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.

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

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)ℛwη′wηw>ϑ,w∈Δ,ℛ1+wη″wη′w>ϑ,w∈Δ,for specific ϑ0≤ϑ<1, respectively, and we express by S∗ϑ 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=w−∑ν=2∞aνwν,aν≥0,w∈Δ,and let T∗ϑ=T∩S∗ϑ,Cϑ=T∩Kϑ. There are interesting properties in the T∗ϑ and Cϑ classes which were thoroughly studied by Silverman [1] and Alessa et al. [2].

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. [3] 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 [3], 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 [4] 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 [4]. In particular, Srivastava et al. [5] 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 [6–13].

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 [14]). 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 [15]:(4)J℘w≔∑ν=0∞−1νw/22ν+℘ν!Γν+℘+1,w∈ℂ,℘∈ℝ,where Γ stands for a function of Gamma. Szasz and Kupan [16] and Thirupathi Reddy and Venkateswarlu [17] have recently explored the univalence of the first-kind normalization Bessel function k℘:Δ⟶ℂ defined by(5)k℘w≔2℘Γ℘+1w1−℘/2J℘w1/2=w+∑ν=2∞−1ν−1Γ℘+14ν−1ν−1!Γν+℘wν,w∈Δ,℘∈ℝ.

For 0<q<1, El-Deeb and Bulboaca [18] defined the q-derivative k℘ operator as follows:(6)∂qk℘w=∂qw+∑ν=2∞−1ν−1Γ℘+14ν−1ν−1!Γν+℘wν≔k℘qw−k℘wwq−1=1+∑ν=2∞−1ν−1Γ℘+14ν−1ν−1!Γν+℘ν,qwν−1,w∈Δ,where(7)ν,q≔1−qν1−q=1+∑j=1ν−1qj,0,q≔0.

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,q≔1,if ν=0,1,q2,q,…,k,q,if ν∈ℕ.

The q-generalized Pochhammer symbol for any positive number r is defined by(9)r,qν≔1,if ν=0,r,qr+1,q,…,r+k−1,,if ν∈ℕ.

For ℘>0,ℓ>−1, and 0<q<1, El-Deeb and Bulboaca [18] defined J℘,qℓ:Δ⟶ℂ by (see [19])(10)J℘,qℓw≔w+∑ν=2∞−1ν−1Γ℘+14ν−1ν−1!Γν+℘ν,q!ℓ+1,qν−1wν,w∈Δ.

A simple computation shows that(11)J℘,qℓw∗ℳq,ℓ+1w=w∂qk℘w,w∈Δ,where the function ℳq,ℓ+1w is supplied with the function(12)ℳq,ℓ+1w≔w+∑ν=2∞ℓ+1,qν−1ν−1,q!wν,w∈Δ.

El-Deeb and Bulboaca [18] introduced the linear operator using the definition of q-derivative along with the idea of convolutions N℘,qℓ:A⟶A defined by(13)N℘,qℓηw≔J℘,qℓw∗ηw=w+∑ν=2∞ϒν,q℘,ℓaνwν,℘>0,ℓ>−1,0<q<1,w∈Δ,where ϒν,q℘,ℓ≔−1ν−1Γ℘+14ν−1ν−1!Γν+℘ν,q!ℓ+1,qν−1.

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 η∈A:(14)ℓ+1,qN℘,qℓηw=ℓ,qN℘,qℓ+1ηw+qℓw∂qℓ+1,qN℘,qℓ+1ηw,w∈Δ,limq⟶1−N℘,qℓηw=J℘,1ℓηw≔J℘ℓηw=w+∑ν=2∞−1ν−1Γ℘+14ν−1ν−1!Γν+℘ν!ℓ+1ν−1aνwν,w∈Δ.

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

Definition 1.

For 0≤ℏ<1,0≤ϑ<1,℘>0,ℓ>−1 and 0<q<1, we say ηw∈A is in ϕ℘,qℓℏ,ϑ if it fulfils the requirement(15)ℛwN℘,qℓηw′+ℏw2N℘,qℓηw′′N℘,qℓηw>ϑ,w∈Δ.

Also, we indicate by Tϕ℘,qℓℏ,ϑ=ϕ℘,qℓℏ,ϑ∩T.

2. Coefficient Inequalities

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

Theorem 1.

A function η∈A is assigned to the class ϕ℘,qℓℏ,ϑ if(16)∑ν=2∞ν+ℏνν−1−ϑϒν,q℘,ℓaν≤1−ϑ.

Proof.

