# A New Subclass ofk-Janowski Type Functions Associated with Ruscheweyh Derivative

Authors: Shahid Mahmood; Sarfraz Nawaz Malik; Saima Mustafa; S. M. Jawwad Riaz
Journal: Journal of Function Spaces
 (2017)

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## Abstract

We introduce and investigate a new subclassVDkA,B,b,δ of analytic functions using Ruscheweyh derivative. We derive the coefficient inequalities and other interesting properties and characteristics for functions belonging to the general class, which we have introduced and studied in this article. We also observe that this class is preserved under the Bernardi integral transform.


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## Body

## 1. Introduction

LetA denote the class of functions f(z) of the form(1)fz=z+∑n=2∞anzn,which are analytic in the unit disc O=z:z<1. Also let S∗β and Cβ denote the well-known classes of starlike and convex functions of order β, respectively. For details, see [1]. For any two analytic functions f(z) and g(z) with (2)fz=z+∑n=2∞anzn,gz=z+∑n=2∞bnzn,forz∈O,their convolution (Hadamard product) is given by (3)f∗gz=z+∑n=2∞anbnzn,forz∈O.In 1975, using the concept of convolution, Ruscheweyh [2] introduced a linear operator Dδ:A→A defined by (4)Dδfz=z1-zδ+1∗fz=z+∑n=2∞φnδanzn,δ>-1, with(5)φnδ=δ+1n-1n-1!,where (ρ)n is a Pochhammer symbol given as (6)ρn=1,n=0,ρρ+1ρ+2⋯ρ+n-1,n∈N.It is obvious thatD0fz=fz,D1fz=zf′z and (7)Dnfz=zzn-1fznn!,∀δ=n∈N0=0,1,2,….The following identity can easily be established:(8)δ+1Dδ+1fz=δDδfz+zDδfz′.The operatorDδfz is called the Ruscheweyh derivative of fz; see [2].Suppose also that, fork≥0, the classes k-CV and k-ST denote the well-known classes of k-uniformly convex and k-starlike functions, respectively. These classes were introduced by Kanas and Wisniowska [3, 4]. For some details see [3–5].Consider the domain(9)Ωk=u+iv;u>ku-12+v2.For fixedk,Ωk represents the conic region bounded successively by the imaginary axis (k=0), the right branch of a hyperbola (0<k<1), a parabola (k=1), and an ellipse (k>1). This domain was studied by Kanas [3–5]. The function pk, with pk0=1,pk′0>0, plays the role of extremal and is given by(10)pkz=1+z1-z,k=0,1+2π2log⁡1+z1-z2,k=1,1+21-k2sinh2⁡2πarccos⁡karctanh⁡z,0<k<1,1+1k2-1sin⁡π2Rt∫0uz/t11-x21-tx2dx+1k2-1,k>1,where u(z)=z-t/1-tz,t∈(0,1),z∈D, and t is chosen such that k=cosh⁡πR′(t)/4R(t), with R(t) being Legendre’s complete elliptic integral of the first kind and R′(t) being complementary integral of R(t) (see [3, 4, 6–8]). Let Pk denote the class of all those functions pz which are analytic in D with p0=1 and pz≺pkz for z∈D. Clearly it can be seen that Pk⊂P where P is the class of functions with positive real part (see [1, 9]). More precisely (11)Pk⊂Pk1+k⊂P, and, for fz∈Pk, we have (12)arg⁡pz≤λπ2, where(13)λ=2πarctan⁡1k.So we can writepz=hλz,hz∈P.Definition 1 (see [10]).
A functionpz is said to be in the class PkA,B, if and only if(14)pz≺pkA,B;z=A+1pkz-A-1B+1pkz-B-1,k≥0,where pk(z) is defined by (10) and -1≤B<A≤1. Geometrically, the function pz∈PkA,B takes all values from the domain ΩkA,B,-1≤B<A≤1,k≥0 which is defined as (15)ΩkA,B=w:Re⁡B-1wz-A-1B+1wz-A+1>kB-1wz-A-1B+1wz-A+1-1 or equivalently(16)ΩkA,B=u+iv:B2-1u2+v2-2AB-1u+A2-12>k2-2B+1u2+v2+2A+B+2u-2A+12+4A-B2v2.
The domainΩkA,B retains the conic domain Ωk inside the circular region defined by ΩkA,B. The impact of ΩkA,B on the conic domain Ωk changes the original shape of the conic regions. The ends of hyperbola and parabola get closer to each other but never meet anywhere and the ellipse gets the shape of oval. When A→1,B→-1, the radius of the circular disk defined by ΩA,B tends to infinity, consequently the arms of hyperbola and parabola expand, and the oval turns into ellipse.
With the help of the above Ruscheweyh derivative, we now define the following class.Definition 2.
A functionf(z)∈A of the form (1) is in the class VDkA,B,b,δ if and only if (17)Re⁡B-11-2/b+2/bDδ+1fz/Dδfz-A-1B+11-2/b+2/bDδ+1fz/Dδfz-A+1>kB-11-2/b+2/bDδ+1fz/Dδfz-A-1B+11-2/b+2/bDδ+1fz/Dδfz-A+1-1, or equivalently, (18)1-2b+2bDδ+1fzDδfz≺pkA,B;z,z∈O,with pkA,B;z being given by (14), k≥0, -1≤B<A≤1, δ>-1, andb∈C-0.
For different permissible choices of parameters, we obtain several known as well as new subclasses of the classA of analytic functions as special cases; for example,(i)
Forδ=0 and b=2, we obtain VDkA,B,2,0=k-STA,B, and for b=1, we have the class k-UCVA,B. These classes are recently introduced and studied by [10].(ii)
VDk1,-1,2,0=k-ST,VDk1,-1,1,0=k-UCV, the well-known classes of k-uniformly convex and k-starlike functions, respectively, introduced by Kanas and Wisniowska [3, 4].(iii)
VDk1-2α,-1,2,0=SDk,α,VDk1-2α,-1,1,0=KDk,α, the classes, introduced by Shams et al. in [11].(iv)
VD0A,B,2,0=S∗A,B,VD0A,B,1,0=CA,B, the well-known classes of Janowski starlike and Janowski convex functions, respectively, introduced by Janowski [12].
Throughout this paper, we assume thatk≥0, δ>-1, -1≤B<A≤1, andb∈C-0 unless otherwise stated.


