The quotients of certain q-hypergeometric functions are presented as g-fractions which converge uniformly in the unit disc. These results lead to the existence of certain q-hypergeometric functions in the class of either q-convex functions, PCq, or q-starlike functions PSq*.

1. Introduction and Preliminaries

Let Ak be the class of analytic functions f in the unit disc U=z:z∈C,|z|<1, normalized by f(0)=f′(0)-1=0 and of the form(1)fz=z+∑k=2∞akzk,z∈U.In this paper, we extend some results obtained in the theory of functions to the q-theory and to achieve this we write out some standard notations and basic definitions used in this paper.

Definition 1.

The q-shift factorial, the multiple q-shift factorial, and the q-binomial coefficients are defined by(2)a1,a2,…,an;qk=1,k=0,j=1,a1=a,∏n=0k-11-aqn,j=1,k≠0,a1=a,n∈N,∏j=1naj;qn,j=1,2,…,n;n∈N,k∈Z,akq=1,k=0,1-qa1-qa-1⋯1-qa-k+1q;qk,k∈N,where a, q∈C.

In [1, 2] Jackson defined the q-derivative operator Dz,q as follows.

Definition 2.

Consider the following:(3)Dz,qfz=fz-fqzz1-q,z∈C-0;0<q<1,Dz,qfzz=0=f′0.From (3), one has (4)Dz,qfz=1+∑k=2∞kqakzk-1,z≠0,where [k]q=1-qk/1-q and as q→1, [k]q→k.

Definition 2 leads to the nth order q-derivative derived by Sofonea [3] as follows: (5)Dz,qDz,qn-1fz=Dz,qnfz=-1n1-q-n·z-nq-nn-1/2∑k=0∞nkq-1kqkk-1/2fqn-kz,where n∈N, (Dz,q0f)(z) denotes the identity operator, and nkq denotes the q-binomial coefficients defined in Definition 1.

Definition 3.

For complex parameters ai, bi, and q with bi∈C∖Z-, i∈N, and |q|<1. rΦs denotes the generalized basic (or q-) hypergeometric functions(6)Φsra1,…,arq,zb1,…,bs=Φsra1,…,ar;b1,…,bs;q,z(7)=a0q+∑k=1∞a1,…,ar;qkq,b1,…,bs;qkzk-qk-1/2ks+1-r.

Let r=2, s=1, r=s+1, |q|<1 with complex parameters a1=a, a2=b, and b1=c∈C∖Z-. Then (6) becomes the basic (q-) hypergeometric functions Φ12 written as (8)Φ12a,b;c,q,z=∑k=0∞a,qkb,qkq,qkc,qkzk,z∈U,where (a,q)k is the q-shifted factorial defined in Definition 1.

In [4], Srivastava and Owa summarized some properties of functions that belong to the class PSq⋆ of q-starlike functions in U, introduced and investigated by Ismail et al. in [5]. Srivastava and Owa [4] further proposed the study of properties of functions that belong to the class PSq⋆(α) of q-starlike functions of order α, 0≤α≤1, and also of the functions that belong to the class PCq of q-convex functions in U. The authors [4] also defined the class of functions PCq, on the function f∈Ak, f(0)=f′(0)-1=0 as follows:(9)zDz,q2fzDz,qfz-11-q≤11-q,0<q<1;z∈U,while [5] defined the class of functions PSq⋆, on the function f∈Ak, f(0)=f′(0)-1=0 as(10)zDz,qfzfz-11-q≤11-q,0<q<1;z∈U.Meanwhile, Agrawal and Sahoo in [6] defined and studied some properties of functions that belong to the class PSq⋆(α) and also Sahoo and Sharma in [7] (see also [8]) defined and studied the class PKq of q-close-to-convex functions. Kanas and Răducanu in [9] also used the Ruscheweyh q-differential operator to introduce and study some properties of (q,k) uniformly starlike functions of order α. Other works related to q-hypergeometric can also be traced in [10–13].

In addition, in [14] Baricz and Swaminathan used the Alexander duality between starlike and convex function to define the following.

Definition 4.

If PCq is the class of q-convex functions, then f∈PCq if and only if zDz,q(f)(z)∈PSq⋆. Hence, as q→1-, zDz,q(f)(z)→zf′(z), and PSq⋆ reduces to S⋆, the class of starlike functions.

They used Definition 4 to establish Theorem 5 as follows.

Theorem 5.

If 0<a,b,c,q<1 satisfies the following two more conditions

Apart from Theorem 5, to the best of our knowledge, other properties of functions in the class of functions PCq are yet to be studied. For more results and further studies on q-calculus see [15–17].

Motivated by the numerous studies, of the abovementioned authors, we aimed, in this paper, at using the parameters a, b, c as real parameters and placing some constraints on quotients of two or more hypergeometric functions to establish the following:

zΦ12[a,b,c;q,z]∈PCq,

zΦ12[a,b,c;q,qz]∈PSq⋆.

We describe the procedures to achieve (⧫i) as follows:

Calculate necessary and sufficient conditions for a function f(z)=2Φ1[a,b,c;q,z] to be in the class PCq.

Calculate suitable, contiguous relations for hypergeometric functions.

Use the contiguous relation to derive equation with a quotient of one or more hypergeometric functions. Let the absolute value of the derived quotient, on the left-hand side of the equation, constrain the function Φ12[a,b,c;q,z] to be in PCq, while the other side of the equation has continued fraction expansion.

Calculate the continued fraction of the hypergeometric quotient on the right-hand side of the equation and convert the continued fractions to a g-fraction which converges uniformly.

Hence, the g-fractions lead to the geometric properties of Pick functions.

Then substituting last result in the third outline with simple calculation on the outcome gives the required results.

The description (i) to (vi) can also be used to establish (⧫ii).

First we write out the known results needed to establish ours. Ismail and Libis noted in [18] that the hypergeometric function (12)ψz≔Φ12a,b;c;q,zsatisfies the q-difference equation(13)abz-cq-1ψq2z-a+bz-1-cq-1ψqz=1-zψz.They rewrite (13) in the form(14)ψqz1-zψz=11-a+bz+cq-1+abz-cq-1ψq2z/ψqz,iterated the functional relationship (14), and got(15)ψqz1-zψz=1b0-a1b1-a2b2-⋯-akbk-cq-1-abzqkψqk+2zψqk+1z.They also noted that if these iterations converge, this will give rise to the continued fraction(16)ψqz1-zψz=1b0-a1b1-a2b2-⋯,where(17)ak+1=cq-1-abzqk1-qk+1z,bk=1+cq-1-a+bzqk,k∈N∪0.They also modified a result in [19] to Theorem 6. Later on, we will modify Theorem 6 to suit our results.

Theorem 6 (see [<xref ref-type="bibr" rid="B28">18</xref>]).

Let |q|≠1 with exception of the zeros of Φ12[aq,bq;cq;q,z]; the continued fraction of a meromorphic function of z, which is equal to the function Φ12(a,b;c;q,z)/Φ12(aq,bq;cq;q,z), is represented by(18)Φ12a,b;c;q,zΦ12aq,bq;cq;q,z=e0+e1f1+e2f2+e3f3+⋯,throughout the z-plane with (19)ek+1=1-aqk+11-bqk+1c-abzqk+1zqk1-cqk1-cqk+1,fk=1-a+b-abqk1+qzqk1-cqk,k∈N∪0.It is equal to this ratio in a neighborhood z at origin and furnishes the analytic continuation of it throughout the finite z-plane.

We review that the sequence of positive numbers, {ek}, k≥0, with ek=1, is called a Hausdorff moment sequence if there exists a positive (Borel) measure ϑ on the close interval [0,1] such that(20)ek=∫01ζdϑζ,k∈N∪0.Equation (20) can also be represented by (21)Θz=∑k=0∞ekzk=∫0111-ζzdϑζ,with Θ analytic in the slit complex domain C∖[1,∞], and also Θ belongs to the set of Pick functions.

Lemma 7 bridges the gaps between total monotone sequence, Hausdorff moment sequence, the set of Pick functions, and g-fractions.

Lemma 7 (see [<xref ref-type="bibr" rid="B27">14</xref>, <xref ref-type="bibr" rid="B16">20</xref>]).

Let {ek}, k≥0 be a real sequence; then the following are equivalent:

{ek}, k≥0 is totally monotone sequence; that is, for Cı,k=ık, k∈N∪{0}, (22)Δıek=e0-Cı,1e1+k+Cı,2e2+k-⋯+-1ıeı+k≥0,ı,k∈N∪0.

{ek}, k≥0 is a Hausdorff moment sequence; that is, there exists a positive (Borel) measure ϑ on the interval [0,1] whose sequence of moment is (23)ek=∫01ζkdϑζ,∀k∈N∪0.

The power series ∑k=0∞ekzk is analytic in U, has the analytic continuation in the complex plane slit C∖[1,∞), and also has a corresponding g-fraction (24)g11+1-g1g2z21+1-g2g3z31+⋯=∫0111-ζzdϑζ,

with 0<gk, gk+1<1, k∈N∪{0}.

For more studies on Hausdorff moment of sequence and hypergeometric mappings see [20–26] and their references. Theorem 8 displays the constraints on the constant g that made the g-fraction uniformly convergent.

Theorem 8 (see [<xref ref-type="bibr" rid="B16">20</xref>]).

Letting g1,g2,g3,… be constant satisfying one of the following conditions:

0<gk≤1, (k∈N),

0≤gk<1, (k∈N),

0<gk<1, (k∈N),

then the continued fraction(25)g11+1-g1g2z21+1-g2g3z31+⋯converges uniformly for |zk|≤1, (k∈N∖{1}).

We write Remark 9, which is some comments made by Wall in [20]. This remark enables us to convert our originally derived continued fraction into a g-fraction which converges uniformly.

Remark 9.

By means of equivalence transformation, the continued fraction of the form(26)a1b1+a2b2+a3b3+⋯can be transformed into continued fraction of the form (25), by first reducing the partial denominator to unity; hence we obtain(27)a1/b11+a2/b1b21+a3/b2b31+⋯.Also (10) converges uniformly if there exist constants gk, gk+1 such that (28)ak+1bkbk+1≤1-gkgk+1,0<gk,gk+1<1,k∈N∪0.

We state necessary and sufficient condition for a function f to be in the class PSq⋆ established in [5].

Lemma 10 (see [<xref ref-type="bibr" rid="B3">5</xref>]).

A function f(z) is in the class PSq⋆ if and only if(29)fqzfz≤1,fz≠0,z∈U.

We calculate the necessary and sufficient condition for a function f to be in the class PCq.

Lemma 11.

A function f(z) is in the class PCq if and only if(30)fq2z-fqzfqz-fz≤q,fqz≠fz,z∈U.

Proof.

The proof of Lemma 11 follows from substituting(31)zDz,q2fzDz,qfz=1q1-qq-fq2z-fqzfqz-fz,in (9).

2. Contiguous Relations and Main Results

In this section we write out some contiguous relations of hypergeometric functions, which can be used to derive a continued fraction expansion, for some quotients of two or more hypergeometric functions. We will also state our results. Suppose(32)fz=ψz≔Φ12a,b;c;q,z,fqz=ψqz≔Φ12a,b;c;q,qz,fq2z=ψq2z≔Φ12a,b;c;q,q2z.Rearranging (13), we have(33)cq-1-abzψq2z-ψqzψqz-ψz=1-z+a-11-bzψqzψz-ψqz.Substituting the contiguous relation,(34)ψz-ψqz-1-a1-b1-czΦ12aq,bq;cq;q,z=0,calculated in [27] (see also [18]) in (32) gives(35)cq-1-abzψq2z-ψqzψqz-ψz=1-a+b-abz-1-cψzΦ12aq,bq;cq;q,z.

Theorem 12.

Let a, b, c, q be nonnegative real numbers with a,b,c,q∈(0,1) and 0<cqk<1 for all k∈N∪{0}. Then hypergeometric quotient Φ12(a,b;c;q,z)/Φ12(aq,bq;cq;q,z) with exception of the zeros of Φ12(aq,bq;cq;q,z) has the continued fraction(36)Φ12a,b;c;q,zΦ12aq,bq;cq;q,z=ρ0+δ1ρ1+δ2ρ2+δ3ρ3+⋯+δk+1ρk+1+⋯,where(37)δk+1=zqk1-aqk+11-bqk+1c-abzqk+11-cqk1-cqk+1,(38)ρk=1-cqk-zqka+b-abqk1+q1-cqk.

then the continued fraction (36) can be represented as(39)Φ12a,b;c;q,zΦ12aq,bq;cq;q,z=ρ0+g11+1-g1g2z21+1-g2g3z31+⋯,zk≤1,k∈N∖1,where (40)ρk+1δkδk+1≤1-gkgk+1,0<gk,gk+1<1,k∈N∪0,where δk is (37) and ρk+1 is (38). Hence the hypergeometric quotient, Φ12(a,b;c;q,z)/Φ12(aq,bq;cq;q,z), converges uniformly.
Theorem 14.

If the hypothesis of Theorem 13 holds and there exists an increasing function, ϑk, mapping [0,1] into itself with ∫01dϑ(ζ)=1, then (41)Φ12a,b;c;q,zΦ12aq,bq;cq;q,z=ρ0+∫0111-ζzdϑζfor all z∈C∖[1,∞). Further,(42)zΦ12a,b;c;q,zΦ12aq,bq;cq;q,z=zρ0+∫01z1-ζzdϑζ,for all z∈C∖[1,∞).

Theorem 15.

If the hypothesis of Theorem 14 and inequality (43) holds,(43)c1+q2+q2abq-1+qab-q2≤4abq1+q2c+q22c-3,then zΦ12[a,b;c;q,z]∈PCq.

Theorem 16 (see [<xref ref-type="bibr" rid="B28">18</xref>]).

Let a,b,c,q∈(0,1), z≠1, with nonzeros of Φ12[a,b;c;q,z]. Then continued fraction of a meromorphic function of qz, Φ12(a,b;c;q,qz)/(1-z)Φ12(a,b;c;q,z), is represented by(44)Φ12a,b;c;q,qz1-zΦ12a,b;c;q,z=1b0-a1b1-a2b2-⋯,where(45)ak+1=cq-1-abzqk1-qk+1z,bk=1+cq-1-a+bzqk,k∈N∪0.

then(46)Φ12a,b;c;q,qz1-zΦ12a,b;c;q,z=g2,11+1-g2,1g2,2z21+1-g2,2g2,3z31+⋯,zk≤1,k∈N∖1with(47)ak+1bkbk+1≤1-g2,kg2,k+1,0<g2,k,g2,k+1<1,k∈N∪0,g2,0=0,where ak+1,bk are as in (45). Hence Φ12(a,b;c;q,qz)/(1-z)Φ12(a,b;c;q,z) converges uniformly.
Theorem 18.

If the hypothesis of Theorem 17 holds and there exists an increasing function, ϑk, mapping [0,1] into itself, with ∫01dϑ(ζ)=1, then(48)Φ12a,b;c;q,qz1-zΦ12a,b;c;q,z=∫0111-ζzdϑζfor all z∈C∖[1,∞). Also,(49)zΦ12a,b;c;q,qz1-zΦ12a,b;c;q,z=∫0zz1-ζzdϑζ.

Theorem 19.

If the hypothesis of Theorem 18 holds, then (50)zΦ12a,b;c;q,qz∈PSq⋆.

3. Proof of the Main ResultsProof of Theorem <xref ref-type="statement" rid="thm2.1">12</xref>.

To calculate the continued fraction of Φ12[a,b;c;q,z]/Φ12[aq,bq;cq;q,z], we let (51)φ0=Φ12a,b;c;q,z,φ1=Φ12aq,bq;cq;q,z,φ2=Φ12aq2,bq2;cq2;q,z,⋮φk=Φ12aqk,bqk;cqk;q,z,k∈N∪0.We use the following (52)1-c1-cqφ0-1-cq1-c-za+b-ab1+qφ1=z1-aq1-bqc-abzqφ2,stated in [18], to calculate the continued fraction of φ0/φ1. Solving (52) gives(53)φ0φ1=δ1φ2φ1+ρ0=ρ0+δ1φ1/φ2,where (54)δ1=z1-aq1-bqc-abzq1-c1-cq,ρ0=1-c-za+b-ab1+q1-c.To calculate φ1/φ2, we set (55)1-cq1-cq2φ1-1-cq21-cq-zqa+b-abq1+qφ2=zq1-aq21-bq2c-abzq2φ3.Solving (55), we obtain(56)φ1φ2=ρ1+δ2φ3φ2=ρ1+δ2φ3φ2=ρ1+δ2φ2/φ3,where (57)ρ1=1-cq-zqa+b-abq1+q1-cq,δ2=zq1-aq21-bq2c-abzq21-cq1-cq2.To calculate φi/φi+1, we set (58)1-cqi1-cqi+1φi-1-cqi+1·1-cqi-zqia+b-abqi1+qφi+1=zqi1-aq1+i1-bq1+ic-abzqi+1φi+2.Assume i=k; we rewrite (58) as(59)φkφk+1=δk+1φk+2φk+1+ρk=ρk+δk+1φk+1/φk+2,k∈N∪0,where δk+1 is (37) and ρk is (38). Substituting (56)–(59) in (53), gives the continued fraction (36).

Proof of Theorem <xref ref-type="statement" rid="thm2.2">13</xref>.

To prove Theorem 13, we first reduce the partial denominator of the continued fraction (36) to unity, by setting (60)δk+1ρkρk+1≤1-sTkQk≤1-gkgk+1,k∈N∪0,where (61)s=gk=1-qk1+qk+1abqk+1-a+b1-cqk1-cqk-2Rezqka+b-abqkq+1+zz¯a+b-abqkq+12,gk+1=+TkQk,0<Tk<Qk,with (62)Tk=cc-2abRezqk+1+zz¯a2b2q2k+1,and also (63)Qk=1-cqk+11-cqk+1-2Rezqk+1a+b-abqk+11+q+zz¯q2k+1a+b-abqk+11+q2.Moreover, (36) can be rewritten as(64)ρ0+δ1/ρ11+δ2/ρ1ρ21+δ3/ρ2ρ31+⋯.Assuming g0=0, we transform (64) by replacing δk+1/ρkρk+1 with (1-gk)gk+1, i, k∈N∪{0}, and obtain(65)Φ12a,b;c;q,zΦ12aq,bq;cq;q,z=ρ0+g11+1-g1g21+⋯.Since |zk|≤1, (k∈N∖{1}) and 0<gk, gk+1<1 (65) can be written as(66)Φ12a,b;c;q,zΦ12aq,bq;cq;q,z=ρ0+g11+1-g1g2z21+⋯.Hence, by Remark 9, the g-fraction of Φ12(a,b;c;q,z)/Φ12(aq,bq;cq;q,z) converges uniformly in U.

Remark 20.

The addition of the leading constant ρ0 does not affect the convergence of the continued fraction Φ12(a,b;c;q,z)/Φ12(aq,bq;cq;q,z). The continued fraction on the right-hand side of (65) converges to the left-hand side provided (a,b,c) belongs to the neighborhood of (0,0,0) with z∈U and z is not a pole of the right-hand side. We also note that the continued fraction (65) is called the infinite g-fraction.

Proof of Theorem <xref ref-type="statement" rid="thm2.3">14</xref>.

By Lemma 7(iii), the coefficients in the power series expansion about z=0, of the analytic function, on the left-hand side of the continued fraction (65), are Hausdorff moments of an increasing function, in the open interval (0,1) with infinitely many points of increase (with total increase less than or equal to 1). Hence, there exists a function ϑ0 mapping [0,1] to itself, satisfying 0=ϑ0(0)≤ϑ0(ξ)≤ϑ0(ζ)≤ϑ0(1) for 0<ξ<ζ<1 and its range contains infinitely many points such that(67)Φ12a,b;c;q,zΦ12aq,bq;cq;q,z=ρ0+∫0111-ζzdϑ0ζ,z∈C∖1,∞,by analytic continuation, with ϑ0(ζ)=ϑ0(a,b;c;q,ζ).

Furthermore, since ∫01ϑ(ζ)=1,(68)zΦ12a,b;c;q,zΦ12aq,bq;cq;q,z=zρ0+∫01z1-ζzdϑ0ζ,for all z∈C∖[1,∞).

Proof of Theorem <xref ref-type="statement" rid="thm2.4">15</xref>.

By (35),(69)zfq2z-fqzfqz-fz=qc-abqzz-a+b-abz2-1-czρ0-1-czφ0φ1.By Theorem 14, (69) can be written as(70)zfq2z-fqzfqz-fz=q2c-abqzz-a+b-abz2+c-1zρ0+c-1∫01z1-ζzdϑζ.For zφ0 to be in PCq, we show that(71)q2c-abqz1-a+b-abz-1-cρ0-1-c∫0111-ζzdϑζ≤1,where ρ0=1-c-z{a+b-ab(1+q)}/1-c.

Applying the triangle inequality to the left-hand side of (71), with |z|≤r<1, we obtain (72)1-r+a+b-abr1-r+1-r1-c-ra+b-ab1+q+1-c≤c-abqr1-rq2,αr2+βr+γ≥0,where (73)α=abq1+q2,β=-c1+q2+q2abq-1+qab-qγ=c+q22c-3.

By hypothesis of Theorem 15 and α>0, we need to show that 4αγ≥β2. Hence, Theorem 15 is established.

Proof of Theorem <xref ref-type="statement" rid="thm2.5">16</xref>.

The calculations in Theorem 16 were established in [18].

Proof of Theorem <xref ref-type="statement" rid="thm2.6">17</xref>.

We then need to reduce the partial denominators of (44) to unity, by setting(74)akbkbk+1=q·ϖkϱklkϱk+1≤1-g2,kg2,k+1,where(75)ϖk=zz¯q2k+1+1-2Rezqk+1,ϱk=q+cq+c-Reza+bqk+1+zz¯a+b2q2k+1,with(76)g2,k=1-q·ϖkϱk,lk=zz¯a2b2q2k+1+cc-2abRezqk+1,ϱk+1=q+cq+c-Reza+bqk+2+zz¯a+b2q2k+2,with(77)g2,k+1=lkϱk+1.By Remark 9, (44) can be written as(78)ψqz1-zψz=1/b01-a1/b0b11-a2/b1b21-⋯.Hence (78) can be written as(79)ψqz1-zψz=g2,11+1-g2,1g2,2z21+⋯.

The proofs of Theorems 18 and 19 are the same procedures as those of Theorems 14 and 15 and hence are omitted in this paper.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The work presented here was fully supported by AP-2013-009.

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