JMATHJournal of Mathematics2314-47852314-4629Hindawi10.1155/2021/55878865587886Research ArticleCertain Class of Analytic Functions Connected with q-Analogue of the Bessel Functionhttps://orcid.org/0000-0003-3283-4870AlessaNazek1VenkateswarluB.2https://orcid.org/0000-0002-6435-2916LoganathanK.3KarthikT.S.4ReddyP. Thirupathi5SujathaG.2LiuMing-Sheng1Department of Mathematical SciencesFaculty of SciencePrincess Nourah Bint Abdulrahman UniversityRiyadhSaudi Arabiapnu.edu.sa2Department of MathematicsGSSGITAM UniversityBengaluru RuralDoddaballapur 562 163KarnatakaIndiagitam.edu3Research and Development WingLive4ResearchTiruppur 638 106TamilnaduIndiasmvdu.ac.in4Department of Electronics and Communication EngineeringAditya College of Engineering and TechnologySurampalem 533 437Andhra PradeshIndiaacet.ac.in5Department of MathematicsKakatiya UniversityWarangal 506 009TelanganaIndiakakatiya.ac.in20211342021202120220211932021273202113420212021Copyright © 2021 Nazek Alessa et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The focus of this article is the introduction of a new subclass of analytic functions involving q-analogue of the Bessel function and obtained coefficient inequities, growth and distortion properties, radii of close-to-convexity, and starlikeness, as well as convex linear combination. Furthermore, we discussed partial sums, convolution, and neighborhood properties for this defined class.

Princess Nourah Bint Abdulrahman University
1. Introduction

Let A specify the category of analytic functions and η represent on the unit disc Δ=w:w<1 with normalization η0=0 and η0=1 such that a function has the extension of the Taylor series on the origin in the form(1)ηw=w+ν=2aνwν.

Indicated by S, the subclass of A is composed of functions that are univalent in Δ.

Then, a ηw function of A is known as starlike and convex of order ϑ if it delights the pursing(2)wηwηw>ϑ,wΔ,1+wηwηw>ϑ,wΔ,for specific ϑ0ϑ<1, respectively, and we express by Sϑ and Kϑ the subclass of A, which is expressed by the aforesaid functions, respectively. Also, indicated by T, the subclass of A is made up of functions of this form(3)ηw=wν=2aνwν,aν0,wΔ,and let Tϑ=TSϑ,Cϑ=TKϑ. There are interesting properties in the Tϑ and Cϑ classes which were thoroughly studied by Silverman  and Alessa et al. .

The intense devotion of scientists has recently fascinated the study of the q-calculus. The great focus in many fields of mathematics and physics is due to its benefits. In the analysis of many subclasses of analytic functions, the importance of the q-derivative operator Dq is very evident from its applications.

The concept of q-star functions was originally proposed by Ismail et al.  in the year 1990. However, in the sense of Geometric Function Theory, a firm basis of the use of the q-calculus was effectively developed. For example, in the rotational flow of Burge’s fluid flowing through an unbounded round channel , it is used to derive velocity and stress.

After that, numerous mathematicians have carried out remarkable studies, which play a significant role in the development of geometric function theory. Furthermore, Srivastava  recently published a survey-cum-expository analysis article that could be useful for researchers and scholars working on these topics. The mathematical description and applications of the fractional q-calculus and fractional q-derivative operators in geometric function theory were systematically investigated in this survey-cum-expository analysis article . In particular, Srivastava et al.  also considered some function groups of conical region related q-star like functions. For other recent investigations involving the q-calculus, one may refer to .

One of the most significant special functions is the Bessel function. As a result, it is important for solving a wide range of problems in engineering, physics, and mathematics (see ). In recent years, several researchers have focused their efforts on forming different types of relationships. Many researchers have recently focused on determining the various conditions under which a Bessel function has geometric properties such as close-to-convexity, starlikeness, and convexity in the frame of a unit disc Δ.

The first-order Bessel function is defined by the infinite series :(4)Jwν=01νw/22ν+ν!Γν++1,w,,where Γ stands for a function of Gamma. Szasz and Kupan  and Thirupathi Reddy and Venkateswarlu  have recently explored the univalence of the first-kind normalization Bessel function k:Δ defined by(5)kw2Γ+1w1/2Jw1/2=w+ν=21ν1Γ+14ν1ν1!Γν+wν,wΔ,.

For 0<q<1, El-Deeb and Bulboaca  defined the q-derivative k operator as follows:(6)qkw=qw+ν=21ν1Γ+14ν1ν1!Γν+wνkqwkwwq1=1+ν=21ν1Γ+14ν1ν1!Γν+ν,qwν1,wΔ,where(7)ν,q1qν1q=1+j=1ν1qj,0,q0.

