Research article

On some geometric properties and Hardy class of q-Bessel functions

  • Received: 12 December 2019 Accepted: 18 March 2020 Published: 25 March 2020
  • MSC : 30C45, 33C10

  • In this paper, we deal with some geometric properties including starlikeness and convexity of order α of Jackson's second and third q-Bessel functions which are natural extensions of classical Bessel function Jν. In additon, we determine some conditions on the parameters such that Jackson's second and third q-Bessel functions belong to the Hardy space and to the class of bounded analytic functions.

    Citation: İbrahim Aktaş. On some geometric properties and Hardy class of q-Bessel functions[J]. AIMS Mathematics, 2020, 5(4): 3156-3168. doi: 10.3934/math.2020203

    Related Papers:

    [1] Pinhong Long, Huo Tang, Wenshuai Wang . Functional inequalities for several classes of q-starlike and q-convex type analytic and multivalent functions using a generalized Bernardi integral operator. AIMS Mathematics, 2021, 6(2): 1191-1208. doi: 10.3934/math.2021073
    [2] Suriyakamol Thongjob, Kamsing Nonlaopon, Sortiris K. Ntouyas . Some (p, q)-Hardy type inequalities for (p, q)-integrable functions. AIMS Mathematics, 2021, 6(1): 77-89. doi: 10.3934/math.2021006
    [3] Muhammad Sabil Ur Rehman, Qazi Zahoor Ahmad, H. M. Srivastava, Nazar Khan, Maslina Darus, Bilal Khan . Applications of higher-order q-derivatives to the subclass of q-starlike functions associated with the Janowski functions. AIMS Mathematics, 2021, 6(2): 1110-1125. doi: 10.3934/math.2021067
    [4] M. Emin Özdemir, Saad I. Butt, Bahtiyar Bayraktar, Jamshed Nasir . Several integral inequalities for (α, s,m)-convex functions. AIMS Mathematics, 2020, 5(4): 3906-3921. doi: 10.3934/math.2020253
    [5] Chunhong Li, Dandan Yang, Chuanzhi Bai . Some Opial type inequalities in (p, q)-calculus. AIMS Mathematics, 2020, 5(6): 5893-5902. doi: 10.3934/math.2020377
    [6] Khadeejah Rasheed Alhindi, Khalid M. K. Alshammari, Huda Ali Aldweby . Classes of analytic functions involving the q-Ruschweyh operator and q-Bernardi operator. AIMS Mathematics, 2024, 9(11): 33301-33313. doi: 10.3934/math.20241589
    [7] Muhammad Amer Latif, Mehmet Kunt, Sever Silvestru Dragomir, İmdat İşcan . Post-quantum trapezoid type inequalities. AIMS Mathematics, 2020, 5(4): 4011-4026. doi: 10.3934/math.2020258
    [8] Arslan Hojat Ansari, Sumit Chandok, Liliana Guran, Shahrokh Farhadabadi, Dong Yun Shin, Choonkil Park . (F, h)-upper class type functions for cyclic admissible contractions in metric spaces. AIMS Mathematics, 2020, 5(5): 4853-4873. doi: 10.3934/math.2020310
    [9] Muhammad Uzair Awan, Nousheen Akhtar, Artion Kashuri, Muhammad Aslam Noor, Yu-Ming Chu . 2D approximately reciprocal ρ-convex functions and associated integral inequalities. AIMS Mathematics, 2020, 5(5): 4662-4680. doi: 10.3934/math.2020299
    [10] Georgia Irina Oros, Gheorghe Oros, Daniela Andrada Bardac-Vlada . Certain geometric properties of the fractional integral of the Bessel function of the first kind. AIMS Mathematics, 2024, 9(3): 7095-7110. doi: 10.3934/math.2024346
  • In this paper, we deal with some geometric properties including starlikeness and convexity of order α of Jackson's second and third q-Bessel functions which are natural extensions of classical Bessel function Jν. In additon, we determine some conditions on the parameters such that Jackson's second and third q-Bessel functions belong to the Hardy space and to the class of bounded analytic functions.