Since 0≤ϑ<1 and ℏ≥0, now if we put(17)ϱw=wN℘,qℓηw′+ℏw2N℘,qℓηw′′N℘,qℓηw,w∈Δ,rhen its just a matter of proving it ϱw−1<1−ϑ,w∈Δ.

Indeed, if ηw=ww∈Δ, then we have ϱw=ww∈Δ.

This implies (16) holds.

If ηw≠ww=r<1, then there exist a coefficient Ων℘,ℓaν≠0 for some ν≥2. The consequence is that ∑ν=2∞ϒν,q℘,ℓaν>0. Further note that(18)∑ν=2∞ν+ℏνν−1−ϑϒν,q℘,ℓaν>1−ϑ∑ν=2∞ϒν,q℘,ℓaν⟹∑ν=2∞ϒν,q℘,ℓaν<1.

By (16), we obtain(19)ϱw−1=∑ν=2∞ν+ℏνν−1−1ϒν,q℘,ℓaνwν−11+∑ν=2∞ϒν,q℘,ℓaνwν−1<∑ν=2∞ν+ℏνν−1−1ϒν,q℘,ℓaν1−∑ν=2∞ϒν,q℘,ℓaν≤∑ν=2∞ν+ℏνν−1−ϑϒν,q℘,ℓaν−1−ϑϒν,q℘,ℓaν1−∑ν=2∞ϒν,q℘,ℓaν≤1−ϑ−1−ϑ∑ν=2∞ϒν,q℘,ℓaν1−∑ν=2∞ϒν,q℘,ℓaν=1−ϑ,w∈Δ.

Hence, we obtain(20)ℛwN℘,qℓηw′+ℏw2N℘,qℓηw′′N℘,qℓηw=ℛϱw>1−1−ϑ=ϑ.

Then, η∈ϕ℘,qℓℏ,ϑ.

Theorem 2.

Let η be given by (3). Then, the function(21)η∈Tϕ℘,qℓℏ,ϑ⟺∑ν=2∞ν+ℏνν−1−ϑϒν,q℘,ℓaν≤1−ϑ.

Proof.

In view of Theorem 1, to examine it, η∈Tϕ℘,qℓℏ,ϑ fulfils the coefficient inequality (16). If η∈Tϕ℘,qℓℏ,ϑ, then the function(22)ϱw=wN℘,qℓηw′+ℏw2N℘,qℓηw′′N℘,qℓηw,w∈Δ,satisfies ℛϱw>ϑ. This implies that(23)N℘,qℓηw=w−∑ν=2∞ϒν,q℘,ℓaνwν≠0,w∈Δ∖0.

Noting that N℘,qℓηr/r in the open interval 0,1, this is the real continuous function with η0=1, and we have(24)N℘,qℓηrr=1−∑ν=2∞ϒν,q℘,ℓaνrν−1>0,0<r<1.

Now,(25)ϑ<ϱr=1−∑ν=2∞ν+ℏνν−1ϒν,q℘,ℓaνrν−11−∑ν=2∞ϒν,q℘,ℓaνrν−1,and consequently, by (24), we obtain(26)∑ν=2∞ν+ℏνν−1−ϑϒν,q℘,ℓaνrν−1≤1−ϑ.

Letting r⟶1, we get ∑ν=2∞ν+ℏνν−1−ϑϒν,q℘,ℓaν≤1−ϑ.

This proves the converse part.

Remark 2.

If a function η of form (3) belongs to the class Tϕ℘,qℓℏ,ϑ, then(27)aν≤1−ϑν+ℏνν−1−ϑϒν,q℘,ℓ,ν≥2.

The equality holds for the functions(28)ηνw=w−1−ϑν+ℏνν−1−ϑϒν,q℘,ℓwν,w∈Δ,ν≥2.

3. Distortion Theorem

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

Theorem 3.

Let η∈Tϕ℘,qℓℏ,ϑ and w=r<1. Then,(29)r−1−ϑ2ℏ−ϑ+2ϒ2,q℘,ℓr2≤ηw≤r+1−ϑ2ℏ−ϑ+2ϒ2,q℘,ℓr2,(30)1−21−ϑ2ℏ−ϑ+2ϒ2,q℘,ℓr≤η′w≤1+21−ϑ2ℏ−ϑ+2ϒ2,q℘,ℓr.

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

Proof.

Since η∈Tϕ℘,qℓℏ,ϑ, we apply Theorem 2 to attain(31)2ℏ−ϑ+2ϒ2,q℘,ℓ∑ν=2∞aν≤∑ν=2∞ν+ℏνν−1−ϑϒν,q℘,ℓaν≤1−ϑ.Thus, ηw≤w+w2∑ν=2∞aν≤r+1−ϑ2ℏ−ϑ+2ϒ2,q℘,ℓr2.Also, we have ηw≤w−w2∑ν=2∞aν≤r−1−ϑ2ℏ−ϑ+2ϒ2,q℘,ℓr2.