Using (7), we are going to define two products in the text:

The q-shifted factorial is given for any nonnegative integer ν:(8)ν,q1,if ν=0,1,q2,q,,k,q,if ν.

The q-generalized Pochhammer symbol for any positive number r is defined by(9)r,qν1,if ν=0,r,qr+1,q,,r+k1,,if ν.

For >0,>1, and 0<q<1, El-Deeb and Bulboaca  defined J,q:Δ by (see )(10)J,qww+ν=21ν1Γ+14ν1ν1!Γν+ν,q!+1,qν1wν,wΔ.

A simple computation shows that(11)J,qwq,+1w=wqkw,wΔ,where the function q,+1w is supplied with the function(12)q,+1ww+ν=2+1,qν1ν1,q!wν,wΔ.

El-Deeb and Bulboaca  introduced the linear operator using the definition of q-derivative along with the idea of convolutions N,q:AA defined by(13)N,qηwJ,qwηw=w+ν=2ϒν,q,aνwν,>0,>1,0<q<1,wΔ,where ϒν,q,1ν1Γ+14ν1ν1!Γν+ν,q!+1,qν1.

Remark 1 (see [<xref ref-type="bibr" rid="B18">18</xref>]).

We can easily verify from the definition relation (13) that the next relation holds for all ηA:(14)+1,qN,qηw=,qN,q+1ηw+qwq+1,qN,q+1ηw,wΔ,limq1N,qηw=J,1ηwJηw=w+ν=21ν1Γ+14ν1ν1!Γν+ν!+1ν1aνwν,wΔ.

Now, we propose a new subclass ϕ,q,ϑ of A concerning q- analogue of the Bessel function as follows.

Definition 1.

For 0<1,0ϑ<1,>0,>1 and 0<q<1, we say ηwA is in ϕ,q,ϑ if it fulfils the requirement(15)wN,qηw+w2N,qηwN,qηw>ϑ,wΔ.

Also, we indicate by Tϕ,q,ϑ=ϕ,q,ϑT.

2. Coefficient Inequalities

This section gives us an adequate requirement for a function η given by (1) to be in ϕ,q,ϑ.

Theorem 1.

A function ηA is assigned to the class ϕ,q,ϑ if(16)ν=2ν+νν1ϑϒν,q,aν1ϑ.

Proof.

Since 0ϑ<1 and 0, now if we put(17)ϱw=wN,qηw+w2N,qηwN,qηw,wΔ,rhen its just a matter of proving it ϱw1<1ϑ,wΔ.

Indeed, if ηw=wwΔ, then we have ϱw=wwΔ.

This implies (16) holds.

If ηwww=r<1, then there exist a coefficient Ων,aν0 for some ν2. The consequence is that ν=2ϒν,q,aν>0. Further note that(18)ν=2ν+νν1ϑϒν,q,aν>1ϑν=2ϒν,q,aνν=2ϒν,q,aν<1.

By (16), we obtain(19)ϱw1=ν=2ν+νν11ϒν,q,aνwν11+ν=2ϒν,q,aνwν1<ν=2ν+νν11ϒν,q,aν1ν=2ϒν,q,aνν=2ν+νν1ϑϒν,q,aν1ϑϒν,q,aν1ν=2ϒν,q,aν1ϑ1ϑν=2ϒν,q,aν1ν=2ϒν,q,aν=1ϑ,wΔ.

Hence, we obtain(20)wN,qηw+w2N,qηwN,qηw=ϱw>11ϑ=ϑ.

Then, ηϕ,q,ϑ.

Theorem 2.

Let η be given by (3). Then, the function(21)ηTϕ,q,ϑν=2ν+νν1ϑϒν,q,aν1ϑ.

Proof.

In view of Theorem 1, to examine it, ηTϕ,q,ϑ fulfils the coefficient inequality (16). If ηTϕ,q,ϑ, then the function(22)ϱw=wN,qηw+w2N,qηwN,qηw,wΔ,satisfies ϱw>ϑ. This implies that(23)N,qηw=wν=2ϒν,q,aνwν0,wΔ0.

Noting that N,qηr/r in the open interval 0,1, this is the real continuous function with η0=1, and we have(24)N,qηrr=1ν=2ϒν,q,aνrν1>0,0<r<1.

Now,(25)ϑ<ϱr=1ν=2ν+νν1ϒν,q,aνrν11ν=2ϒν,q,aνrν1,and consequently, by (24), we obtain(26)ν=2ν+νν1ϑϒν,q,aνrν11ϑ.