    Special functions appears in many branches of mathematics and applied sciences. One of the most important special functions is Bessel function of the first kind Jν. The Bessel function of the first kind Jν is a particular solution of the following homogeneous differential equation:

    z2w(z)+zw(z)+(z2ν2)w(z)=0,

    which is known as Bessel differential equation. Also, the function Jν has the following power series representation:

    Jν(z)=n0(1)n(z2)2n+νn!Γ(n+ν+1),

    where Γ(z) denotes Euler's gamma function. Comprehensive information about the Bessel function can be found in Watson's treatise [1]. On the other hand, there are some q-analogues of the Bessel functions in the literature. At the beginning of the 19 century, with the help of the q-calculus, famous English mathematician Frank Hilton Jackson has defined some functions which are known as Jackson's q-Bessel functions. The Jackson's second and third q-Bessel functions are defined by (see [2,3,4])

    J(2)ν(z;q)=(qν+1;q)(q;q)n0(1)n(z2)2n+ν(q;q)n(qν+1;q)nqn(n+ν) (1.1)

    and

    J(3)ν(z;q)=(qν+1;q)(q;q)n0(1)nz2n+ν(q;q)n(qν+1;q)nq12n(n+1), (1.2)

    where zC,ν>1,q(0,1) and

    (a;q)0=1,(a;q)n=nk=1(1aqk1),(a,q)=k1(1aqk1).

    Here we would like to say that Jackson's third q-Bessel function is also known as Hahn-Exton q-Bessel function due to their contributions to the theory of qBessel functions (see [5,6]). In addition, it is known from [2] that these q-analogues satisfy the following limit relations:

    limq1J(2)ν((1q)z;q)=Jν(z) and limq1J(3)ν((1q)z;q)=Jν(2z).

    Also, it is important to mention here that the notations (1.1) and (1.2) are from Ismail (see [3,7]) and they are different from the Jackson's original notations. Results on the properties of Jackson's second and third q-Bessel functions may be found in [2,6,7,8,9,10,11,12,13,14,15] and the references therein, comprehensively.

    This paper is organized as follow: The rest of this section is devoted to some basic concepts and results needed for the proof of our main results. In Section 2, we deal with some geometric properties including starlikeness and convexity of order α of normalized q-Bessel functions. Also, we present some results regarding Hardy space of normalized q-Bessel functions.

    Let U={zC:|z|<1} be the open unit disk and H be the set of all analytic functions on the open unit disk U. We denote by A the class of analytic functions f:UC, normalized by

    f(z)=z+n2anzn. (1.3)

    By S we mean the class of functions belonging to A which are univalent in U. Also, for 0α<1, S(α) and C(α) denote the subclasses of S consisting of all functions in A which are starlike of order α and convex of order α in the open unit disk U, respectively. When α=0, we denote the classes S(α) and C(α) by S and C, respectively. The analytic characterizations of these subclasses can be given as follows:

    S(α)={f:fA and (zf(z)f(z))>α for zU} (1.4)

    and

    C(α)={f:fA and (1+zf(z)f(z))>α for zU}, (1.5)

    respectively. In [16], for α<1, the author introduced the classes:

    P(α)={pH:ηR such that p(0)=1,[eiη(p(z)α)]>0,zU} (1.6)

    and

    R(α)={fA:ηR such that [eiη(p(z)α)]>0,zU}. (1.7)

    When η=0, the classes P(α) and R(α) will be denoted by P0(α) and R0(α), respectively. Also, for α=0 we denote P0(α) and R0(α) simply by P and R, respectively. In addition, the Hadamard product (or convolution) of two power series

    f1(z)=z+n2anzn and f2(z)=z+n2bnzn,

    is defined by

    (f1f2)(z)=z+n2anbnzn.

    Let Hp (0<p) denote the Hardy space of all analytic functions f(z) in U, and define the integral means Mp(r,f) by

    Mp(r,f)={(12π2π0|f(reiθ)|pdθ)1p, if 0<p<sup0θ<2π|f(reiθ)|, if p=. (1.8)

    An analytic function f(z) in U, is said to belong to the Hardy space Hp where 0<p, if the set {Mp(r,f):r[0,1)} is bounded. It is important to remind here that Hp is a Banach space with the norm defined by (see [17])

    ||f||p=limr1Mp(r,f)

    for 1p. On the other hand, we know that H is the class of bounded analytic functions in U, while H2 is the class of power series anzn such that |an|2<. In addition, it is known from [17] that Hq is a subset of Hp for 0<pq. Also, two well-knonw results about the Hardy space Hp are the following (see [17]):

    f(z)>0{fHq,q<1fHq1q,q(0,1). (1.9)

    In the recent years, the authors in [18,19,20] proved several interesting results involving univalence, starlikeness, convexity and close-to-convexity of functions fA. Later on, many authors investigated geometric properties of some special functions such as Bessel, Struve, Lommel, Mittag-Leffler and Wright by using the above mentioned results. Furthermore, Eenigenburg and Keogh determined some conditions on the convex, starlike and close-to-convex functions to belong to the Hardy space Hp in [21]. On the other hand, the authors in [16,22,23,24,25,26,27] studied the Hardy space of some special functions (like Hypergeometric, Bessel, Struve, Lommel and Mittag-Leffler) and analytic function families.

