O3Jqα,f;ς−γO1+O3−1−Reλ−1O1Jqα,f;ς−γ−1O1λO1+O3Jqα,f;ς−γO1+O3−1≤m+1λ−1O1Jqα,f;ς−γ−1O1λO1+O3Jqα,f;ς−γO1+O3−1=4m+11−αςDqfςDqfς+αfςDqςDqfς−fςDqfς1−αλO1+O3ςDqfςDqfς+αλO1+O3fςDqςDqfς−γO1+O3fςDqfς.
We have(33)ςDqfςDqfς=∑n=0∞nqanςn∑n=0∞nqanςn−1,a0=0,a1=0q=1q=1=∑n=0∞∑ι=0nιqn−ιqaιan−ιςn−1=ς+∑n=2∞∑ι=1n+1ιqn+1−ιqaιan+1−ιςn=ς+∑n=2∞2nqan+∑ι=2n−1ιqn+1−ιqaιan+1−ιςn.
Similarly,(34)DqςDqfς=ς+∑n=2∞nq2+1an+∑ι=2n−1n+1−ιq2aιan+1−ιςn,(35)fςDqfς=ς+∑n=2∞nq+1an+∑ι=2n−1n+1−ιqaιan+1−ιςn.
From inequality (32) and equations (33) and (34), we arrive at(36)mλ−1O1Jqα,f;ς−γ−1O1λO1+O3Jqα,f;ς−γO1+O3−1−Reλ−1O1Jqα,f;ς−γ−1O1λO1+O3Jqα,f;ς−γO1+O3−1≤4m+1∑n=2∞nq−11+αnq−1an+∑ι=2n−1ιq−1+αn+1−ιq−ιqn+1−ιqaιan+1−ιςn×λ−γO1ς+∑n=2∞λO1+O32nq+αnq−12−γO1+O3nq+1an+∑ι=2n−1λO1+O3ιq+αn+1−ιq−ιq−γO1+O3n+1−ιqaιan+1−ιςn−1|≤4m+1∑n=2∞nq−11+αnq−1an+∑ι=2n−1ιq−1+αn+1−ιq−ιqn+1−ιqaιan+1−ι×λ−γO1−∑n=2∞|λO1+O32nq+αnq−12−γO1+O3nq+1|an−∑ι=2n−1|λO1+O3ιq+αn+1−ιq−ιq−γO1+O3|n+1−ιqaιan+1−ι−1.
The last inequality is bounded by 1 if (29) is satisfied. This completes the proof.
Asq⟶1− in Theorem 1, we are led to Theorem 1 in [21].Corollary 1.
A functionfς of type (1) is in m−UMα,γ,λ if it satisfies the condition(37)ϕnα,m,γ,λ<λ−γ,where(38)ϕnα,m,γ,λ=∑n=2∞2m+1n−11+αn−1+|λ+12n+αn−12−γ+1n+1|an+∑n=2∞∑ι=2n−12m+1ι−1+αn+1−2ι+λ+1ι+αn+1−2ι−γ+1n+1−ιaιan+1−ι.Settingα=0 in Theorem 1, we obtain a variance version of the result presented in [17].Corollary 2.
Iffς having representation (1) satisfies the condition(39)∑n=2∞4qm+1n−1q+λO1+O3nq−γO1+O3an<λ−γO1,then f∈m−USTqγ,λ.If we chooseα=0 and allow q⟶1− in the above theorem, our investigation comes down to Theorem 1 in [22].Corollary 3.
A functionfς demonstrated by (1) is in the class m−USTγ,λ if it satisfies the condition(40)∑n=2∞2m+1n−1+λ+1n−γ+1an<λ−γ.If we chooseα=0,λ=−1,andγ=1 and allow q⟶1− in the above theorem, our investigation comes down to Theorem 1 in [22].Corollary 4 (see [11]).
Iffς∈A satisfies the condition(41)∑n=2∞mn−1+nan<1,then f∈m−USTq⟶1−0,1,−1=m−UST.Theorem 2.