Letting r1, we get ν=2ν+νν1ϑϒν,q,aν1ϑ.

This proves the converse part.

Remark 2.

If a function η of form (3) belongs to the class Tϕ,q,ϑ, then(27)aν1ϑν+νν1ϑϒν,q,,ν2.

The equality holds for the functions(28)ηνw=w1ϑν+νν1ϑϒν,q,wν,wΔ,ν2.

3. Distortion Theorem

In the section, the distortion limits of the functions are owned by the class Tϕ,q,ϑ.

Theorem 3.

Let ηTϕ,q,ϑ and w=r<1. Then,(29)r1ϑ2ϑ+2ϒ2,q,r2ηwr+1ϑ2ϑ+2ϒ2,q,r2,(30)121ϑ2ϑ+2ϒ2,q,rηw1+21ϑ2ϑ+2ϒ2,q,r.

The approximation is sharp, with the η2w extreme function indicated by (28).

Proof.

Since ηTϕ,q,ϑ, we apply Theorem 2 to attain(31)2ϑ+2ϒ2,q,ν=2aνν=2ν+νν1ϑϒν,q,aν1ϑ.Thus, ηww+w2ν=2aνr+1ϑ2ϑ+2ϒ2,q,r2.Also, we have ηwww2ν=2aνr1ϑ2ϑ+2ϒ2,q,r2.

Equation (29) follows. In a similar way, for η, the inequalities(32)ηw1+ν=2νaνwν11+wν=2νaν,ν=2νaν21ϑ2ϑ+2ϒ2,q,,are satisfied, which leads to (30).

4. Radii of Close-to-Convexity and Starlikeness

A close-to-convex and star-like radius of this class Tϕ,q,ϑ is obtained in this section.

Theorem 4.

Let η be specified by (3) in Tϕ,q,ϑ. Then, η is the order of close-to-convex 0<1 in the disc w<t1, where(33)t1=infν21ν+νν1ϑϒν,q,ν1ϑ1/ν1.

The estimate is sharp with the extremal function ηw is indicated by (28).

Proof.

If ηT and η is order of close-to-convex , then we obtain(34)ηw11.

For the L.H.S of (34), we obtain(35)ηw1ν=2νaνwν1<1ν=2ν1aνwν11.

We know that(36)ηwTϕ,q,ϑν=2ν+νν1ϑϒν,q,1ϑaν1.

Thus, (34) holds true if(37)ν1wν1ν+νν1ϑϒν,q,1ϑ,w1ν+νν1ϑϒν,q,ν1ϑ1/ν1,and hence proved.

Theorem 5.

Let ηTϕ,q,ϑ. Then, η is order of starlike 0<1 in the disc w<t2, where(38)t2=infν21ν+νν1ϑϒν,q,ν1ϑ1/ν1.

The estimate is sharp with the extremal function ηw indicated by (28).

Proof.

We have ηT and η is order of starlike , and we have(39)wηwηw1<1.

For the L.H.S of (39), we have(40)wηwηw1ν=2ν1aνwν11ν=2aνwν1.

1 is bigger than the R.H.S of the left relation if(41)ν=2ν1aνwν1<1.

We know that(42)ηTϕ,q,ϑν=2ν+νν1ϑϒν,q,1ϑaν1.

Thus, (39) is true if(43)ν1wν1ν+νν1ϑϒν,q,1ϑw1ν+νν1ϑϒν,q,ν1ϑ1/ν1.

It yields starlikeness of the family.

5. Convex Linear CombinationsTheorem 6.

Let η1w=w and(44)ηνw=w1ϑν+νν1ϑϒν,q,wν,wΔ,ν2.

Then, ηTϕ,q,ϑη in the way it can be expressed:(45)ηw=ν=1μνηνw,μν0,and ν=1μν=1.

Proof.

If a function η is of the form ηw=ν=1μνηνw,μν0 and ν=1μν=1, then(46)ν=2ν+νν1ϑϒν,q,aν=ν=2ν+νν1ϑϒν,q,1ϑμνν+νν1ϑϒν,q,=ν=21ϑμν=1μ11ϑ1ϑ,which provides (21), and hence ηTϕ,q,ϑ, by Theorem 2.

On the contrary, if η is in the class ηTϕ,q,ϑ, then we may set(47)μν=ν+νν1ϑϒν,q,1ϑaν,ν2,and μ1=1ν=2μν.

Then, the function η is of form (45).

6. Partial Sums

Silverman  examined partial sums η for the function ηA given by (1) and established through(48)η1w=w,ηmw=w+ν=2maνwν,m=2,3,4,.