## 2. Preliminary Results

We need the following lemmas to obtain our results.Lemma 3 (see [5]).
Letσ,λ with any complex numbers with λ≠0 and 0≤γ≤Reλk/k+1+σ. If ϕz is analytic in O with ϕ0=1 and satisfies(19)ϕz+zϕ′zλϕz+σ≺pk,γzand φk,γz is an analytic solution of (20)φk,γz+zφk,γ′zλφk,γz+σ=pk,γz, then φk,γz is univalent, (21)ϕz≺φk,γz≺pk,γz, and φk,γz is the best dominant of (19), where φk,γz is given as (22)φk,γz=∫01exp⁡∫ttzpk,γu-1ududt-1+σλ.Lemma 4 (see [10]).
Letpz=1+∑n=1∞‍cnzn∈k-PA,B. Then (23)cn≤QAB,where(24)QAB=A-BQk2,(25)Qk=8cos-1⁡k2π21-k2,0≤k<1,8π2,k=1,π24tk2-1R2t1+t,k>1.Lemma 5 (see [13]).
Letpz=1+∑n=1∞‍pnzn≺Fz=1+∑n=1∞‍dnzn in O. If Fz is univalent in O and FO is convex, then (26)pn≤d1,n≥1.Lemma 6 (see [14]).
Letβ0>0, β0+γ0>0 and α∈α0,1, where (27)α0=max⁡β0-γ0-12β0,-γ0β0.
Lethz∈A in O with h0=1 and let (28)hz+zh′zβ0hz+γ0∈Pα,0≤α<1.
Then(29)Re⁡hz>β0+γ0β0F122β01-α,1,β0+γ0+1,1/2-γ0β0,and the bound in (29) is sharp, the extremal functions being (30)h0=1β0g0z-γ0β0,with(31)g0z=∫011-z1-tz2β01-αtβ0+γ0-1dt.


