JFS Journal of Function Spaces 2314-8888 2314-8896 Hindawi Publishing Corporation 10.1155/2016/7390410 7390410 Research Article Subclasses of Analytic Functions Defined by Generalized Hypergeometric Functions Alkahtani Badr S. 1 Mustafa Saima 2 Bulboacă Teodor 3 Fiorenza Alberto 1 Mathematics Department College of Science King Saud University P.O. Box 1142 Riyadh 11989 Saudi Arabia ksu.edu.sa 2 Department of Mathematics Pir Mehr Ali Shah Arid Agriculture University Rawalpindi Pakistan uaar.edu.pk 3 Faculty of Mathematics and Computer Science Babeş-Bolyai University 400084 Cluj-Napoca Romania ubbcluj.ro 2016 3112016 2016 02 09 2015 15 12 2015 17 12 2015 3112016 2016 Copyright © 2016 Badr S. Alkahtani 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.

We introduce a new subclass of analytic functions in the unit disk U, defined by using the generalized hypergeometric functions, which extends some previous well-known classes defined by different authors. Inclusion results, radius problems, and some connections with the Bernardi-Libera-Livingston integral operator are discussed.

1. Introduction

Let A denote the class of functions f of the form (1)fz=z+k=1ak+1zk+1,zU,which are analytic in the unit disk U.

The convolution (or the Hadamard product) of two functions f and g, where f is given by (1) and g(z)=z+k=1bk+1zk+1, zU, is defined as(2)fgz=z+k=1ak+1bk+1zk+1,zU.

For the complex parameters αi with βjCZ0-, where Z0-=0,-1,-2,, i1,2,,q, and j1,2,,s, we define the generalized hypergeometric function Fsq(α1,,αq;β1,,βs;z) as follows (see [1, 2]): (3)Fsqα1,,αq;β1,,βs;z=k=0α1kαqkβ1kβskzkk!,zU,  qs+1;  q,sN0=N0;  N=1,2,,where (λ)k (or the shifted factorial) is defined by (4)λ0=1,1k=k!,kN0,λk=λλ+1λ+k-1,kN.

Using the function F(α1,,αq;β1,,βs;z) defined by(5)Fα1,,αq;β1,,βs;z=z-m·Fsqα1,,αq;β1,,βs;z,Liu and Srivastava  introduced and studied the properties of the linear operator H(α1,,αq;β1,,βs), defined by the Hadamard (or convolution) product(6)Hα1,,αq;β1,,βsfz=Fα1,,αq;β1,,βs;zfz,where the function f is analytic and m-valent in the punctured unit disk U˙=U0 and has the form f(z)=z-m+k=1akzk-m. Note that linear operator H(α1,,αq;β1,,βs) was motivated essentially by the work of Dziok and Srivastava .

Corresponding to the function F(α1,,αq;β1,,βs;z) defined by(7)Fα1,,αq;β1,,βs;z=z·Fsqα1,,αq;β1,,βs;z,we introduce a function Fp(α1,,αq;β1,,βs;z) given by(8)Fα1,,αq;β1,,βs;zFpα1,,αq;β1,,βs;z=z1-zp,where p>0.

Analogous to H(α1,,αq;β1,,βs) defined by (6), we now define the linear operator Hp(α1,,αq;β1,,βs) on A as follows:(9)Hpα1,,αq;β1,,βsfz=Fpα1,,αq;β1,,βs;zfz,αj,βjCZ0-;  i1,2,,q;  j1,2,,s;  p>0.For convenience, we write(10)Hp,q,sα1Hpα1,,αq;β1,,βs.

Remark 1.

The linear operator Hp(α1,,αq;β1,,βs) includes various other linear operators which were considered in some earlier works:

In particular, for p=1, q=2, and s=1, we obtain the linear operator H1(α1,α2;β1) which was defined by Hohlov .

Moreover, putting p=1, q=2, s=1, and α2=1, we obtain the well-known Carlson-Shaffer operator L(α1,β1)=H1(α,1;β1) (see [6, 7]).

From definitions (8) and (9), using notation (10), it is easy to prove the differentiation formula(11)zHp,q,sλfz=pHp+1,q,sλfz-p-1Hp,q,sλfz.

We note that the operator Hp,q,s(λ) is closely related to the Choi-Saigo  operator for analytic functions, which includes the integral studied by Liu  and K. I. Noor and M. A. Noor .

Let Pkρ be the class of functions h, analytic in the unit disk U, satisfying the condition h(0)=1 and(12)02πRehz-ρ1-ρdθkπ,where z=reiθ, k2, and 0ρ<1. This class was introduced in , and as a special case we note that the class Pk(0) was defined by Pinchuk in . Moreover, P(ρ)P2(ρ) is the class of analytic functions with the real part greater than ρ.

Remark 2.

(i) Like in [13, 14], it can easily be seen that the function h, analytic in U, with h(0)=1, belongs to Pk(ρ) if and only if there exists the functions h1,h2P(ρ) such that(13)hz=k4+12h1z-k4-12h2z,zU.

(ii) It is known from  that the class Pk(ρ) is a convex set.

We will assume throughout our discussions, unless otherwise stated, that αi,βjCZ0-, i1,2,,q, j1,2,,s, with qs+1, p>0, and all the powers represent the principal branches; that is, log1=0.

Using the linear operator Hp,q,s(λ), we will define the following classes of analytic functions.

Definition 3.

Let 0ρ<1, c>0, and let b be a complex number such that Reb0. A function fA is said to be in the class Hp,q,sb(λ,ρ,c,δ,k) if and only if(14)1-bHp,q,sλfzHp,q,sλgzc+bHp+1,q,sλfzHp+1,q,sλgzHp,q,sλfzHp,q,sλgzc-1Pkρ,where k2 and gA satisfies the condition(15)βz=Hp+1,q,sλgzHp,q,sλgzPδ,0δ<1.

Remark 4.

From the above definition, the following subclasses of A emerge as special cases:

For  p=q=s=1, k=2, b=0, c=1, g(z)=z, and λ=1, we have(16)H1,1,101,ρ,1,0,2=fA:fzzPρ,  0ρ<1,and this class was studied by Chen .

When  p=q=s=1, g(z)=z, and λ=1, then H1,1,1b(1,ρ,c,δ,k) reduces to the class studied by Noor .

For  p=q=s=1, b=1, k=2, g(z)=z, and λ=1, we obtain the class(17)H1,1,111,ρ,c,0,2=fA:fzfzzc-1Pρ,  0ρ<1,which was studied by Ponnusamy and Karunakaran .

We will use the following lemmas to prove our main results.

Lemma 5 (see [<xref ref-type="bibr" rid="B20">18</xref>]).

If h is an analytic function in U, with h(0)=1, and if λ1 is a complex number satisfying Reλ10, λ10, then(18)Rehz+λ1zhz>ρ,zU,  0ρ<1implies(19)Rehz>γ,zU,where γ is given by(20)γ=ρ+1-ρ2γ1-1,γ1=011+tReλ1-1dt,and γ1 is an increasing function of Reλ1, and 1/2γ1<1. The estimate is sharp in the sense that the bound cannot be improved.

Lemma 6 (see [<xref ref-type="bibr" rid="B12">19</xref>]).

Let u=u1+iu2, v=v1+iv2, and let ψ(u,v) be a complex-valued function satisfying the following conditions:

ψ(u,v) is continuous in a domain DC2.

(1,0)D and Reψ(1,0)>0.

Reψ(iu2,v1)0, whenever (iu2,v1)D and v1-(1/2)1+u22.

If p(z)=1+m=1cmzm is an analytic function in U, such that (p(z),zp(z))D for all zU, and Reψ(p(z),zp(z))>0 for all zU, then Rep(z)>0 for all zU.

In this paper, we investigate several properties of the class Hp,q,sb(λ,ρ,c,δ,k), like inclusion results and radius problems; moreover, a connection with the Bernardi-Libera-Livingston integral operator is also discussed.

2. Main Results Theorem 7.

If b0, then Hp,q,sb(λ,ρ,c,δ,k)Hp,q,s0(λ,ρ,c,δ,k).

Proof.

Let an arbitrary function fHp,q,sb(λ,ρ,c,δ,k), and denote(21)hzHp,q,sλfzHp,q,sλgzc,where h is analytic in U, with h(0)=1, and g satisfies condition (15). From part (i) of Remark 2, we have that fHp,q,s0(λ,ρ,c,δ,k), if and only if(22)hz=k4+12h1z-k4-12h2z,where h1,h2P(ρ).

Using the differentiation formula (11) together with (15), after an elementary computation, we obtain(23)1-bHp,q,sλfzHp,q,sλgzc+bHp+1,q,sλfzHp+1,q,sλgzHp,q,sλfzHp,q,sλgzc-1=hz+bzhzpcβz,where β is given by (15).

Now, using the representation formula (22), we have(24)hz+bzhzpcβz=k4+12h1z+bzh1zpcβz-k4-12h2z+bzh2zpcβz.

Since fHp,q,sb(λ,ρ,c,δ,k), from relations (23) and (24), it follows that(25)hiz+bzhizpcβzPρ,i=1,2,and using the substitution  Hi(z)hi(z)-ρ, i=1,2, the above relation becomes(26)Hiz+bzHizpcβzP0,i=1,2.

To prove our result, we need to show that (26) implies HiP(0), i=1,2. We will define the functional ψ(u,v) by taking u=Hi(z), and v=zHi(z), and thus we have(27)ψu,v=u+bvpcβz.

It is easy to see that the first two conditions of Lemma 6 are satisfied; hence, we proceed to verify condition (iii). Since βP(δ), that is, Reβ(z)>δ for all zU, 0δ<1, it follows that(28)Reψiu2,v1=Rebv1pcβz-b1+u22δ2pcβz20,whenever v1-(1/2)1+u22. Using Lemma 6, we conclude that HiP(0), for i=1,2, which completes our proof.

Theorem 8.

If 0b1<b2, then Hp,q,sb2(λ,ρ,c,δ,k)Hp,q,sb1(λ,ρ,c,δ,k).

Proof.

If we consider an arbitrary function fHp,q,sb2(λ,ρ,c,δ,k), then φ2Pk(ρ), where(29)φ2z1-b2Hp,q,sλfzHp,q,sλgzc+b2Hp+1,q,sλfzHp+1,q,sλgzHp,q,sλfzHp,q,sλgzc-1.

According to Theorem 7, we have(30)φ1zHp,q,sλfzHp,q,sλgzcPkρ,and a simple computation shows that(31)1-b1Hp,q,sλfzHp,q,sλgzc+b1Hp+1,q,sλfzHp+1,q,sλgzHp,q,sλfzHp,q,sλgzc-1=1-b1b2φ1z+b1b2φ2z.Since the class Pk(ρ) is a convex set (see (ii) from Remark 2), it follows that the right-hand side of (31) belongs to Pk(ρ) for 0b1<b2, which implies that fHp,q,sb1(λ,ρ,c,δ,k).

Now, let us define the operator Jc:AA by(32)Jcfz=c+1zc0ztc-1ftdtc>-1.For cN, the operator Jc was introduced by Bernardi , while the special case J1 was previously studied by Libera  and Livingston .

Theorem 9.

If fA, Jc(f) is given by (32), and bC with Reb>0, then(33)1-bHp,q,sλJcfzz+bHp,q,sλfzzPkρimplies that(34)Hp,q,sλJcfzzPkγ,where γ is given by (20), with λ1=b/(c+1).

Proof.

Differentiating relation (32), we have(35)zJcfz=c+1fz-cJcfz,and using definition (9), this implies(36)zHp,q,sλJcfz=c+1Hp,q,sλfz-cHp,q,sλJcfz.If we let(37)hzHp,q,sλJcfzz,according to part (i) of Remark 2, we need to prove that h is of the form(38)hz=k4+12h1z-k4-12h2z,where h1,h2P(γ).

Using (36), from the above relation, we have (39)1-bHp,q,sλJcfzz+bHp,q,sλfzz=hz+bzhzc+1=k4+12h1z+bzh1zc+1-k4-12h2z+bzh2zc+1Pkρ.Thus, from part (i) of Remark 2, it follows that(40)hiz+bzhizc+1Pρ,i=1,2,and from Lemma 5, we conclude that hiP(γ), i=1,2, with γ given by (20) and λ1=b/(c+1).

The next result deals with the converse of Theorem 7.

Theorem 10.

Let b>0 and 0<δ<1. If fHp,q,s0(λ,ρ,c,δ,k), then fHp,q,sb(λ,ρ,c,δ,k) for z<R, where(41)R=min-b+b2+p2c2δ2pcδ;pcρδ-b+pcρδ-b2+p2c2δ21-ρ2pcδ1-ρ.

Proof.

For arbitrary fHp,q,s0(λ,ρ,c,δ,k), let us define the function h as in (43). Thus, it follows that(42)Hp,q,sλfzHp,q,sλgzc=hzPkρ,and gA satisfies the condition(43)βz=Hp+1,q,sλgzHp,q,sλgzPδ.

From part (i) of Remark 2, we have that (42) holds if and only if(44)hz=k4+12h1z-k4-12h2z,where h1,h2P(ρ).

Using the above representation formula, similar to the proof of Theorem 7, we deduce that (45)1-bHp,q,sλfzHp,q,sλgzc+bHp+1,q,sλfzHp+1,q,sλgzHp,q,sλfzHp,q,sλgzc-1=k4+12h1z+bzh1zpcβz-k4-12h2z+bzh2zpcβz,and substituting Hi(z)hi(z)-ρ, i=1,2, we finally obtain (46)1-bHp,q,sλfzHp,q,sλgzc+bHp+1,q,sλfzHp+1,q,sλgzHp,q,sλfzHp,q,sλgzc-1=k4+12H1z+ρ+bzH1zpcβz-k4-12H2z+ρ+bzH2zpcβz,where H1,H2P(0).

To prove our result, we need to determine the value of R, such that(47)ReHiz+ρ+bzHizpcβz>0,for  z<R,  i=1,2,whenever  H1,H2P(0).

Using the well-known estimates for the class P(0) , that is, (48)Hiz2ReHiz1-r2,zr<1,  i=1,2,ReHiz1-r1+r,zr<1,  i=1,2,and according to (43), we obtain(49)ReHiz+ρ+bzHizpcβzρ+ReHiz-bpczHizβzρ+ReHiz-bpczHizδρ+ReHiz1-2bpcδr1-r2,for all zr<1 and i=1,2.

A simple computation shows that 1-2b/pcδr/1-r20 (0r<1) if and only if(50)r0,-b+b2+p2c2δ2pcδ.Assuming that (50) holds, from (49), we deduce that(51)ReHiz+ρ+bzHizpcβzρ+1-r1+r1-2bpcδr1-r2,zr<1,for i=1,2. It is easy to see that the right-hand side of the above inequality is greater than or equal to zero if and only if(52)r0,min1;pcρδ-b+pcρδ-b2+p2c2δ21-ρ2pcδ1-ρ,and combining this with (50), we obtain our result.

Remark 11.

We note the following special case obtained from the above result: for ρ=0, formula (41) reduces to(53)R=-b+b2+p2c2δ2pcδ.

Conflict of Interests

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

Acknowledgments

The work of the third author was entirely supported by the grant given by Babeş-Bolyai University, dedicated for Supporting the Excellence Research 2015. The first author, Badr S. Alkahtani, is grateful to King Saud University, Deanship of Scientific Research, College of Science Research Center, for supporting this project.

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