In this paragraph, in the class ϕ,q,ϑ, partial function sums can be considered and sharp lower limits can be reached for the function. True component ratios are η to ηm and η to ηm.

Theorem 7.

Let ηϕ,q,ϑ fulfil (16). Then,(49)ηwηmw11dm+1,wΔ,m,where(50)dν=ν+νν1ϑ1ϑ.

Proof.

Clearly, dν+1>dν>1,ν=2,3,4,.

Thus, by Theorem 1, we obtain(51)ν=2aν+dm+1ν=2aνν=2dνaν1.(52)Setting gw=dm+1ηwηmw11dm+1,gw=1+dm+1ν=m+1aνwν11+ν=2maνwν1,and it is good enough to show gw>0,wΔ. Applying (51), we think that(53)gw1gw+1dm+1ν=2aν22ν=2maνdm+1ν=m+1aν1,which gives(54)ηwηmw11dm+1,and hence proved.

Theorem 8.

Let η in Tϕ,q,ϑ fulfil (16). Then,(55)ηmwηwdm+11+dm+1,wΔ,m,where(56)dν=ν+νν1ϑ1ϑ.

Proof.

Clearly, dν+1>dν>1,ν=2,3,4,.

Thus, by Theorem 1, we obtain(57)ν=2aν+dm+1ν=m+1aνν=2dνaν1,(58)Setting hw=1+dm+1ηmwηwdm+11+dm+1,hw=11+dm+1ν=m+1aνwν11+ν=2maνwν1,to show hw>0,wΔ. Implementing (57), we attain(59)hw1hw+11+dm+1ν=2aν22ν=2aν1+dm+1ν=m+1aν1,which gives(60)ηmwηwdm+11+dm+1,and hence proved.

Theorem 9.

Let η in Tϕ,q,ϑ fulfil (16). Then,(61)ηwηmw1m+1dm+1,wΔ,m,ηmwηwdm+1m+1+dm+1,wΔ,m,where(62)dν=ν+νν1ϑ1ϑ.

Proof.

By setting(63)gw=dm+1ηwηmw1m+1dm+1,wΔ,hw=m+1+dm+1ηmwηwdm+1m+1+dm+1,wΔ,the evidence is close to that of (51) and (52) theorems, so the specifics are omitted.

7. Convolution Properties

We will prove in this section that the Tϕ,q,ϑ class is closed by convolution.

Theorem 10.

Let gw of the form,(64)gw=wν=2bνwν,be regular in Δ. If ηTϕ,q,ϑ, then the function ηg is in the class Tϕ,q,ϑ. Here, the symbol denotes to the Hadamard product.

Proof.

Since ηTϕ,q,ϑ, we have(65)ν=2ν+νν1ϑϒν,q,aν1ϑ.

Employing the last inequality and the fact that(66)ηgw=wν=2aνbνwν,we obtain(67)ν=2ν+νν1ϑϒν,q,aνbνν=2ν+νν1ϑϒν,q,aν1ϑ,and hence, in view of Theorem 1, the result follows.

8. Neighborhood Property

Following [21, 22], we defined the α-neighbourhood of the function ηwT by(68)αη=gT:gw=wν=2bνwν and ν=2νaνbνα.

Definition 2.

The function ηA is defined in the class Tϕ,q,ϑ if the function hTϕ,q,ϑ occurs in such a way that the function is hTϕ,q,ϑ:(69)ηwhw1<1γ,wΔ,0γ<1.

Theorem 11.

If hTϕ,q,ϑ and(70)γ=1α2ϑ+2ϒ2,q,2ϑ+2ϒ2,q,1+ϑ,then αhTϕ,q,γ,ϑ.

Proof.

Let ηαh. We then find from (71)ν=2νaνbνα,which easily implies the coefficient inequality(72)ν=2aνbναν.

Since hTϕ,q,ϑ, we have from equation (16) that(73)ν=2aν1ϑ2ϑ+2ϒ2,q,,ηwhw1<ν=2νaνbν1ν=2bνα22ϑ+2ϒ2,q,2ϑ+2ϒ2,q,1+ϑ=1γ,and hence proved.

9. Conclusions

This research has introduced q-analogue of the Bessel function and studied some basic properties of geometric function theory. Accordingly, some results related to coefficient estimates, growth and distortion properties, convex linear combination, partial sums, radii of close-to-convexity and starlikeness, convolution, and neighborhood properties have also been considered, inviting future research for this field of study.

Data Availability

No data were used to support the findings of the study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to this work. And, all the authors have read and approved the final version manuscript.

Acknowledgments

This research was funded by the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University through the Fast-track Research Funding Program.

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