    Motivated by the above studies, our main aim is to determine some conditions on the parameters such that Jackson's second and third qBessel functions are starlike of order α and convex of order α, respectively. Also, we find some conditions for the hadamard products h(2)ν(z;q)f(z) and h(3)ν(z;q)f(z) to belong to HR, where h(k)ν(z;q) are Jackson's normalized q-Bessel functions which are given by (2.1) and (2.2) for k{2,3}, and f is an analytic function in R. Morever, we investigate the Hardy space of the mentioned qBessel functions.

    The next results will be used in order to prove several theorems.

    Lemma 1. [19] Let f is of the form (1.3). If

    n=2(nα)|an|1α, (1.10)

    then the function f(z) is in the class S(α).

    Lemma 2. [19] Let f is of the form (1.3). If

    n=2n(nα)|an|1α, (1.11)

    then the function f(z) is in the class C(α).

    Lemma 3. [21] Let α[0,1). If the function fC(α) is not of the form

    {f(z)=k+lz(1zeiθ)2α1,α12f(z)=k+llog(1zeiθ),α=12 (1.12)

    for some k,lC and θR, then the following statements hold:

    a. There exist δ=δ(f)>0 such that fHδ+12(1α).

    b. If α[0,12), then there exist τ=τ(f)>0 such that fHτ+112α.

    c. If α12, then fH.

    Lemma 4. [28] P0(α)P0(β)P0(γ), where γ=12(1α)(1β). The value of γ is the best possible.

    In addition to the above Lemmas, in proving our assertions we will use some inequalities and series sums. It is easy to see that the following inequalites

    q(n1)(n1+ν)q(n1)ν, (1.13)
    q12(n1)nq12(n1), (1.14)
    (q;q)n1=n1k=1(1qk)>(1q)n1, (1.15)

    and

    (qν+1;q)n1=n1k=1(1qν+k)>(1qν)n1 (1.16)

    hold true for n2, q(0,1) and ν>1. Furthermore, it can be easily shown that the following series sums

    n2rn1=r1r, (1.17)
    n2nrn1=r(2r)(1r)2, (1.18)
    n2n2rn1=r(r23r+4)(1r)3 (1.19)

    and

    n2rnn=log11rr (1.20)

    hold true for |r|<1.

    In this section we present our main results. Due to the functions defined by (1.1) and (1.2) do not belong to the class A, we consider following normalized forms of the q-Bessel functions:

    h(2)ν(z;q)=2νcν(q)z1ν2J(2)ν(z;q)=z+n2(1)n1q(n1)(n1+ν)4n1(q;q)n1(qν+1;q)n1zn (2.1)

    and

    h(3)ν(z;q)=cν(q)z1ν2J(3)ν(z;q)=z+n2(1)n1q12(n1)n(q;q)n1(qν+1;q)n1zn, (2.2)

    where cν(q)=(q;q)/(qν+1;q). As a result, these functions are in the class A. Now, we are ready to present our main results related to the some geometric properties and Hardy class of Jackson's second and third q-Bessel functions.

    Theorem 1. Let α[0,1), q(0,1), ν>1 and

    4(1q)(1qν)qν>0. (2.3)

    The following assertions are true:

    a. If the inequality

    αq2ν+8(1q)2(1qν)28qν(1q)(1qν)q2ν+8(1q)2(1qν)26qν(1q)(1qν) (2.4)

    holds, then the normalized qBessel function zh(2)ν(z;q) is starlike of order α in U.

    b. If the inequality

    α2q3ν24q2ν(1q)(1qν)+112qν(1q)2(1qν)264(1q)3(1qν)32q3ν24q2ν(1q)(1qν)+80qν(1q)2(1qν)264(1q)3(1qν)3 (2.5)

    holds, then the normalized qBessel function zh(2)ν(z;q) is convex of order α in U.

