IJMMSInternational Journal of Mathematics and Mathematical Sciences1687-04250161-1712Hindawi Publishing Corporation45063210.1155/2009/450632450632Research ArticleSome Properties of Fractional Calculus and Linear Operators Associated with Certain Subclass of Multivalent FunctionsKhosravianarabSh.1KulkarniS. R.2AhujaO. P.3GovilNarendra Kumar1Department of MathematicsPune University411007 PuneIndiaunipune.ernet.in2Department of MathematicsFergusson College411004 PuneIndiafergusson.edu3Department of Mathematical SciencesKent State University14111 Claridon Troy RoadBurton, Ohio 44021-9500USAkent.edu200927072009200927032009250720092009Copyright © 2009This 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 investigate several distortion inequalities involving fractional calculus, Ruscheweyh derivatives, and some well-known integral operators. In special cases, the results presented in this paper provide new approaches to several previously known results.

1. Introduction

Let 𝒜(p) denote the class of functions f of the form f(z)=zp+k=1ap+kzp+k(p={1,2,3,}), which are analytic in the open unit disc 𝒰={z:|z|<1}. Also, let 𝒦(p) denote the subclass of 𝒜(p) consisting of all functions f of the form f(z)=zp-k=1ap+kzp+k(p={1,2,3,},ap+k0), For functions f,g𝒜(p), given by f(z):=zp+k=1ap+kzp+k,      g(z):=zp+k=1bp+kzp+k, the Hadamard product (or convolution) of f and g is defined by (f*g)(z):=zp+k=1ap+kbp+kzp+k. The Ruscheweyh derivative of f of order δ+p-1 is defined by Dδ+p-1f(z):=zp(1-z)δ+p*f(z)=zp+k=1(δ+p)k(1)kap+kzp+k, where f is given by (1.1) and δ>-p. The Ruscheweyh derivative Dδ+p-1 has been studied by several authors; for example, see [1, 2].

For β<1, γ0, p, and δ>-p, let 𝒯δ,p,γ(β) consist of functions f𝒦(p) so that {eiη((1-γ)Dδ+p-1f(z)zp+γ(Dδ+p-1f(z))pzp-1-β)}>0,z𝒰, for some η. In , the authors obtained four containment results for the class 𝒯δ,p,γ(β). We denote 𝒯0,1,γ(β)=𝒯γ(β). The class 𝒯γ(β) was studied by Swaminathan , Barnard et al. , Kim and Rønning , and others.

In the present paper, we investigate several distortion inequalities involving fractional calculus, Ruscheweyh derivative, and some well-known integral operators defined on the class 𝒯δ,p,γ(β). In special cases, the results presented here provide new approaches to some previously known results.

Remark 1.1.

Throughout this section, we assume that δ+p1.

2. Definitions and Lemmas

For the function f given by (1.2), we define qδ,p,γ(z)=(1-γ)Dδ+p-1f(z)zp+γ(Dδ+p-1f(z))pzp-1. It is easy to verify that qδ,p,γ(z)=1-k=1(δ+p)k(p+kγ)p(1)kap+kzk,      ap+k0.

Lemma 2.1.

Let the function f be given by (1.2). Then f𝒯δ,p,γ(β) if and only if k=1(δ+p)k(p+kγ)p(1)kap+k12(|1+eiη(1-β)|-|1-eiη(1-β)|) for some η.

Proof.