## 3. Main Results

Theorem 7.
A functionfz∈A and of the form (1) is in the class VDkA,B,b,δ, if it satisfies the condition(32)∑n=2∞4n-1k+1δ+1+2B+1δ+nδ+1+bB-A-2B+1φnδan<bB-A.Proof.
Let we note that(33)B-11-2/b+2/bDδ+1fz/Dδfz-A-1B+11-2/b+2/bDδ+1fz/Dδfz-A+1-1=4Dδ+1fz-Dδfz2B+1Dδ+1fz+bB-A-2B+1Dδfz=∑n=2∞4n-1/δ+1φnδanznbB-Az+∑n=2∞2B+1δ+n/δ+1+bB-A-2B+1φnδanzn≤∑n=2∞4n-1/δ+1φnδanbB-A-∑n=2∞2B+1δ+n/δ+1+bB-A-2B+1φnδan,because from (32) it follows that (34)bB-A-∑n=2∞2B+1δ+nδ+1+bB-A-2B+1φnδan>0.
To show thatfz∈VDkA,B,b,δ it suffices that (35)kB-11-2/b+2/bDδ+1fz/Dδfz-A-1B+11-2/b+2/bDδ+1fz/Dδfz-A+1-1-ReB-11-2/b+2/bDδ+1fz/Dδfz-A-1B+11-2/b+2/bDδ+1fz/Dδfz-A+1<1.
From (33), we have (36)kB-11-2/b+2/bDδ+1fz/Dδfz-A-1B+11-2/b+2/bDδ+1fz/Dδfz-A+1-1-ReB-11-2/b+2/bDδ+1fz/Dδfz-A-1B+11-2/b+2/bDδ+1fz/Dδfz-A+1≤k+1B-11-2/b+2/bDδ+1fz/Dδfz-A-1B+11-2/b+2/bDδ+1fz/Dδfz-A+1-1≤∑n=2∞4n-1k+1/δ+1φnδanbB-A-∑n=2∞2B+1δ+n/δ+1+bB-A-2B+1φnδan.
The last expression is bounded above by1 if (37)∑n=2∞4n-1k+1δ+1+2B+1δ+nδ+1+bB-A-2B+1φnδan<bB-A, and this completes the proof.When we putδ=0 and b=2 in the above theorem, we obtain the following known result, proved by Noor and Malik in [10].Corollary 8.
A functionf(z)∈A and of the form (1) is in the class k-STA,B, if it satisfies the condition (38)∑n=2∞2n-1k+1+B+1n-1+Aφnδan<B-A.ForA=1-2α, B=-1 with 0≤α<1, δ=0, and b=2, Theorem 7 reduces to the following known result, proved by Shams et al. [11].Corollary 9.
A functionfz∈A and of the form (1) is in the class SD(k,α), if it satisfies the condition (39)∑n=2∞nk+1-k+αan<1-α.Theorem 10.
Letfz∈VDkA,B,b,δ. Then(40)an≤σn-1n-1!φnδ,forn≥2,where(41)σ=bQABδ+12,where QAB and φnδ are given by (24) and (5).Proof.
Set(42)1-2b+2bDδ+1fzDδfz=pz,so that pz∈PkA,B. Let pz=1+∑n=1∞cnzn. Then (42) can be written as (43)2Dδ+1fz-Dδfz=bDδfz∑n=1∞cnzn, which implies that (44)2φnδn-1anδ+1=bcn-1+φ2δa2cn-2+⋯+φn-1δan-1c1.
Using the coefficient estimatescn≤QAB for the class PkA,B (see [10]), we obtain (45)an≤bQABδ+12n-1φnδ1+φ2δa2+⋯+φn-1δan-1.
Forn=2, (46)a2≤bQAB2.
Therefore (40) holds for n=2. Assume that (40) is true for n=m and consider (47)am+1≤bQABδ+12mφm+1δ1+φ2δa2+⋯+φn-1δam≤bQABδ+12mφm+1δ+bQABδ+12!1+bQABδ+12+⋯+bQABδ+1m-1!∏j=1m-21+bQABδ+12j=bQABδ+12mφm+1δ∏j=1m-11+bQABδ+12j=σmm!φm+1δ.
Therefore, the result is true forn=m+1. Using mathematical induction, (40) holds true for all n≥2.Corollary 11 (see [10]).
Iffz∈VDkA,B,2,0=k-STA,B, then (48)an≤∏j=0n-2QkA-β-2jB2j+1,forn≥2, where Qk is defined by (8).WhenA=1, B=-1, b=2, and δ=0, we obtain the following coefficient inequality for the class k-ST, introduced by Kanas and Wisniowska [4].Corollary 12.
Iffz∈k-ST, then (49)an≤∏j=0n-2Qk+jj+1,forn≥2.
This result is sharp.By taking the valuesA=1-2α with 0≤α<1,B=-1, b=2, and δ=0, we obtain the coefficient inequality of the class SDk,α, introduced by Shams et al. [11].Corollary 13.
Iffz∈SDk,α, then (50)an≤∏j=0n-2Qk1-α+jj+1,forn≥2.
This result is sharp.Theorem 14.
For realb>0, let fz∈VDkA,B,b,δ+1. Then fz∈VDkA,B,b+1,δ for z∈O.Proof.
Supposefz∈VDkA,B,b,δ+1 and set(51)pz=1-2b+1+2b+1Dδ+1fzDδfz,where pz is analytic in O with p0=1. Then simple computations, together with (51) and (8), yield(52)1-2b+2bDδ+2fzDδ+1fz=1-λ1+λ1pz+λ2zp′zpz+λ3,with λ1=δ+1b+1/bδ+2,λ2=2/δ+1b+1,λ3=2/b+1-1. Since fz∈VDkA,B,b,δ+1, it follows that (53)1-λ1+λ1pz+λ2zp′zpz+λ3=h0z∈PkA,B, or, equivalently, (54)pz+zp′z1/λ2pz+λ3/λ2=1λ1h0z+1-1λ1p0z,p0z=1.
Sinceh0z,p0z∈PkA,B and PkA,B is a convex set (see [10]), it follows that 1/λ1h0z+1-1/λ1p0z, with p0z=1, belong to PkA,B in O and hence p(z)+zp′z/1/λ2pz+λ3/λ2∈PkA,B. We now use the Lemma 3 with η=1/λ2,σ=λ3/λ2, and Reηk/k+1+σ>0 to obtain p(z)∈PkA,B and hence fz∈VDkA,B,b+1,δ. This complete the proof.For a functionfz∈A, we consider the integral operator(55)Fz=Icfz=c+1zc∫0ztc-1ftdt,c>-1.The operatorIc was introduced by Bernardi [15] for c∈N. In particular, the operator I1 was studied earlier by Libera [16] and Livingston [17].Theorem 15.
Letfz∈VDkA,B,b,δ and let F(z) be defined by (55). Then F(z)∈VDkA,B,b,δ.Proof.
Let(56)1-2b+2bDδ+1FzDδFz=pz,where pz is analytic in O, p0=1. Then using (8), we have (57)zDδFz′DδFz=apz-a+1, where a=bδ+1/2. Simple computation and use of (8), (55), and (56), we have(58)1-2b+2bDδ+1fzDδfz=pz+zp′zapz-a+1+c≺pkA,B;z.
Apply Lemma3 with η=a,σ=1-a+c and Reηk/k+1+σ>0 to obtain p(z)≺pkA,B;z and consequently Fz∈VDkA,B,b,δ.Corollary 16.
Letb>0, fz∈VDkA,B,b,δ⊂VDα1b,δ, and α1=2k+1-A/2k+1-B and let F(z) be defined by (55). Then F(z)∈VDβb,δ, where c>-1, and(59)β=c+1aF122a1-α1,1;c+2;1/2-1+c-aa.Proof.
Proceeding as in Theorem15, it follows from (58) that (60)1-2b+2bDδ+1fzDδfz=pz+zp′zapz-a+1+c∈Pα1,α1=2k+1-A2k+1-B.
Applying Lemma6, we obtain pz∈Pβ, where β is given by (59). This proves that F(z)∈VDβb,δ in O.Theorem 17.
Iffz is of the form (1) belonging to VDkA,B,b,δ andFz=z+∑n=2∞‍bnzn, where Fz is the integral operator defined by (55), then (61)bn≤c+1c+nσn-1n-1!φnδ,forn≥2.Proof.
From (55), we obtain (62)c+1fz=cFz+zF′z.
Using the series for the functionsfz and Fz, we obtain (63)1+cz+∑n=2∞1+canzn=cz+∑n=2∞cbnzn+z+∑n=2∞nbnzn, and thus (64)n+cbn=1+can,n≥2.
From the above we have(65)bn≤c+1c+nan,n≥2.
Using the estimates from Theorem10, we obtain the required result.



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