    Proof. a. By virtue of the Silverman's result which is given in Lemma 1, in order to prove the starlikeness of order α of the function, zh(2)ν(z;q) it is enough to show that the following inequality

    κ1=n2(nα)|(1)n1q(n1)(n1+ν)4n1(q;q)n1(qν+1;q)n1|1α (2.6)

    holds true under the hypothesis. Considering the inequalities (1.13), (1.15) and (1.16) together with the sums (1.17) and (1.18), we may write that

    κ1=n2(nα)|(1)n1q(n1)(n1+ν)4n1(q;q)n1(qν+1;q)n1|=n2(nα)q(n1)(n1+ν)4n1(q;q)n1(qν+1;q)n1n2(nα)q(n1)ν4n1(1q)n1(1qν)n1=n2(nα)[qν4(1q)(1qν)]n1=n2n[qν4(1q)(1qν)]n1αn2[qν4(1q)(1qν)]n1=qν(8(1q)(1qν)qν)(4(1q)(1qν)qν)2αqν4(1q)(1qν)qν.

    The inequality (2.4) implies that the last sum is bounded above by 1α. As a result, κ11α and so the function zh(2)ν(z;q) is starlike of order α in U.

    b. It is known from the Lemma 2 that to prove the convexity of order α of the function zh(2)ν(z;q), it is enough to show that the following inequality

    κ2=n2n(nα)|(1)n1q(n1)(n1+ν)4n1(q;q)n1(qν+1;q)n1|1α (2.7)

    is satisfied under our assumptions. Now, if we consider the inequalities (1.13), (1.15) and (1.16) together with the sums (1.18) and (1.19) then we may write that

    κ2=n2n(nα)|(1)n1q(n1)(n1+ν)4n1(q;q)n1(qν+1;q)n1|=n2n(nα)q(n1)(n1+ν)4n1(q;q)n1(qν+1;q)n1n2(n2nα)q(n1)ν4n1(1q)n1(1qν)n1=n2(n2nα)[qν4(1q)(1qν)]n1=n2n2[qν4(1q)(1qν)]n1αn2n[qν4(1q)(1qν)]n1=qν(64(1q)2(1qν)212qν(1q)(1qν)+q2ν)(4(1q)(1qν)qν)3αqν(8(1q)(1qν)qν)(4(1q)(1qν)qν)2.

    The inequality (2.5) implies that the last sum is bounded above by 1α. As a result, κ21α and so the function zh(2)ν(z;q) is convex of order α in U.

    Theorem 2. Let α[0,1), q(0,1), ν>1 and

    (1q)(1qν)q>0. (2.8)

    The next two assertions are hold:

    a. If the inequality

    α2q(1q)(1qν)q((1q)(1qν)q)2((1q)(1qν)q)(2q(1q)(1qν)) (2.9)

    holds, then the normalized qBessel function zh(3)ν(z;q) is starlike of order α in U.

    b. If the inequality

    α4q(1q)2(1qν)23q(1q)(1qν)+qq((1q)(1qν)q)3((1q)(1qν)q)(2q(1q)(1qν)q((1q)(1qν)q)2) (2.10)

    holds, then the normalized qBessel function zh(3)ν(z;q) is convex of order α in U.

    Proof. a. By virtue of the Silverman's result which is given in Lemma 1, in order to prove the starlikeness of order α of the function, zh(3)ν(z;q) it is enough to show that the following inequality

    κ3=n2(nα)|(1)n1q12n(n1)(q;q)n1(qν+1;q)n1|1α (2.11)

    holds true under the hypothesis. Considering the inequalities (1.14), (1.15) and (1.16) together with the sums (1.17) and (1.18), we may write that

    κ3=n2(nα)|(1)n1q12n(n1)(q;q)n1(qν+1;q)n1|=n2(nα)q12n(n1)(q;q)n1(qν+1;q)n1n2(nα)q12(n1)(1q)n1(1qν)n1=n2n[q(1q)(1qν)]n1αn2[q(1q)(1qν)]n1=2q(1q)(1qν)q((1q)(1qν)q)2αq(1q)(1qν)q

    The inequality (2.9) implies that the last sum is bounded above by 1α. As a result, κ31α and so the function zh(3)ν(z;q) is starlike of order α in U.