Equation (29) follows. In a similar way, for η′, the inequalities(32)η′w≤1+∑ν=2∞νaνwν−1≤1+w∑ν=2∞νaν,∑ν=2∞νaν≤21−ϑ2ℏ−ϑ+2ϒ2,q℘,ℓ,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℘,qℓℏ,ϑ is obtained in this section.

Theorem 4.

Let η be specified by (3) in Tϕ℘,qℓℏ,ϑ. Then, η is the order of close-to-convex ℓ0≤ℓ<1 in the disc w<t1, where(33)t1=infν≥21−ℓν+νℏν−1−ϑϒν,q℘,ℓν1−ϑ1/ν−1.

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

Proof.

If η∈T and η is order of close-to-convex ℓ, then we obtain(34)η′w−1≤1−ℓ.

For the L.H.S of (34), we obtain(35)η′w−1≤∑ν=2∞νaνwν−1<1−ℓ⇒∑ν=2∞ν1−ℓaνwν−1≤1.

We know that(36)ηw∈Tϕ℘,qℓℏ,ϑ⟺∑ν=2∞ν+νℏν−1−ϑϒν,q℘,ℓ1−ϑaν≤1.

Let η∈Tϕ℘,qℓℏ,ϑ. Then, η is order of starlike ℓ0≤ℓ<1 in the disc w<t2, where(38)t2=infν≥21−ℓν+νℏν−1−ϑϒν,q℘,ℓν−ℓ1−ϑ1/ν−1.

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

Proof.

We have η∈T and η is order of starlike ℓ, and we have(39)wη′wηw−1<1−ℓ.

For the L.H.S of (39), we have(40)wη′wηw−1≤∑ν=2∞ν−1aνwν−11−∑ν=2∞aνwν−1.

1−ℓ is bigger than the R.H.S of the left relation if(41)∑ν=2∞ν−ℓ1−ℓaνwν−1<1.

We know that(42)η∈Tϕ℘,qℓℏ,ϑ⟺∑ν=2∞ν+νℏν−1−ϑϒν,q℘,ℓ1−ϑaν≤1.

Thus, (39) is true if(43)ν−ℓ1−ℓwν−1≤ν+νℏν−1−ϑϒν,q℘,ℓ1−ϑ⟹w≤1−ℓν+νℏν−1−ϑϒν,q℘,ℓν−ℓ1−ϑ1/ν−1.

It yields starlikeness of the family.

5. Convex Linear CombinationsTheorem 6.

Let η1w=w and(44)ηνw=w−1−ϑν+ℏνν−1−ϑϒν,q℘,ℓwν,w∈Δ,ν≥2.

Then, η∈Tϕ℘,qℓℏ,ϑ⟺η in the way it can be expressed:(45)ηw=∑ν=1∞μνηνw,μν≥0,and ∑ν=1∞μν=1.

Proof.

If a function η is of the form ηw=∑ν=1∞μνηνw,μν≥0 and ∑ν=1∞μν=1, then(46)∑ν=2∞ν+ℏνν−1−ϑϒν,q℘,ℓaν=∑ν=2∞ν+ℏνν−1−ϑϒν,q℘,ℓ1−ϑμνν+ℏνν−1−ϑϒν,q℘,ℓ=∑ν=2∞1−ϑμν=1−μ11−ϑ≤1−ϑ,which provides (21), and hence η∈Tϕ℘,qℓℏ,ϑ, by Theorem 2.

On the contrary, if η is in the class η∈Tϕ℘,qℓℏ,ϑ, then we may set(47)μν=ν+ℏνν−1−ϑϒν,q℘,ℓ1−ϑaν,ν≥2,and μ1=1−∑ν=2∞μν.

Then, the function η is of form (45).

6. Partial Sums

Silverman [20] examined partial sums η for the function η∈A given by (1) and established through(48)η1w=w,ηmw=w+∑ν=2maνwν,m=2,3,4,….

In this paragraph, in the class ϕ℘,qℓℏ,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 η∈ϕ℘,qℓℏ,ϑ fulfil (16). Then,(49)ℛηwηmw≥1−1dm+1,w∈Δ,m∈ℕ,where(50)dν=ν+ℏνν−1−ϑ1−ϑ.

Proof.

Clearly, dν+1>dν>1,ν=2,3,4,….