Letf∈A. Then, f∈m−UQqα,γ,λ if inequality (42) is satisfied,(42)∑n=2∞∏ι=0n−24m+1+γO1+O3Q1O1γ−λ−4qιqλ4qι+1q+∑n=2∞4m+1+λO1+O31+αqn−1qnqan<λ−γO1,where Q1 is given by (23).Proof.
Assuming (42) holds, then it suffices to establish that(43)mλ−1O1Jqα,f;ς−γ−1O1λO1+O3Jqα,f;ς−γO1+O3−1−Reλ−1O1Jqα,f;ς−γ−1O1λO1+O3Jqα,f;ς−γO1+O3−1<1.
Following the same process in the proof of Theorem1, we have(44)mλ−1O1Jqα,f;ς−γ−1O1λO1+O3Jqα,f;ς−γO1+O3−1−Reλ−1O1Jqα,f;ς−γ−1O1λO1+O3Jqα,f;ς−γO1+O3−1≤4m+11−αDqfς+αDqςDqfς−DqgςλO1+O31−αDqfς+αDqςDqfς−γO1+O3Dqgς,whereg∈m−UCVqγ,λ≤4m+1∑n=2∞1+αnq−1nqan+nqbnλ−γO1−∑n=2∞λO1+O31+αnq−1nq−γO1+O3∑n=2∞nqbn≤4m+1∑n=2∞1+αnq−1nqan+∑n=2∞∏ι=0n−2Q1O1γ−λ−4qιq/4qι+1qλ−γO1−∑n=2∞λO1+O31+αnq−1nqan−γO1+O3∑n=2∞∏ι=0n−2Q1O1γ−λ−4qιq/4qι+1q,where we have used Lemma 3. Thus, the last inequality is bounded by 1 if (42) is satisfied. Hence, we complete the proof.
Asq⟶1− in Theorem 2, we obtain the similar result proved in [18].Corollary 5.
Letf∈A. Then, f∈m−UQα,γ,λ if(45)∑n=2∞∏ι=0n−22m+1+γ+1Q1O1γ−λ−2ιλ2ι+1+∑n=2∞2m+1+λ+11+αn−1nan<λ−γ,holds.Forα=0 in Theorem 2, we get the result established by Naeem et al. [20].Corollary 6.
Letf∈A. Then, f∈m−UKVqγ,λ if inequality (46) is satisfied,(46)∑n=2∞∏ι=0n−24m+1+γO1+O3Q1O1γ−λ−4qιqλ4qι+1q+∑n=2∞4m+1+λO1+O3nqan<λ−γO1.Forα=0 and α=1 as q⟶1− in Theorem 2, we obtain the following results established in [16].Corollary 7.
Letf∈A. Then, f∈m−UKVγ,λ if(47)∑n=2∞∏ι=0n−22m+1+γ+1Q1O1γ−λ−2ιλ2ι+1+∑n=2∞2m+1+λ+1nan<λ−γis satisfied.Corollary 8.
Letf∈A. Then, f∈m−UKVγ,λ if(48)∑n=2∞∏ι=0n−22m+1+γ+1Q1O1γ−λ−2ιλ2ι+1+∑n=2∞2m+1+λ+1n2an<λ−γ,is satisfied.


### 4.2. Fekete Szegö Inequality

Theorem 3.