    b. It is known from the Lemma 2 that to prove the convexity of order α of the function zh(3)ν(z;q), it is enough to show that the following inequality

    κ4=n2n(nα)|(1)n1q12n(n1)(q;q)n1(qν+1;q)n1|1α (2.12)

    is satisfied under the assumptions of Theorem 2. Now, if we consider the inequalities (1.14), (1.15) and (1.16) together with the sums (1.18) and (1.19) then we may write that

    κ4=n2n(nα)|(1)n1q12n(n1)(q;q)n1(qν+1;q)n1|=n2n(nα)q12n(n1)(q;q)n1(qν+1;q)n1n2(n2nα)(q)n1(1q)n1(1qν)n1=n2n2[q(1q)(1qν)]n1αn2n[q(1q)(1qν)]n1=4q(1q)2(1qν)23q(1q)(1qν)+qq((1q)(1qν)q)3α2q(1q)(1qν)q((1q)(1qν)q)2.

    The inequality (2.10) implies that the last sum is bounded above by 1α. As a result, κ41α and so the function zh(3)ν(z;q) is convex of order α in U.

    Theorem 3. Let α[0,1), q(0,1) and ν>1. The following assertions hold true:

    a. If the inequality (2.3) is satisfied and

    α<4(1q)(1qν)2qν4(1q)(1qν)qν, (2.13)

    then the function h(2)ν(z;q)z is in the class P0(α).

    b. If the inequality (2.8) is satisfied and

    α<(1q)(1qν)2q(1q)(1qν)q, (2.14)

    then the function h(3)ν(z;q)z is in the class P0(α).

    Proof. a. In order to prove h(2)ν(z;q)zP0(α), it is enough to show that (h(2)ν(z;q)z)>α. For this purpose, consider the function p(z)=11α(h(2)ν(z;q)zα). It can be easly seen that |p(z)1|<1 implies (h(2)ν(z;q)z)>α. Now, using the inequalities (1.13), (1.15), (1.16) and the well known geometric series sum, we have

    |p(z)1|=|11α[1+n2(1)n1q(n1)(n1+ν)4n1(q;q)n1(qν+1;q)n1zn1α]1|11αn2(qν4(1q)(1qν))n1=qν4(1α)(1q)(1qν)n0(qν4(1q)(1qν))n=qν(1α)[4(1q)(1qν)qν].

    It follows from the inequality (2.13) that |p(z)1|<1, and hence h(2)ν(z;q)zP0(α).

    b. Similarly, let define the function t(z)=11α(h(3)ν(z;q)zα). By making use of the inequalities (1.14), (1.15), (1.16) and the geometric series sum, we can write that

    |t(z)1|=|11α[1+n2(1)n1q12n(n1)(q;q)n1(qν+1;q)n1zn1α]1|11αn2(q(1q)(1qν))n1=q(1α)(1q)(1qν)n0(q(1q)(1qν))n=q(1α)[(1q)(1qν)q].

    But, the inequality (2.14) implies that |t(z)1|<1. Therefore, we get h(3)ν(z;q)z is in the class P0(α), and the proof is completed.

    Setting α=0 and α=12 in the Theorem 3, respectively, we have:

    Corollary 1. Let q(0,1) and ν>1. The next claims are true:

    i. If 2(1q)(1qν)qν>0, then h(2)ν(z;q)zP.

    ii. If (1q)(1qν)2q>0, then h(3)ν(z;q)zP.

    iii. If 4(1q)(1qν)3qν>0, then h(2)ν(z;q)zP0(12).

    vi. If (1q)(1qν)3q>0, then h(3)ν(z;q)zP0(12).

    Theorem 4. Let α[0,1), q(0,1), ν>1. If the inequalities (2.3) and (2.5) are satisfied, then h(2)ν(z;q)H112α for α[0,12) and h(2)ν(z;q)H for α(12,1).

    Proof. It is known that Gauss hypergeometric function is defined by

    2F1(a,b,c;z)=n0(a)n(b)n(c)nznn!. (2.15)

    Now, using the equality (2.15) it is possible to show that the function zh(2)ν(z;q) can not be written in the forms which are given by (1.12) for corresponding values of α. More precisely, we can write that the following equalities:

    k+lz(1zeiθ)12α=k+ln0(12α)nn!eiθnzn+1 (2.16)

    and

    k+llog(1zeiθ)=kln01n+1eiθnzn+1 (2.17)

    hold true for k,lC and θR. If we consider the series representation of the function zh(2)ν(z;q) which is given by (2.1), then we see that the function zh(2)ν(z;q) is not of the forms (2.16) for α12 and (2.17) for α=12, respectively. On the other hand, the part b. of Theorem 1 states that the function zh(2)ν(z;q) is convex of order α under hypothesis. Therefore, the proof is completed by applying Lemma 3.