Using the fact that Re(ω)0 if and only if |1+ω||1-ω|, it suffices to show that |1+eiη(qδ,p,γ(z)-β)|-|1-eiη(qδ,p,γ(z)-β)|0, where qδ,p,γ(z) is defined by (2.1). Letting 𝒜k=(δ+p)k(p+kγ)p(1)k, and assuming (2.3), we obtain |1+eiη(qδ,p,γ(z)-β)|-|1-eiη(qδ,p,γ(z)-β)|=|1+eiη(1-β)-eiηk=1Akap+kzk|-|1-eiη(1-β)+eiηk=1Akap+kzk||1+eiη(1-β)|-|eiη||k=1Akap+kzk|-|1+eiη(1-β)|-|eiη||k=1Akap+kzk||1+eiη(1-β)|-|1-eiη(1-β)|-2k=1Akap+k0, where z𝒰={z;    zand    |z|=1}. By (2.3), the desired inequality (2.4) follows at once. Conversely, if f𝒯δ,p,γ(β), then Re{eiη(qδ,p,γ(z)-β)}>0, or, equivalently (2.4). This yields |1+eiη(1-k=1Akap+kzk-β)||1-eiη(1-k=1Akap+kzk-β)|, which implies that |1+eiη(1-β)-eiηk=1Akap+kzk||1-eiη(1-β)+eiηk=1Akap+kzk|. Squaring the above inequality, choosing the value of z on the half line z=re-iθ(0r<1), and letting r1- through this line, we obtain |1+eiη(1-β)|2+|eiη|2|k=1Akap+kzk|2-2|1+eiη(1-β)||k=1Akap+kzk||1-eiη(1-β)|2+|eiη|2|k=1Akap+kzk|2+2|1-eiη(1-β)||k=1Akap+kzk|. Hence we get 2|k=1Akap+kzk|(|1-eiη(1-β)|+|1+eiη(1-β)|)|1+eiη(1-β)|2-|1-eiη(1-β)|2=(|1+eiη(1-β)|-|1-eiη(1-β)|)(|1+eiη(1-β)|+|1-eiη(1-β)|), which reduces to 2k=1Akap+k|1+eiη(1-β)|-|1-eiη(1-β)|. So the desired inequality (2.3) follows upon using (2.5).

Setting η=δ=0 and p=1 in Lemma 2.1, we get the following result.

Corollary 2.2 ([<xref ref-type="bibr" rid="B11">5</xref>, Theorem  2.4]).

Let f(z) be of the form (1.2). Then necessary and sufficient condition for f to be in 𝒯γ(β) is k=2[1+γ(k-1)]ak1-β,ak0.

Throughout this paper, we define Eη,β:=|1+eiη(1-β)|-|1-eiη(1-β)|.

As an immediate consequence of Lemma 2.1, we have the following corollary.

Corollary 2.3.

Let the function f be defined by (1.2). If f𝒯δ,p,γ(β), then ap+kp(1)kEη,β2(δ+p)k(p+kγ) for some η.

Let α1,α2,,αp and β1,β2,,βq (p,q{0}, pq+1) be complex numbers such that βk0,-1,-2,for k{1,2,,q}. The generalized hypergeometric function pFq is given by pFq(α1,,αp;β1,,βq;z):=n=0(α1)n(α2)n(αp)n(β1)n(β2)n(βq)nznn!,z𝒰, where (x)n denotes the Pochhammer symbol defined by (x)n=Γ(x+n)Γ(x)=x(x+1)(x+2)(x+n-1),forn,(x)0=1. The operator pFq has recently been studied by several authors; for example, [3, 5]. For p=q+1=2, the above series give rise to the Gaussian hypergeometric series F(a,b;c;z).

In , Hohlov introduced the convolution operator Ha,b;c by Ha,b;c(f)(z):=zF(a,b;c;z)*f(z),f𝔸(1). Motivated by the operator Ha,b;c, the authors in  defined the convolution operators Gfp and Ha,b;cp,d as follows: Gfp(a,b;z):=G(z):=(k=1(1+a)(1+b)(k+a)(k+b)zk+p-1)*f(z),(a>-1,b>-1),Ha,b;cp,d(f)(z)=H(z)=zp  3F2(a,b,1+pd;pd,c;z)*f(z), where f𝒜(p), p, and d0. For p=1, the operator Gf1(a,b;z) was introduced in .

In Section 3, we will make use of the following well-known fractional calculus operators Dz-μ, Dzμ, and Dzn+μ. For an analytic function f defined in a simply connected region of the complex z-plane containing the origin, these operators are defined as follows (See [1, 10]): Dz-μf(z)=1Γ(μ)0zf(t)(z-t)1-μdt(μ>0), where multiplicity of (z-t)μ-1 is removed by requiring log(z-t) to be real when z-t>0; Dzμf(z)=1Γ(1-μ)0zf(t)(z-t)μdt(0μ<1), where the multiplicity of (z-t)-μ is removed, as in the definition of Dz-μf(z); Dzn+μ  f(z):=dndznDzμ  f(z)(0μ<1;n0={0}). By virtue of (2.21), (2.22), (2.23) and in terms of Gamma function, it is wellknown (see for details ) that

Dz-μzk=Γ(k+1)Γ(k+μ+1)zk+μ(k,μ>0),Dzμzk=Γ(k+1)Γ(k-μ+1)zk-μ(k,0μ<1),Dzq+μzk=dqdzqDzμzk=Γ(k+1)Γ(k-q-μ+1)zk-(q+μ), where q0, k, 0μ<1, and qk for μ=0.

In Section 4, we will investigate the integral operator Jδ,p defined by (Jδ,pf)(z)=δ+pzp0ztδ-1f(t)dt, where f𝒜(p), δ>-p, and p. For p=1 and δ=0, the operator was first defined by Bernardi . Later on several authors studied the operator Jδ,p; for example, see [1, 5].

3. Distortion Inequalities of Convolution OperatorsTheorem 3.1.

Let the function f defined by (1.2) be in the class 𝒯δ,p,γ(β). Then |qδ,p,γ(z)|1-12Eη,β|z|,|qδ,p,γ(z)|1+12Eη,β|z| for some η. Here, qδ,p,γ(z) and Eη,β are defined, respectively, by (2.1) and (2.14).

Proof.

From (2.2), we have |qδ,p,γ(z)|1-|k=1(δ+p)k(p+kγ)p(1)kap+kzk|1-k=1(δ+p)k(p+kγ)p(1)kap+k|z|k1-|z|k=1(δ+p)k(p+kγ)p(1)kap+k. Making use of Lemma 2.1, we get |qδ,p,γ(z)|1-12Eη,β|z| for some η. Similarly, |qδ,p,γ(z)|1+|z|k=1(δ+p)k(1+(kγ/p))(1)kap+k1+12Eη,β|z| for some η. This completes the proof.

We next obtain distortion inequalities for the fractional operaters Dzμ and Dz-μ.

Theorem 3.2.

Suppose μ(1+k+p)/(k+b+2) and p<b+1. If f𝒯δ,p,γ(β), then for some η, one has |DzμG(z)|Γ(p+1)Γ(p-μ+1)|z|p-μ(1-p(1+a)(1+b)(1+p)2(2+a)(2+b)(1-μ+p)(δ+p)(γ+p)|z|Eη,β),|DzμG(z)|Γ(p+1)Γ(p-μ+1)|z|p-μ(1+p(1+a)(1+b)(1+p)2(2+a)(2+b)(1-μ+p)(δ+p)(γ+p)|z|Eη,β), where 0μ<1, z𝒰, p, and the operator G(z):=Gfp(a,b;z) was defined by (2.19).

Proof.

By using (2.19), we deduce that DzμG(z)=Γ(p+1)Γ(p-μ+1)zp-μ-k=1(1+a)(1+b)Γ(k+p+1)(k+a+1)(k+b+1)Γ(k+p-μ+1)ap+kzp+k-μ. Then Γ(p-μ+1)Γ(p+1)zμ-pDzμG(z)=1-k=1θ(k)ap+kzk, where θ(k)=(1+a)(1+b)Γ(k+p+1)Γ(p-μ+1)(k+a+1)(k+b+1)Γ(k+p-μ+1)Γ(p+1)(k,p,0μ<1). Since θ(k) is a decreasing function of k, when μ(1+k+p)/(k+b+2), then 0<θ(k)θ(1)=(1+a)(1+b)(1+p)(2+a)(2+b)(1-μ+p). Also, according to Lemma 2.1 and δ+p1, we have 1p(δ+p)(p+γ)k=1ap+k=Γ(δ+p+1)(1+(γ/p))Γ(δ+p)k=1ap+kk=1(δ+p)k(1+(kγ/p))(1)kap+k12Eη,β for some η. Then k=1ap+kp2(δ+p)(p+γ)Eη,β for some η. From (3.7) and (3.9), we obtain Γ(p-μ+1)Γ(p+1)zμ-pDzμG(z)1-θ(1)k=1ap+kzk. In view of (3.11), we conclude that |Γ(p-μ+1)Γ(p+1)zμ-pDzμG(z)|1-p(1+a)(1+b)(1+p)2(2+a)(2+b)(1-μ+p)(δ+p)(γ+p)|z|Eη,β for some η, and |Γ(p-μ+1)Γ(p+1)zμ-pDzμG(z)|1+p(1+a)(1+b)(1+p)2(2+a)(2+b)(1-μ+p)(δ+p)(γ+p)|z|Eη,β for some η, which yield (3.5).

By letting p=1 and b=a-1>0 in Theorem 3.2, we deduce the following consequence.

Corollary 3.3.

If f𝒯δ,1,b(β), then for 0μ<1, z𝒰, δ0 and some η|DzμG(z)|1Γ(2-μ)|z|1-μ(1-1(3+b)(2-μ)(1+δ)|z|Eη,β),|DzμG(z)|1Γ(2-μ)|z|1-μ(1+1(3+b)(2-μ)(1+δ)|z|Eη,β).

Theorem 3.4.

Let μ>0, z𝒰 and p. If f𝒯δ,p,γ(β), then |Dz-μG(z)|Γ(p+1)Γ(p+μ+1)|z|p+μ(1-p(1+a)(1+b)(1+p)2(2+a)(2+b)(1-μ+p)(δ+p)(γ+p)|z|Eη,β),|Dz-μG(z)|Γ(p+1)Γ(p+μ+1)|z|p+μ(1+p(1+a)(1+b)(1+p)2(2+a)(2+b)(1-μ+p)(δ+p)(γ+p)|z|Eη,β) for some η. The operator G(z):=Gfp(a,b;z) was defined by (2.19).

Proof.

In view of (2.19) and (2.22), we have Γ(p+μ+1)Γ(p+1)z-(μ+p)Dz-μG(z)=1-k=1τ(k)ap+kzk, where τ(k)=(1+a)(1+b)Γ(k+p+1)Γ(p+μ+1)(k+a+1)(k+b+1)Γ(k+p+μ+1)Γ(p+1). Since τ is a decreasing function of k, it follows that 0<τ(k)τ(1)=(1+a)(1+b)(1+p)(2+a)(2+b)(1+μ+p). By using (3.11), (3.18), and (3.20), we get |Γ(p+μ+1)Γ(p+1)z-(μ+p)Dz-μG(z)|1-p(1+a)(1+b)(1+p)2(2+a)(2+b)(1+μ+p)(δ+p)(γ+p)|z|Eη,β,|Γ(p+μ+1)Γ(p+1)z-(μ+p)Dz-μG(z)|1+p(1+a)(1+b)(1+p)2(2+a)(2+b)(1+μ+p)(δ+p)(γ+p)|z|Eη,β for some η. The last two inequalities yield (3.16) and (3.17), respectively.

Letting δ=0, p=1, and a=b+1=μ+2 in Theorem 3.4, we get the following result.

Corollary 3.5.

Let Gf1(μ+2,μ+1;z) be defined by (2.19). If f𝒯γ(β), then |Dz-μGf1(μ+2,μ+1;z)|1Γ(μ+2)|z|μ+1(1-1(μ+4)(γ+1)|z|Eη,β),|Dz-μGf1(μ+2,μ+1;z)|1Γ(μ+2)|z|μ+1(1+1(μ+4)(γ+1)|z|Eη,β) for some η, μ>0, z𝒰, and p.

We next prove the distortion theorems involving fractional calculus and generalized convolution operator defined by (2.20).

Theorem 3.6.

Suppose 0μ<1, z𝒰, p, and η. Also, let a1, bp-μ+1, and cp+1. If f𝒯δ,p,γ(β), then |DzμH(f)(z)|Γ(p+1)Γ(p-μ+1)|z|p-μ(1-ab(p+1)2c(1-μ+p)(δ+p)|z|Eη,β),|DzμH(f)(z)|Γ(p+1)Γ(p-μ+1)|z|p-μ(1+ab(p+1)2c(1-μ+p)(δ+p)|z|Eη,β) for some η. Here, the operator H(f)(z) is defined by (2.20).

Proof.

By making use of (2.20), we have DzμH(f)(z)=Dzμ{zp-k=1(a)k(b)k(1+(p/γ))k(p/γ)k(c)k(1)kap+kzp+k}=Γ(p+1)Γ(p-μ+1)zp-μ-k=1(a)k(b)k(1+(p/γ))kΓ(p+k+1)(p/γ)k(c)k(1)kΓ(p+k-μ+1)ap+kzp+k-μ. It is easy to verify that 1+kγp=(1+(p/γ))k(p/γ)k. This implies that Γ(p-μ+1)Γ(p+1)zμ-pDzμH(f)(z)=1-k=1λ(k)(1+kγp)ap+kzk, where λ(k)=(a)k(b)kΓ(p+k+1)Γ(p-μ+1)(c)k(1)kΓ(p+k-μ+1)Γ(p+1). Since λ is a decreasing function of k, when a1, cp+1 and bp-μ+1, we get 0<λ(k)λ(1)=ab(p+1)c(p-μ+1). From Lemma 2.1 and δ+p1, we obtain (δ+p)k=1(1+kγp)ap+kk=1(δ+p)k(1+(kγ/p))(1)kap+k12Eη,β for some η. It follows from (3.26) and (3.28) that |Γ(p-μ+1)Γ(p+1)zμ-pDzμH(f)(z)|1-ab(p+1)2c(p-μ+1)(δ+p)|z|Eη,β,|Γ(p-μ+1)Γ(p+1)zμ-pDzμH(f)(z)|1+ab(p+1)2c(p-μ+1)(δ+p)|z|Eη,β for some η, which yield (3.23).

We state an obvious variant of Theorem 3.6 as follows.

Corollary 3.7.

Let the function f defined by (1.2) be in the class 𝒯δ,p,γ(β). Also let a1, bp-μ+1, cp+1, and δ+p>1. Then |DzμH(f)(z)|Γ(p+1)Γ(p-μ+1)|z|p-μ(1-12|z|Eη,β),|DzμH(f)(z)|Γ(p+1)Γ(p-μ+1)|z|p-μ(1+12|z|Eη,β) for some η, 0μ<1, z𝒰, and p.

The proof of Theorem 3.8 is much akin to that of Theorem 3.6, and so it is omitted here.

Theorem 3.8.

Let a1, bp+μ+1 and cp+1. Also, let μ>0, p, and z𝒰. If f𝒯δ,p,γ(β), then |Dz-μH(f)(z)|Γ(p+1)Γ(p+μ+1)|z|p+μ(1-ab(p+1)2c(1+μ+p)(δ+p)|z|Eη,β),|Dz-μH(f)(z)|Γ(p+1)Γ(p+μ+1)|z|p+μ(1+ab(p+1)2c(1+μ+p)(δ+p)|z|Eη,β) for some η.

Next we prove the following.

Theorem 3.9.

Let a1, bp-μ+1, and cp+1. Also, let 0μ<1, p, and z𝒰. If f𝒯δ,p,γ(β), then |Dzμ(Dδ+p-1H(f)(z))|Γ(p+1)Γ(p-μ+1)|z|p-μ(1-ab(p+1)2c(1-μ+p)|z|Eη,β),|Dzμ(Dδ+p-1H(f)(z))|Γ(p+1)Γ(p-μ+1)|z|p-μ(1+ab(p+1)2c(1-μ+p)|z|Eη,β) for some η.

Proof.

We have Dδ+p-1H(f)(z)=zp-k=1Akzp+k, where Ak=(δ+p)k(a)k(b)k(1+(p/γ))k(1)k(p/γ)k(c)k(1)kap+k. Therefore Dzμ(Dδ+p-1H(f)(z))=Γ(p+1)Γ(p-μ+1)zp-μ-k=1AkΓ(p+k+1)Γ(p+k-μ+1)zp+k-μ. So, from (3.25), we have Γ(p-μ+1)Γ(p+1)zμ-pDzμ(Dδ+p-1H(f)(z))=1-k=1(δ+p)k(p+kγ)p(1)kλ(k)ap+kzk, where λ(k) is defined by (3.27). Since λ is a decreasing function of k, when a1,  bp-μ+1 and cp+1, then 0<λ(k)λ(1)=ab(p+1)c(p-μ+1). From (3.37), (3.38), and Lemma 2.1, we find that |Γ(p-μ+1)Γ(p+1)zμ-pDzμ(Dδ+p-1H(f)(z))|1-ab(p+1)2c(1-μ+p)|z|Eη,β,|Γ(p-μ+1)Γ(p+1)zμ-pDzμ(Dδ+p-1H(f)(z))|1+ab(p+1)2c(1-μ+p)|z|Eη,β for some η. The above inequalities lead us to the desired inequalities (3.33).

The proof of the following theorem is similar to Theorem 3.9, and so it is omitted here.

Theorem 3.10.

Let a1, bp+μ+1, and cp+1. Also, let μ>0, z𝒰, and p. If f𝒯δ,p,γ(β), then |Dz-μ(Dδ+p-1H(f)(z))|Γ(p+1)Γ(p+μ+1)|z|p+μ(1-ab(p+1)2c(1+μ+p)|z|Eη,β),|Dz-μ(Dδ+p-1H(f)(z))|Γ(p+1)Γ(p+μ+1)|z|p+μ(1+ab(p+1)2c(1+μ+p)|z|Eη,β) for some η.

Upon setting δ=0 and p=1 in Theorems 3.6, 3.8, 3.9, and 3.10, we arrive at the following result.

Corollary 3.11.

Let a1, c2, μ>0, z𝒰, and p. If f𝒯δ,p,γ(β), then |DzμH(f)(z)|1Γ(2-μ)|z|1-μ(1-abc(2-μ)|z|Eη,β),|DzμH(f)(z)|1Γ(2-μ)|z|1-μ(1+abc(2-μ)|z|Eη,β) for some η and b2-μ. Furthermore |Dz-μH(f)(z)|1Γ(2+μ)|z|1+μ(1-abc(2+μ)|z|Eη,β),|Dz-μH(f)(z)|1Γ(2+μ)|z|1+μ(1+abc(2+μ)|z|Eη,β) for some η and b2+μ.

Remark 3.12.

Under the hypothesis of Corollary 3.11, DzμH(f)(z) and Dz-μH(f)(z) are included in disks with its center at origin and radii r and R, respectively, given by r=1Γ(2-μ)(1+abc(2-μ)Eη,β)(0μ<1,b2-μ,forsomeη),R=1Γ(2+μ)(1+abc(2+μ)Eη,β)(μ>0,b2+μ,forsomeη).

4. Distortion Inequalities of Integral Operator

In this section, we obtain the distortion theorems involving the integral operator Jδ,p of functions in the class 𝒯δ,p,γ(β) and fractional calculus operator.

Theorem 4.1.

Let 0μ<1/(δ+p+1), z𝒰, and p. If f𝒯δ,p,γ(β), then |Dzμ(z-δ(Jδ,pf)(z))|1Γ(2-μ)|z|-μ(1-p2(δ+p+1)(p+γ)|z|Eη,β),|Dzμ(z-δ(Jδ,pf)(z))|1Γ(2-μ)|z|-μ(1+p2(δ+p+1)(p+γ)|z|Eη,β) for some η. The operator (Jδ,pf)(z) is defined by (2.21).

Proof.

Using the definition (2.25), for function f𝒦(p) of the form (1.2), we have (Jδ,pf)(z)=zδ-k=1δ+pp+k+δap+k. So Dzμ(z-δ(Jδ,pf)(z))=1Γ(2-μ)z-μ-k=1(δ+p)Γ(k+1)(δ+p+k)Γ(k-μ+1)ap+kzk-μ. Therefore, we obtain Γ(2-μ)zμDzμ(z-δ(Jδ,pf)(z))=1-k=1ϕ(k)ap+kzk, where ϕ(k)=(δ+p)Γ(k+1)Γ(2-μ)(δ+p+k)Γ(k-μ+1). Since ϕ(k) is a decreasing function of k, when μ<1/(δ+p+1), then 0<ϕ(k)ϕ(1)=δ+pδ+p+1. By using (3.11), (4.4), and (4.6), we get |Γ(2-μ)zμDzμ(z-δ(Jδ,pf)(z))|1-p2(δ+p+1)(p+γ)|z|Eη,β,|Γ(2-μ)zμDzμ(z-δ(Jδ,pf)(z))|1+p2(δ+p+1)(p+γ)|z|Eη,β for some η, which prove the inequalities (4.1).

The proof of the following theorem is similar to Theorem 4.1, and so it is omitted here.

Theorem 4.2.

Let μ>0, p, and z𝒰. If f𝒯δ,p,γ(β), then |Dz-μ(z-δ(Jδ,pf)(z))|1Γ(μ+1)|z|μ(1-p2(δ+p+1)(μ+1)(p+γ)|z|Eη,β),|Dz-μ(z-δ(Jδ,pf)(z))|1Γ(μ+1)|z|μ(1+p2(δ+p+1)(μ+1)(p+γ)|z|Eη,β) for some η.

Acknowledgment

The authors thank the referee for some useful suggestions for improvement of the article.

AoufM. K.SilvermanH.SrivastavaH. M.Some families of linear operators associated with certain subclasses of multivalent functionsComputers & Mathematics with Applications2008553535549MR2384165ZBL1155.30309SrivastavaH. M.OwaS.AhujaO. P.A new class of analytic functions associated with the Ruscheweyh derivativesProceedings of the Japan Academy. Series A19886411720MR95375510.3792/pjaa.64.17ZBL0621.30010Khosravian-ArabSh.KulkarniS. R.JahangiriJ. M.Certain properties of multivalent functions associated with Ruscheweyh derivativeProceeding Book of the International Symposium on 2007 “Geometric Function Theory and Applications“2008Istanbul Kültür University Publications153160SwaminathanA.Inclusion theorems of convolution operators associated with normalized hypergeometric functionsJournal of Computational and Applied Mathematics200619711528MR225604810.1016/j.cam.2005.10.025ZBL1104.30008SwaminathanA.Certain sufficiency conditions on Gaussian hypergeometric functionsJournal of Inequalities in Pure and Applied Mathematics200454, article 83110MR2112436ZBL1126.30010SwaminathanA.Sufficiency for hypergeometric transforms to be associated with conic regionsMathematical and Computer Modelling2006443-4276286MR223905610.1016/j.mcm.2005.11.012ZBL1139.30308BarnardR. W.NaikS.PonnusamyS.Univalency of weighted integral transforms of certain functionsJournal of Computational and Applied Mathematics20061932638651MR222956510.1016/j.cam.2005.06.025ZBL1098.30016KimY. C.RønningF.Integral transforms of certain subclasses of analytic functionsJournal of Mathematical Analysis and Applications20012582466489MR183555410.1006/jmaa.2000.7383ZBL0982.44001HohlovY. E.Convolution operators preserving univalent functionsUkrainskiĭ Matematicheskiĭ Zhurnal1985372220226MR787030SekineT.OwaS.TsurumiK.Integral means of certain analytic functions for fractional calculusApplied Mathematics and Computation20071871425432MR232359810.1016/j.amc.2006.08.142ZBL1128.30012SrivastavaH. M.OwaS.Univalent Functions, Fractional Calculus, and Their Applications1989New York, NY, USAEllis Horwood, Chichester, UK; John Wiley & Sons404Ellis Horwood Series: Mathematics and Its ApplicationsMR1199135BernardiP. K.Convex and starlike univalent functionsTransactions of the American Mathematical Society1969135429446MR023292010.2307/1995025ZBL0172.09703