Letf∈m−UMqα,γ,λ. Then, for any real number δ, we have(49)a3−δa22≤γ−λQ18q1+αq2q×Λ1α,m,q,γ,λ,δ<ρ1,2,ρ1≤δ≤ρ2,Λ2α,m,q,γ,λ,δ>ρ2,where(50)ρ1=12q1+αq2q8q1+αq2γ−λO1Q1Q2−Q12Q1−λO1+O3Q18+1+αq2q+1,ρ2=12q1+αq2q8q1+αq2γ−λO1Q1Q2+Q12Q1−λO1+O3Q18+1+αq2q+1,Λ1α,m,q,γ,λ=4Q2−λO1+O3Q122Q1+1+αq2q+1−δ2q1+αq2qγ−λO1Q12q1+αq2,Λ2α,m,q,γ,λ=λO1+O3Q12−4Q22Q1+δ2q1+αq2q−1+αq2q+1γ−λO1Q12q1+αq2,and Q1andQ2 are given by (23) and (24). This result cannot be improved. Then, fς∈m−UMqα,γ,λ implies there exists a Schwarz function wς such that(51)Jqα,f;ς=O1γ+O3pmwς−γ−1O1O1λ+O3pmwς−λ−1O1.Using the representation forpmς in Lemma 3 and the relationship between wς and pς=1+c1ς+c2ς2+⋯∈P, we can write(52)wς=pς−1pς+1=c12ς+c2−c122ς2+⋯,pmς=1+c1Q12ς+2c2−c12Q1+c12Q24ς2+⋯.Thus,(53)Jqα,f;ς=4+γO1+O3c1Q1/2ς+γO1+O3/42c2−c12Q1+c21Q2ς2+⋯4+λO1+O3c1Q1/2ς+λO1+O3/42c2−c12Q1+c21Q2ς2+⋯=1+γ−λc1O1Q18ς+γ−λ16O12c2−c12Q1+c12Q2−γ−λλO1+O3c12O1Q1264ς2+⋯.However,(54)Jqα,f;ς=1−αςDqfςfς+αDqςDqfςDqfς=1+q1+αqa2ς+q2q1+αq2qa3−1+αq2q+1a22ς2+⋯.On comparing the coefficients ofς and ς2 of (53) and (54), we obtain(55)a2=γ−λc1O1Q18q1+αq,a3=1q2q1+αq2qγ−λ16O12c2−c12Q1+c12Q2−γ−λλO1+O3c12O1Q1264+1+αq2q+1a22q2q1+αq2q.Now, for a real numberδ, we have(56)a3−δa22=1q2q1+αq2qγ−λ8O1Q1c2+Q2−Q12−λO1+O3Q128c12+γ−λO1Q12c12γ−B8O11+αq2q+1−δ2q1+αq2q8q21+αq22q1+αq2q=γ−λO1Q18q2q1+αq2q|c2−Q1−Q22Q1+λO1+O3Q18−1+αq2q+1−δ2q1+αq2qγ−λO1Q18q1+αq2c12|=γ−λO1Q18q2q1+αq2qc2−βc12,where(57)β=Q1−Q22Q1+λO1+O3Q18−1+αq2q+1−δ2q1+αq2qγ−λO1Q18q1+αq2.Hence, the result follows from Lemma2.Ifα=0 and 0<m<1, then Theorem 3 reduces to Theorem 10 in [17].Corollary 9.
Let0<m<1 and f∈m−USTqγ,λ be of form (1). Then, for any real number δ, we have(58)a3−δa22≤γ−λT24q1−m2×Λ1m,q,γ,λ,δ<λ1,2,λ1≤δ≤λ2,Λ2m,q,γ,λ,δ>λ2,where(59)λ1=4q1−m2γ−λ1+q2T2T2−16−T2λ1+q+3−q41−m2+11+q,where(60)λ2=4q1−m2γ−λ1+q2T2T2+56−T2λ1+q+3−q41−m2+11+q,Λ1m,q,γ,λ=22+T23−T2λ1+q+3−q1−m2+T21−δ1+q1+qγ−λq1−m2,Λ2m,q,γ,λ=T2λ1+q+3−q1−m2−22+T23+T2δ1+q−11+qγ−λq1−m2.The result is sharp.Proof.
The proof is straightforward from Theorem3 and Lemma 1.
Ifα=0 in Theorem 3, the case m=1 is contained in the following corollary.Corollary 10.
Letm=1, and fς of form (1) belongs to m−USTqγ,λ. Then, for a real number δ, we have the following sharp inequality:(61)a3−δa22≤γ−λqπ2×Λ1q,γ,λ,δ<ν1,2,ν1≤δ≤ν2,Λ2q,γ,λ,δ>ν2,where(62)ν1=11+q−qπ2γ−λ1+q216+λ1+q+3−qπ2,where(63)ν2=11+q−qπ2γ−λ1+q256−λ1+q+3−qπ2,Λ1q,γ,λ=43−4λ1+q+3−qπ2+1−δ1+q1+qγ−λqπ2,Λ2m,q,γ,λ=4λ1+q+3−qπ2−43+δ1+q−11+qγ−λqπ2.The result is sharp.Theorem3 becomes Theorem 1 in [28] when α=m=0.Corollary 11.
Letfς of the series representation (1) be in STqγ,λ. Then, we have the sharp inequality:(64)a3−δa22≤γ−λ4q2×γ−λ+γ−2λ−1q+1−λq2−δγ−λ1+q2,δ<σ1,2q,σ1≤δ≤σ2,λ−γ+2λ−γ+1q+λ−1q2+δγ−λ1+q2,δ>σ2,where(65)σ1=11+q1−qλ1+q+3−qγ−λ1+q,σ2=σ1+4qγ−λ1+q2.


### 4.3. Covering Theorems

Theorem 4.
The range of every univalent functionsf∈m−UMqα,γ,λ contains the disc:(66)ς<4q1+αq8q1+αq+γ−λO1Q1.Proof.
From the proof of Theorem3, we can see that(67)a2≤γ−λO1Q14q1+αq.
Since the Koebe one-quater theorem asserted that each omitted valuew of the univalent function fς of form (1) satisfies(68)w>12+a2≥4q1+αq8q1+αq+γ−λO1Q1.
Thus, we have the result.Corollary 12.
The range of every univalent functionf∈STqγ,λ contains the disc:(69)ς<2q4q+γ−λO1.Theorem 5.
The range of every univalent functionf∈m−UQqα,γ,λ contains the same disc given by (66).Proof.
Letwς be a Schwarz function. We note first in Theorem 3 that(70)O1γ+O3pmwς−γ−1O1O1λ+O3pmwς−λ−1O1=1+γ1ς+γ2ς2+⋯,where(71)γ1=γ−λc1O1Q18,γ2=γ−λ16O12c2−c12Q1+c12Q2−γ−λλO1+O3c12O1Q1264.
Sincef∈m−UQqα,γ,λ, then for some gς=ς+b2ς2+b3ς3+⋯∈m−UCVqγ,λ, we have(72)1−αDqfς+αDqςDqfς=O1γ+O3pmwς−γ−1O1O1λ+O3pmwς−λ−1O1Dqgς,which in turn implies(73)1+2q1+αqa2ς+⋯=1+γ1+2qb2ς⋯.
It is easy to see that(74)b2=γ1q2q.
Therefore, comparing the coefficients ofς of (73) and applying (74), we obtain(75)a2=γ1q1+αq,such that(76)a2≤γ−λO1Q14q1+αq.
Now, proceeding the same way as in the proof of Theorem4, we have the required result.


## 4.1. Sufficient Conditions

Theorem 1.
A functionfς of form (1) belongs to the class m−UMqα,γ,λ if it satisfies the condition(29)ϕnα,m,q,γ,λ<λ−γO1,where(30)ϕnα,m,q,γ,λ=∑n=2∞4m+1nq−11+αnq−1+λO1+O32nq+αnq−12−γO1+O3nq+1|an+∑n=2∞∑ι=2n−14m+1ιq−1+αn+1−ιq−ιq+|λO1+O3ιq+αn+1−ιq−ιq−γO1+O3|n+1−ιqaιan+1−ι.Proof.