    Theorem 5. Let α[0,1), q(0,1), ν>1. If the inequalities (2.8) and (2.10) are satisfied, then h(3)ν(z;q)H112α for α[0,12) and h(3)ν(z;q)H for α(12,1).

    Proof. From the power series representation of the function zh(3)ν(z;q) which is given by (2.2) it can be easily seen that this function is not of the forms (2.16) for α12 and (2.17) for α=12, respectively. Also, we know from the second part of Theorem 2 that the function zh(3)ν(z;q) is convex of order α under our asumptions. As consequences, by applying Lemma 3 we have the desired results.

    Theorem 6. Let q(0,1), ν>1 and f(z)R be of the form (1.3). The following statements are hold:

    a. If 4(1q)(1qν)3qν>0, then the hadamard product u(z)=h(2)ν(z;q)f(z)HR.

    b. If (1q)(1qν)3q>0, then the hadamard product v(z)=h(3)ν(z;q)f(z)HR.

    Proof. a. Suppose that the function f(z) is in R. Then, from the definition of the class R we can say that the function f(z) is in P. On the other hand, from the equality u(z)=h(2)ν(z;q)f(z) we can easily see that u(z)=h(2)ν(z;q)zf(z). It is known from part ⅲ. of the Corollary 1 that the function h(2)ν(z;q)zP0(12). So, it follows from Lemma 4 that u(z)P. This means that u(z)R and (u(z))>0. If we consider the result which is given by (1.9), then we have u(z)Hp for p<1 and u(z)Hq1q for 0<q<1, or equivalently, u(z)Hp for all 0<p<.

    Now, from the known upper bound for the Caratreodory functions (see [29]), we have that, if the function f(z)R, then n|an|2 for n2. Using this fact together with the inequalities (1.13), (1.15), (1.16) and the sum (1.20) we get

    |u(z)|=|h(2)ν(z;q)f(z)|=|z+n2(1)n1q(n1)(n1+ν)4n1(q;q)n1(qν+1;q)n1anzn|1+2ρ1n2ρn1n=1+2ρ1(log11ρ1ρ1),

    where ρ1=qν4(1q)(1qν). This means that the function zu(z) is convergent absolutely for |z|=1 under the hypothesis. On the other hand, we know from [17] that u(z)Hq implies the function u(z) is continuous in ¯U, where ¯U is closure of U. Since ¯U is a compact set, u(z) is bounded in U, that is, u(z)H.

    b. If f(z)R, then f(z)P. Also, v(z)=h(3)ν(z;q)f(z) implies v(z)=h(3)ν(z;q)zf(z). We known from part ⅵ. of the Corollary 1 that the function h(3)ν(z;q)zP0(12). So, by applying Lemma 4 we get v(z)P. That is, v(z)R and (v(z))>0. Now, from (1.9), we have that v(z)Hp for p<1 and v(z)Hq1q for 0<q<1, or equivalently, v(z)Hp for all 0<p<. Using the well-known upper bound for the Caratreodory functions together with the inequalities (1.14), (1.15), (1.16) and the sum (1.20) we have

    |v(z)|=|h(3)ν(z;q)f(z)|=|z+n2(1)n1q12n(n1)(q;q)n1(qν+1;q)n1anzn|1+2ρ2n2ρn2n=1+2ρ2(log11ρ2ρ2),

    where ρ2=q(1q)(1qν). So, we can say that the function zv(z) is convergent absolutely for |z|=1 under the stated conditions. Also, it is known from [17] that v(z)Hq implies the function v(z) is continuous in ¯U. Since ¯U is a compact set, we may write that v(z) is bounded in U. Hence v(z)H and the proof is completed.

    Theorem 7. Let α[0,1), β<1, γ=12(1α)(1β), q(0,1) and ν>1. Suppose that the function f(z) of the form (1.3) is in the class R0(β). The following statements hold true:

    a. If the inequalities (2.3) and (2.13) are hold, then u(z)=h(2)ν(z;q)f(z)R0(γ).

    b. If the inequalities (2.8) and (2.14) are hold, then v(z)=h(3)ν(z;q)f(z)R0(γ).