Thus, by Theorem 1, we obtain(51)∑ν=2∞aν+dm+1∑ν=2∞aν≤∑ν=2∞dνaν≤1.(52)Setting gw=dm+1ηwηmw−1−1dm+1,gw=1+dm+1∑ν=m+1∞aνwν−11+∑ν=2maνwν−1,and it is good enough to show ℛgw>0,w∈Δ. Applying (51), we think that(53)gw−1gw+1≤dm+1∑ν=2∞aν2−2∑ν=2maν−dm+1∑ν=m+1∞aν≤1,which gives(54)ℛηwηmw≥1−1dm+1,and hence proved.

Theorem 8.

Let η in Tϕ℘,qℓℏ,ϑ fulfil (16). Then,(55)ℛηmwηw≥dm+11+dm+1,w∈Δ,m∈ℕ,where(56)dν=ν+ℏνν−1−ϑ1−ϑ.

Proof.

Clearly, dν+1>dν>1,ν=2,3,4,….

Thus, by Theorem 1, we obtain(57)∑ν=2∞aν+dm+1∑ν=m+1∞aν≤∑ν=2∞dνaν≤1,(58)Setting hw=1+dm+1ηmwηw−dm+11+dm+1,hw=1−1+dm+1∑ν=m+1∞aνwν−11+∑ν=2maνwν−1,to show ℛhw>0,w∈Δ. Implementing (57), we attain(59)hw−1hw+1≤1+dm+1∑ν=2∞aν2−2∑ν=2∞aν−1+dm+1∑ν=m+1∞aν≤1,which gives(60)ℛηmwηw≥dm+11+dm+1,and hence proved.

Theorem 9.

Let η in Tϕ℘,qℓℏ,ϑ fulfil (16). Then,(61)ℛη′wηm′w≥1−m+1dm+1,w∈Δ,m∈ℕ,ℛηm′wη′w≥dm+1m+1+dm+1,w∈Δ,m∈ℕ,where(62)dν=ν+ℏνν−1−ϑ1−ϑ.

Proof.

By setting(63)gw=dm+1η′wηm′w−1−m+1dm+1,w∈Δ,hw=m+1+dm+1ηm′wη′w−dm+1m+1+dm+1,w∈Δ,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℘,qℓℏ,ϑ class is closed by convolution.

Theorem 10.

Let gw of the form,(64)gw=w−∑ν=2∞bνwν,be regular in Δ. If η∈Tϕ℘,qℓℏ,ϑ, then the function η∗g is in the class Tϕ℘,qℓℏ,ϑ. Here, the symbol ∗ denotes to the Hadamard product.

Proof.

Since η∈Tϕ℘,qℓℏ,ϑ, we have(65)∑ν=2∞ν+ℏνν−1−ϑϒν,q℘,ℓaν≤1−ϑ.

Employing the last inequality and the fact that(66)η∗gw=w−∑ν=2∞aνbνwν,we obtain(67)∑ν=2∞ν+ℏνν−1−ϑϒν,q℘,ℓaνbν≤∑ν=2ν+ℏνν−1−ϑϒν,q℘,ℓaν≤1−ϑ,and hence, in view of Theorem 1, the result follows.

8. Neighborhood Property

Following [21, 22], we defined the α-neighbourhood of the function ηw∈T by(68)ℵαη=g∈T:gw=w−∑ν=2∞bνwν and ∑ν=2∞νaν−bν≤α.

Definition 2.

The function η∈A is defined in the class Tϕ℘,qℓℏ,ϑ if the function h∈Tϕ℘,qℓℏ,ϑ occurs in such a way that the function is h∈Tϕ℘,qℓℏ,ϑ:(69)ηwhw−1<1−γ,w∈Δ,0≤γ<1.

Theorem 11.

If h∈Tϕ℘,qℓℏ,ϑ and(70)γ=1−α2ℏ−ϑ+2ϒ2,q℘,ℓ2ℏ−ϑ+2ϒ2,q℘,ℓ−1+ϑ,then ℵαh⊆Tϕ℘,qℓ,γℏ,ϑ.

Proof.

Let η∈ℵαh. We then find from (71)∑ν=2∞νaν−bν≤α,which easily implies the coefficient inequality(72)∑ν=2∞aν−bν≤αν.

Since h∈Tϕ℘,qℓℏ,ϑ, we have from equation (16) that(73)∑ν=2∞aν≤1−ϑ2ℏ−ϑ+2ϒ2,q℘,ℓ,ηwhw−1<∑ν=2∞νaν−bν1−∑ν=2∞bν≤α22ℏ−ϑ+2ϒ2,q℘,ℓ2ℏ−ϑ+2ϒ2,q℘,ℓ−1+ϑ=1−γ,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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