Suppose condition (29) holds. Then, we need to prove that(31)mλ−1O1Jqα,f;ς−γ−1O1λO1+O3Jqα,f;ς−γO1+O3−1−Reλ−1O1Jqα,f;ς−γ−1O1λO1+O3Jqα,f;ς−γO1+O3−1<1.
Therefore,(32)mλ−1O1Jqα,f;ς−γ−1O1λO1+O3Jqα,f;ς−γO1+O3−1−Reλ−1O1Jqα,f;ς−γ−1O1λO1+O3Jqα,f;ς−γO1+O3−1≤m+1λ−1O1Jqα,f;ς−γ−1O1λO1+O3Jqα,f;ς−γO1+O3−1=4m+11−αςDqfςDqfς+αfςDqςDqfς−fςDqfς1−αλO1+O3ςDqfςDqfς+αλO1+O3fςDqςDqfς−γO1+O3fςDqfς.
We have(33)ςDqfςDqfς=∑n=0∞nqanςn∑n=0∞nqanςn−1,a0=0,a1=0q=1q=1=∑n=0∞∑ι=0nιqn−ιqaιan−ιςn−1=ς+∑n=2∞∑ι=1n+1ιqn+1−ιqaιan+1−ιςn=ς+∑n=2∞2nqan+∑ι=2n−1ιqn+1−ιqaιan+1−ιςn.
Similarly,(34)DqςDqfς=ς+∑n=2∞nq2+1an+∑ι=2n−1n+1−ιq2aιan+1−ιςn,(35)fςDqfς=ς+∑n=2∞nq+1an+∑ι=2n−1n+1−ιqaιan+1−ιςn.
From inequality (32) and equations (33) and (34), we arrive at(36)mλ−1O1Jqα,f;ς−γ−1O1λO1+O3Jqα,f;ς−γO1+O3−1−Reλ−1O1Jqα,f;ς−γ−1O1λO1+O3Jqα,f;ς−γO1+O3−1≤4m+1∑n=2∞nq−11+αnq−1an+∑ι=2n−1ιq−1+αn+1−ιq−ιqn+1−ιqaιan+1−ιςn×λ−γO1ς+∑n=2∞λO1+O32nq+αnq−12−γO1+O3nq+1an+∑ι=2n−1λO1+O3ιq+αn+1−ιq−ιq−γO1+O3n+1−ιqaιan+1−ιςn−1|≤4m+1∑n=2∞nq−11+αnq−1an+∑ι=2n−1ιq−1+αn+1−ιq−ιqn+1−ιqaιan+1−ι×λ−γO1−∑n=2∞|λO1+O32nq+αnq−12−γO1+O3nq+1|an−∑ι=2n−1|λO1+O3ιq+αn+1−ιq−ιq−γO1+O3|n+1−ιqaιan+1−ι−1.
The last inequality is bounded by 1 if (29) is satisfied. This completes the proof.
Asq⟶1− in Theorem 1, we are led to Theorem 1 in [21].Corollary 1.
A functionfς of type (1) is in m−UMα,γ,λ if it satisfies the condition(37)ϕnα,m,γ,λ<λ−γ,where(38)ϕnα,m,γ,λ=∑n=2∞2m+1n−11+αn−1+|λ+12n+αn−12−γ+1n+1|an+∑n=2∞∑ι=2n−12m+1ι−1+αn+1−2ι+λ+1ι+αn+1−2ι−γ+1n+1−ιaιan+1−ι.Settingα=0 in Theorem 1, we obtain a variance version of the result presented in [17].Corollary 2.
Iffς having representation (1) satisfies the condition(39)∑n=2∞4qm+1n−1q+λO1+O3nq−γO1+O3an<λ−γO1,then f∈m−USTqγ,λ.If we chooseα=0 and allow q⟶1− in the above theorem, our investigation comes down to Theorem 1 in [22].Corollary 3.
A functionfς demonstrated by (1) is in the class m−USTγ,λ if it satisfies the condition(40)∑n=2∞2m+1n−1+λ+1n−γ+1an<λ−γ.If we chooseα=0,λ=−1,andγ=1 and allow q⟶1− in the above theorem, our investigation comes down to Theorem 1 in [22].Corollary 4 (see [11]).
Iffς∈A satisfies the condition(41)∑n=2∞mn−1+nan<1,then f∈m−USTq⟶1−0,1,−1=m−UST.Theorem 2.
Letf∈A. Then, f∈m−UQqα,γ,λ if inequality (42) is satisfied,(42)∑n=2∞∏ι=0n−24m+1+γO1+O3Q1O1γ−λ−4qιqλ4qι+1q+∑n=2∞4m+1+λO1+O31+αqn−1qnqan<λ−γO1,where Q1 is given by (23).Proof.
Assuming (42) holds, then it suffices to establish that(43)mλ−1O1Jqα,f;ς−γ−1O1λO1+O3Jqα,f;ς−γO1+O3−1−Reλ−1O1Jqα,f;ς−γ−1O1λO1+O3Jqα,f;ς−γO1+O3−1<1.
Following the same process in the proof of Theorem1, we have(44)mλ−1O1Jqα,f;ς−γ−1O1λO1+O3Jqα,f;ς−γO1+O3−1−Reλ−1O1Jqα,f;ς−γ−1O1λO1+O3Jqα,f;ς−γO1+O3−1≤4m+11−αDqfς+αDqςDqfς−DqgςλO1+O31−αDqfς+αDqςDqfς−γO1+O3Dqgς,whereg∈m−UCVqγ,λ≤4m+1∑n=2∞1+αnq−1nqan+nqbnλ−γO1−∑n=2∞λO1+O31+αnq−1nq−γO1+O3∑n=2∞nqbn≤4m+1∑n=2∞1+αnq−1nqan+∑n=2∞∏ι=0n−2Q1O1γ−λ−4qιq/4qι+1qλ−γO1−∑n=2∞λO1+O31+αnq−1nqan−γO1+O3∑n=2∞∏ι=0n−2Q1O1γ−λ−4qιq/4qι+1q,where we have used Lemma 3. Thus, the last inequality is bounded by 1 if (42) is satisfied. Hence, we complete the proof.
Asq⟶1− in Theorem 2, we obtain the similar result proved in [18].Corollary 5.
Letf∈A. Then, f∈m−UQα,γ,λ if(45)∑n=2∞∏ι=0n−22m+1+γ+1Q1O1γ−λ−2ιλ2ι+1+∑n=2∞2m+1+λ+11+αn−1nan<λ−γ,holds.Forα=0 in Theorem 2, we get the result established by Naeem et al. [20].Corollary 6.
Letf∈A. Then, f∈m−UKVqγ,λ if inequality (46) is satisfied,(46)∑n=2∞∏ι=0n−24m+1+γO1+O3Q1O1γ−λ−4qιqλ4qι+1q+∑n=2∞4m+1+λO1+O3nqan<λ−γO1.Forα=0 and α=1 as q⟶1− in Theorem 2, we obtain the following results established in [16].Corollary 7.
Letf∈A. Then, f∈m−UKVγ,λ if(47)∑n=2∞∏ι=0n−22m+1+γ+1Q1O1γ−λ−2ιλ2ι+1+∑n=2∞2m+1+λ+1nan<λ−γis satisfied.Corollary 8.
Letf∈A. Then, f∈m−UKVγ,λ if(48)∑n=2∞∏ι=0n−22m+1+γ+1Q1O1γ−λ−2ιλ2ι+1+∑n=2∞2m+1+λ+1n2an<λ−γ,is satisfied.


## 4.2. Fekete Szegö Inequality

Theorem 3.
Letf∈m−UMqα,γ,λ. Then, for any real number δ, we have(49)a3−δa22≤γ−λQ18q1+αq2q×Λ1α,m,q,γ,λ,δ<ρ1,2,ρ1≤δ≤ρ2,Λ2α,m,q,γ,λ,δ>ρ2,where(50)ρ1=12q1+αq2q8q1+αq2γ−λO1Q1Q2−Q12Q1−λO1+O3Q18+1+αq2q+1,ρ2=12q1+αq2q8q1+αq2γ−λO1Q1Q2+Q12Q1−λO1+O3Q18+1+αq2q+1,Λ1α,m,q,γ,λ=4Q2−λO1+O3Q122Q1+1+αq2q+1−δ2q1+αq2qγ−λO1Q12q1+αq2,Λ2α,m,q,γ,λ=λO1+O3Q12−4Q22Q1+δ2q1+αq2q−1+αq2q+1γ−λO1Q12q1+αq2,and Q1andQ2 are given by (23) and (24). This result cannot be improved. Then, fς∈m−UMqα,γ,λ implies there exists a Schwarz function wς such that(51)Jqα,f;ς=O1γ+O3pmwς−γ−1O1O1λ+O3pmwς−λ−1O1.Using the representation forpmς in Lemma 3 and the relationship between wς and pς=1+c1ς+c2ς2+⋯∈P, we can write(52)wς=pς−1pς+1=c12ς+c2−c122ς2+⋯,pmς=1+c1Q12ς+2c2−c12Q1+c12Q24ς2+⋯.Thus,(53)Jqα,f;ς=4+γO1+O3c1Q1/2ς+γO1+O3/42c2−c12Q1+c21Q2ς2+⋯4+λO1+O3c1Q1/2ς+λO1+O3/42c2−c12Q1+c21Q2ς2+⋯=1+γ−λc1O1Q18ς+γ−λ16O12c2−c12Q1+c12Q2−γ−λλO1+O3c12O1Q1264ς2+⋯.However,(54)Jqα,f;ς=1−αςDqfςfς+αDqςDqfςDqfς=1+q1+αqa2ς+q2q1+αq2qa3−1+αq2q+1a22ς2+⋯.On comparing the coefficients ofς and ς2 of (53) and (54), we obtain(55)a2=γ−λc1O1Q18q1+αq,a3=1q2q1+αq2qγ−λ16O12c2−c12Q1+c12Q2−γ−λλO1+O3c12O1Q1264+1+αq2q+1a22q2q1+αq2q.Now, for a real numberδ, we have(56)a3−δa22=1q2q1+αq2qγ−λ8O1Q1c2+Q2−Q12−λO1+O3Q128c12+γ−λO1Q12c12γ−B8O11+αq2q+1−δ2q1+αq2q8q21+αq22q1+αq2q=γ−λO1Q18q2q1+αq2q|c2−Q1−Q22Q1+λO1+O3Q18−1+αq2q+1−δ2q1+αq2qγ−λO1Q18q1+αq2c12|=γ−λO1Q18q2q1+αq2qc2−βc12,where(57)β=Q1−Q22Q1+λO1+O3Q18−1+αq2q+1−δ2q1+αq2qγ−λO1Q18q1+αq2.Hence, the result follows from Lemma2.Ifα=0 and 0<m<1, then Theorem 3 reduces to Theorem 10 in [17].Corollary 9.
Let0<m<1 and f∈m−USTqγ,λ be of form (1). Then, for any real number δ, we have(58)a3−δa22≤γ−λT24q1−m2×Λ1m,q,γ,λ,δ<λ1,2,λ1≤δ≤λ2,Λ2m,q,γ,λ,δ>λ2,where(59)λ1=4q1−m2γ−λ1+q2T2T2−16−T2λ1+q+3−q41−m2+11+q,where(60)λ2=4q1−m2γ−λ1+q2T2T2+56−T2λ1+q+3−q41−m2+11+q,Λ1m,q,γ,λ=22+T23−T2λ1+q+3−q1−m2+T21−δ1+q1+qγ−λq1−m2,Λ2m,q,γ,λ=T2λ1+q+3−q1−m2−22+T23+T2δ1+q−11+qγ−λq1−m2.The result is sharp.Proof.
The proof is straightforward from Theorem3 and Lemma 1.
Ifα=0 in Theorem 3, the case m=1 is contained in the following corollary.Corollary 10.
Letm=1, and fς of form (1) belongs to m−USTqγ,λ. Then, for a real number δ, we have the following sharp inequality:(61)a3−δa22≤γ−λqπ2×Λ1q,γ,λ,δ<ν1,2,ν1≤δ≤ν2,Λ2q,γ,λ,δ>ν2,where(62)ν1=11+q−qπ2γ−λ1+q216+λ1+q+3−qπ2,where(63)ν2=11+q−qπ2γ−λ1+q256−λ1+q+3−qπ2,Λ1q,γ,λ=43−4λ1+q+3−qπ2+1−δ1+q1+qγ−λqπ2,Λ2m,q,γ,λ=4λ1+q+3−qπ2−43+δ1+q−11+qγ−λqπ2.The result is sharp.Theorem3 becomes Theorem 1 in [28] when α=m=0.Corollary 11.
Letfς of the series representation (1) be in STqγ,λ. Then, we have the sharp inequality:(64)a3−δa22≤γ−λ4q2×γ−λ+γ−2λ−1q+1−λq2−δγ−λ1+q2,δ<σ1,2q,σ1≤δ≤σ2,λ−γ+2λ−γ+1q+λ−1q2+δγ−λ1+q2,δ>σ2,where(65)σ1=11+q1−qλ1+q+3−qγ−λ1+q,σ2=σ1+4qγ−λ1+q2.


## 4.3. Covering Theorems

Theorem 4.
The range of every univalent functionsf∈m−UMqα,γ,λ contains the disc:(66)ς<4q1+αq8q1+αq+γ−λO1Q1.Proof.
From the proof of Theorem3, we can see that(67)a2≤γ−λO1Q14q1+αq.
Since the Koebe one-quater theorem asserted that each omitted valuew of the univalent function fς of form (1) satisfies(68)w>12+a2≥4q1+αq8q1+αq+γ−λO1Q1.
Thus, we have the result.Corollary 12.
The range of every univalent functionf∈STqγ,λ contains the disc:(69)ς<2q4q+γ−λO1.Theorem 5.
The range of every univalent functionf∈m−UQqα,γ,λ contains the same disc given by (66).Proof.
Letwς be a Schwarz function. We note first in Theorem 3 that(70)O1γ+O3pmwς−γ−1O1O1λ+O3pmwς−λ−1O1=1+γ1ς+γ2ς2+⋯,where(71)γ1=γ−λc1O1Q18,γ2=γ−λ16O12c2−c12Q1+c12Q2−γ−λλO1+O3c12O1Q1264.
Sincef∈m−UQqα,γ,λ, then for some gς=ς+b2ς2+b3ς3+⋯∈m−UCVqγ,λ, we have(72)1−αDqfς+αDqςDqfς=O1γ+O3pmwς−γ−1O1O1λ+O3pmwς−λ−1O1Dqgς,which in turn implies(73)1+2q1+αqa2ς+⋯=1+γ1+2qb2ς⋯.
It is easy to see that(74)b2=γ1q2q.
Therefore, comparing the coefficients ofς of (73) and applying (74), we obtain(75)a2=γ1q1+αq,such that(76)a2≤γ−λO1Q14q1+αq.
Now, proceeding the same way as in the proof of Theorem4, we have the required result.


## 5. Conclusion

Using the concept ofq-calculus, we have introduced some new subclasses of analytic functions in the unit disc related to Janowski class of functions. In addition, sufficient conditions, Fekete–Szegö inequality as well as covering results for functions belonging to these new classes were established. Consequently, many remarkable special cases of our findings which were studied in the previous work were obtained [19, 26].



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