    Proof. a. If f(z)R0(β), then this implies that f(z)P0(β). We know from the first part of Theorem 3 that the function h(2)ν(z;q)z is in the class P0(α). Since u(z)=h(2)ν(z;q)zf(z), taking into acount the Lemma 4 we may write that u(z)P0(γ). This implies that u(z)R0(γ).

    b. Similarly, f(z)R0(β) implies that f(z)P0(β). It is known from the second part of Theorem 3 that the function h(3)ν(z;q)z is in the class P0(α). Using the fact that v(z)=h(3)ν(z;q)zf(z) and Lemma 4, we have v(z)P0(γ). As a result, v(z)R0(γ).

    In this paper, some geometric properties like starlikeness and convexity of order α of Jackson's second and third q-Bessel functions are investigated. Also, some immediate applications of convexity involving Jackson's second and third q-Bessel functions associated with the Hardy space of analytic functions are presented. In addition, some conditions for the mentioned functions to belong to the class of bounded analytic functions are obtained.

    The author declares that there is no conflict of interest.



    [1] G. N. Watson, A treatise on the theory of Bessel functions, Cambridge: Cambridge University Press, 1944.
    [2] M. H. Annaby, Z. S. Mansour, q-Fractional Calculus and Equations (Lecture Notes in Mathematics 2056), Berlin: Springer-Verlag, 2012.
    [3] M. E. H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge: Cambridge University Press, 2005.
    [4] F. H. Jackson, The basic Gamma-Function and the Elliptic functions, Proc. R. Soc. Lond. A, 76 (1905), 127-144. doi: 10.1098/rspa.1905.0011
    [5] F. H. Jackson, The applications of basic numbers to Bessel's and Legendre's equations, Proc. Lond. Math. Soc., 3 (1905), 1-23. doi: 10.1112/plms/s2-3.1.1
    [6] H. T. Koelink, R. F. Swarttouw, On the zeros of the Hahn-Exton q-Bessel function and associated q-Lommel polynomials, J. Math. Anal. Appl., 186 (1994), 690-710. doi: 10.1006/jmaa.1994.1327
    [7] M. E. H. Ismail, The zeros of basic Bessel function, the functions Jν+αx(x), and associated orthogonal polynomials, J. Math. Anal. Appl., 86 (1982), 1-19. doi: 10.1016/0022-247X(82)90248-7
    [8] L. D. Abreu, A q-sampling theorem related to the q-Hankel transform, Proc. Amer. Math. Soc., 133 (2005), 1197-1203. doi: 10.1090/S0002-9939-04-07589-6
    [9] İ. Aktaş, Á. Baricz, Bounds for the radii of starlikeness of some q-Bessel functions, Results Math., 72 (2017), 947-963. doi: 10.1007/s00025-017-0668-6
    [10] İ. Aktaş, H. Orhan, On Partial sums of Normalized q-Bessel Functions, Commun. Korean Math. Soc., 33 (2018), 535-547.
    [11] İ. Aktaş, H. Orhan, Bounds for the radii of convexity of some q-Bessel functions, Bull. Korean Math. Soc., 57 (2020), 355–369.
    [12] M. H. Annaby, Z. S. Mansour, O. A. Ashour, Sampling theorems associated with biorthogonal q-Bessel functions, J. Phys. A, 43 (2010), Art. No. 295204. doi: 10.1088/1751-8113/43/29/295204
    [13] M. E. H. Ismail, M. E. Muldoon, On the variation with respect to a parameter of zeros of Bessel and q-Bessel functions, J. Math. Anal. Appl., 135 (1988), 187-207. doi: 10.1016/0022-247X(88)90148-5
    [14] T. H. Koornwinder, R. F. Swarttouw, On q-analogues of the Hankel and Fourier transforms, Trans. Amer. Math. Soc., 333 (1992), 445-461.
    [15] Á. Baricz, D. K. Dimitrov, I. Mező, Radii of starlikeness and convexity of some q-Bessel functions, J. Math. Anal. Appl., 435 (2016), 968-985. doi: 10.1016/j.jmaa.2015.10.065
    [16] Á. Baricz, Bessel transforms and Hardy space of generalized Bessel functions, Mathematica, 48 (2006), 127-136.
    [17] P. L. Duren, Theory of Hp spaces, A series of Monographs and Textbooks in Pure and Applied Mathematics, vol. 38, New York and London: Academic Press, 1970.
    [18] S. Owa, M. Nunokawa, H. Saitoh, et al. Close-to-convexity, starlikeness, and convexity of certain analytic functions, App. Math. Letter., 15 (2002), 63-69. doi: 10.1016/S0893-9659(01)00094-5
    [19] H. Silverman, Univalent functions with negative coefficients, Proc. Am. Math. Soc., 51 (1975), 109-116. doi: 10.1090/S0002-9939-1975-0369678-0
    [20] R. Singh, S. Singh, Some sufficient conditions for univalence and starlikeness, Colloq. Math., 47 (1982), 309-314. doi: 10.4064/cm-47-2-309-314
    [21] P. J. Eenigenburg, F. R. Keogh, The Hardy class of some univalent functions and their derivatives, Michigan Math. J., 17 (1970), 335-346. doi: 10.1307/mmj/1029000519
    [22] I. B. Jung, Y. C. Kim, H. M. Srivastava, The Hardy space of analytic functions associated with certain one-parameter families of integral operators, J. Math. Anal. Appl., 176 (1993), 138-147. doi: 10.1006/jmaa.1993.1204
    [23] Y. C. Kim, H. M. Srivastava, Some families of generalized Hypergeometric functions associated with the Hardy space of analytic functions, Prod. Japan Acad., Seri A, 70 (1994), 41-46. doi: 10.3792/pjaa.70.41
    [24] S. Ponnusamy, The Hardy space of hypergeometric functions, Complex Var. Elliptic Equ., 29 (1996), 83-96.
    [25] J. K. Prajapat, S. Maharana, D. Bansal, Radius of Starlikeness and Hardy Space of Mittag-Leffer Functions, Filomat, 32 (2018), 6475-6486. doi: 10.2298/FIL1818475P
    [26] N. Yağmur, Hardy space of Lommel functions, Bull. Korean Math. Soc., 52 (2015), 1035-1046. doi: 10.4134/BKMS.2015.52.3.1035
    [27] N. Yağmur, H. Orhan, Hardy space of generalized Struve functions, Complex Var. Elliptic Equ., 59 (2014), 929-936. doi: 10.1080/17476933.2013.799148
    [28] J. Stankiewich, Z. Stankiewich, Some applications of Hadamard convolutions in the theory of functions, Ann. Univ. Mariae Curie-Sklodowska, 40 (1986), 251-265.
    [29] T. H. MacGregor, Functions whose derivative has a positive real part, Trans. Amer. Math. Soc., 104 (1962), 532-537. doi: 10.1090/S0002-9947-1962-0140674-7
  • This article has been cited by:

    1. Karima M. Oraby, Zeinab S. I. Mansour, Starlike and convexity properties ofq-Bessel-Struve functions, 2022, 55, 2391-4661, 61, 10.1515/dema-2022-0004
    2. Muhey U. Din, Mohsan Raza, Qin Xin, Sibel Yalçin, Sarfraz Nawaz Malik, Close-to-Convexity of q-Bessel–Wright Functions, 2022, 10, 2227-7390, 3322, 10.3390/math10183322
    3. Khaled Mehrez, Mohsan Raza, The Mittag-Leffler-Prabhakar Functions of Le Roy Type and its Geometric Properties, 2024, 2731-8095, 10.1007/s40995-024-01738-1
    4. Abdulaziz Alenazi, Khaled Mehrez, Certain properties of a class of analytic functions involving the Mathieu type power series, 2023, 8, 2473-6988, 30963, 10.3934/math.20231584
    5. İbrahim Aktaş, Luminiţa-Ioana Cotîrlâ, Certain Geometrical Properties and Hardy Space of Generalized k-Bessel Functions, 2024, 16, 2073-8994, 1597, 10.3390/sym16121597
    6. İbrahim Aktaş, Büşra Korkmaz, Some geometric properties of the generalized \(k\)-Bessel functions, 2025, 13, 2321-5666, 15, 10.26637/mjm1301/003
    7. Yücel Özkan, Semra Korkmaz, Erhan Deniz, The Monotony of the q−Bessel functions, 2025, 0022247X, 129439, 10.1016/j.jmaa.2025.129439
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(4462) PDF downloads(364) Cited by(7)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog