# Class of Multivalent Analytic Functions Defined by a Linear Operator

Authors: B. A. Frasin
Journal: Journal of Complex Analysis
 (2013)

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## Abstract

Making use of the linear operatorJpm(λ,l) defined in (Prajapat, 2012), we introduce the class 𝔹pm(λ,l,μ,α) of analytic and p-valent functions in the open unit disk 𝒰. Furthermore, we obtain some sufficient conditions for starlikeness and close-to-convexity and some angular properties for functions belonging to this class. Several corollaries and consequences
of the main results are also considered.


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## Body

## 1. Introduction and Definitions

Let𝒜p(n) denote the class of functions f(z) of the form
(1)f(z)=zp+∑k=p+n∞‍akzk,(p,n∈ℕ:={1,2,3,…}),
which are analytic and p-valent in the open unit disk 𝒰={z:z∈ℂ and |z|<1}. In particular, we set 𝒜1(1)=:𝒜. A function f(z)∈𝒜p(n) is said to be in the class 𝒮p*(n,α) of p-valently starlike of order α in 𝒰 if and only if it satisfies the inequality
(2)Re(zf′(z)f(z))>α,(z∈𝒰;0≤α<p).Furthermore, a functionf(z)∈𝒜p(n) is said to be in the class 𝒞p(n,α) of p-valently close-to-convex of order α in 𝒰 if and only if it satisfies the inequality
(3)Re(f′(z)zp-1)>α,(z∈𝒰;≤α<p).In particular, we write𝒮1*(1,0)=:𝒮* and 𝒞1(1,0)=:𝒞, where 𝒮* and 𝒞 are the usual subclasses of 𝒜 consisting of functions which are starlike and close-to-convex in 𝒰, respectively.In [1], Prajapat define a generalized multiplier transformation operator Jpm(λ,l) as follows:
(4)Jp-m(λ,l)f(z)=p+lλzp-(p+l)/λ×∫0zt(p+l)/λ-p-1Jp-(m-1)(λ,l)f(t)dt,(z∈𝒰),⋮Jp-2(λ,l)f(z)=p+lλzp-(p+l)/λ×∫0zt(p+l)/λ-p-1Jp-1(λ,l)f(t)dt,(z∈𝒰),Jp-1(λ,l)f(z)=p+lλzp-(p+l)/λ∫0zt(p+l)/λ-p-1f(t)dt,(z∈𝒰),Jp0(λ,l)f(z)=f(z),Jp1(λ,l)f(z)=λp+lz1+p-(p+l)/λ(z(p+l)/λ-pf(z))′,(z∈𝒰),Jp2(λ,l)f(z)=λp+lz1+p-(p+l)/λ×(z(p+l)/λ-pJp1(λ,l)f(z))′,(z∈𝒰),⋮Jpm(λ,l)f(z)=λp+lz1+p-(p+l)/λ×(z(p+l)/λ-pJpm-1(λ,l)f(z))′,(z∈𝒰).We see that forf(z)∈𝒜p(n), we have
(5)Jpm(λ,l)f(z)=zp+∑k=p+n∞‍(p+l+λ(k-p)p+l)makzk,
where λ≥0,l>-p,p∈ℕ,m∈ℤ={0,±1,…} and z∈𝒰. It is readily verified from (5) that
(6)λz(Jpm(λ,l)f(z))′=(l+p)Jpm+1(λ,l)f(z)-(l+p(1-λ))Jpm(λ,l)f(z),(λ>0),λz(Jpm(λ,l)f(z))′′=(l+p)Jpm+1(λ,l)f′(z)-(l+λ+p(1-λ))Jpm(λ,l)f′(z),(λ>0).We observe that the operatorJpm(λ,l) generalize several previously studied familiar operators, and we will show some of the interesting particular cases as follows:(i)
Jpm(λ,l)f(z)=Ipm(λ,l)f(z)(l≥0,m∈ℕ0=ℕ∪{0}) (see [2]);(ii)
Jpm(1,l)f(z)=Ip(m,l)f(z)(l≥0,m∈ℕ0) (see [3, 4]);(iii)
Jpm(1,0)f(z)=Dpmf(z)(m∈ℕ0) (see [5–7]);(iv)
J1m(1,l)f(z)=Ilmf(z)(l≥0,m∈ℕ0) (see [8, 9]);(v)
J1m(1,0)f(z)=Dmf(z)(m∈ℕ0) (see [10]);(vi)
J1m(λ,0)f(z)=Dλmf(z)(m∈ℕ0) (see [11]);(vii)
J1m(1,1)f(z)=𝒟mf(z)(m∈ℕ0) (see [12]);(viii)
Jp-n(λ,l)f(z)=Jpn(λ,l)f(z)(n∈ℕ0) (see [13]).(For other generalizations of the operator Jpm(λ,l), see [1]).Making use of the above operatorJpm(λ,l), we introduce the class 𝔹pm(λ,l,μ,α) of analytic and p-valent functions defined as follows.Definition 1.
A functionf(z)∈𝒜p(n) is said to be a member of the class 𝔹pm(n,λ,l,μ,α) if and only if
(7)|(zpJpm(λ,l)f(z))μ-1z1-p(Jpm(λ,l)f(z))′-p|<p-α,(p∈ℕ),
for some α  (0≤α<p),   μ≥0, λ≥0,   l>-p,  p∈ℕ,m∈ℤ and for all z∈𝒰.Note that condition (7) implies that
(8)Re((zpJpm(λ,l)f(z))μ-1z1-p(Jpm(λ,l)f(z))′)>α.We note that𝔹p0(n,λ,l,μ,α)≡𝔹p1(n,0,l,μ,α)≡ℬ(p,n,μ,α), the class which has been introduced and studied by the author in [14]. Also, we have 𝔹p0(n,λ,l,2,α)≡𝒮p*(n,α), 𝔹p0(λ,l,1,α)≡𝒞p(n,α). The class 𝔹10(1,λ,l,3,α)≡ℬ(α) is the class which has been introduced and studied by Frasin and Darus [15] (see also [16, 17]).In this paper, we obtain some sufficient conditions and some angular properties for functions belonging to the class𝔹pm(n,λ,l,μ,α). Several corollaries and consequences of the main results are also considered.In order to derive our main results, we have to recall the following lemmas.Lemma 2 (see [18]).
Letw(z) be analytic in 𝒰 and such that w(0)=0. Then if |w(z)| attains its maximum value on circle |z|=r<1 at a point zo∈𝒰, one has
(9)zow′(zo)=kw(zo),
where  k≥1  is a real number.Lemma 3 (see [19]).
LetΩ be a set in the complex plane ℂ and suppose that Φ(z) is a mapping from ℂ2×𝒰 to ℂ which satisfies Φ(ix,y;z)∉Ω for z∈𝒰, and for all real x,y such that y≤-n(1+x2)/2. If the function q(z)=1+qnzn+qn+1zn+1+⋯ is analytic in 𝒰 such that Φ(q(z),zq′(z);z)∈Ω  for all z∈𝒰, then Req(z)>0.Lemma 4 (see [20]).
Letq(z) be analytic in 𝒰 with q(0)=1 and q(z)≠0 for all z∈𝒰. If there exist two points z1,   z2∈𝒰 such that
(10)-πα12=argq(z1)<argq(z)<argq(z2)=πα22
for α1>0,  α2>0, and for  |z|<|z1|=|z2|,  then we have
(11)z1q′(z1)q(z1)=-i(α1+α22)β,z2q′(z2)q(z2)=i(α1+α22)β,
where
(12)β≥1-|a|1+|a|,a=itanπ4(α2-α1α1+α2).


## 2. Sufficient Conditions for Starlikeness and Close-to-Convexity

Unless otherwise mentioned, we shall assume in the remainder of this paper that(13)μ,γ≥0;λ>0;l>-p;p,n∈ℕ,m∈ℤ.
Making use of Lemma 2, we first prove the following.Theorem 5.
Iff(z)∈𝒜p(n) satisfies
(14)|l+pλ(Jpm+1(λ,l)f′(z)Jpm(λ,l)f′(z)-(μ-1)Jpm+1(λ,l)f(z)Jpm(λ,l)f(z)+μ-2)+γ((zpJpm(λ,l)f(z))μ-1z1-p(Jpm(λ,l)f(z))′-p)|<(p-α)(γ(2p-α)+1)2p-α,(z∈𝒰),
for some α  (0≤α<p),  then f(z)∈𝔹pm(n,λ,l,μ,α).Proof.
Define the functionw(z) by
(15)(zpJpm(λ,l)f(z))μ-1z1-p(Jpm(λ,l)f(z))′=p+(p-α)w(z).
Then w(z) is analytic in 𝒰 and w(0)=0. It follows from (15) and the identities (6) and (1.6) that
(16)l+pλ(Jpm+1(λ,l)f′(z)Jpm(λ,l)f′(z)-(μ-1)Jpm+1(λ,l)f(z)Jpm(λ,l)f(z)+μ-2)+γ((zpJpm(λ,l)f(z))μ-1z1-p(Jpm(λ,l)f(z))′-p)=γ(p-α)w(z)+(p-α)zw′(z)p+(p-α)w(z).
Suppose that there exists z0∈𝒰 such that
(17)max|z|<z0|w(z)|=|w(z0)|=1.
Then from Lemma 2, we have (9). Therefore, letting w(z0)=eiθ,  we obtain that
(18)|l+pλ(Jpm+1(λ,l)f′(z0)Jpm(λ,l)f′(z0)-(μ-1)Jpm+1(λ,l)f(z0)Jpm(λ,l)f(z0)+μ-2)+γ((z0pJpm(λ,l)f(z0))μ-1z1-p(Jpm(λ,l)f(z0))′-p)|=|γ(p-α)w(z0)+(p-α)zw′(z0)p+(p-α)w(z0)|≥Re{γ(p-α)+(p-α)kp+(p-α)w(z0)}>γ(p-α)+p-α2p-α=(p-α)(γ(2p-α)+1)2p-α.
Which contradicts our assumption (14). Therefore we have |w(z)|<1 in 𝒰. Finally, we have
(19)|(zpf(z))μ-1z1-pf′(z)-p|=(p-α)|w(z)|<p-α(z∈𝒰),
that is, f(z)∈𝔹pm(n,λ,l,μ,α). This completes the proof of the theorem.Puttingm=l=0 and λ=1 in Theorem 5, we obtain the following.Corollary 6.
Iff(z)∈𝒜p(n) satisfies
(20)|1+zf′′(z)f′(z)-p+(μ-1)×(p-zf′(z)f(z))+γ((zpf(z))μ-1z1-pf′(z)-p)|<(p-α)(γ(2p-α)+1)2p-α,(z∈𝒰),
for some α(0≤α<p), then f(z)∈ℬ(p,n,μ,α).Puttingμ=λ=1 and m=l=0 in Theorem 5, one  obtains the following.Corollary 7.
Iff(z)∈𝒜p(n) satisfies
(21)|1+zf′′(z)f′(z)-p+γ(z1-pf′(z)-p)|<(p-α)(γ(2p-α)+1)2p-α(z∈𝒰),
for some    α(0≤α<p), then f(z)∈𝒞p(n,α).Lettingγ=p=n=1 in Corollary 7, one  hasCorollary 8.
Iff(z)∈𝒜 satisfies
(22)|zf′′(z)f′(z)+f′(z)-1|<(1-α)(3-α)2-α(z∈𝒰),
for some  α(0≤α<1), then f(z)∈𝒞(α). In particular, if f(z)∈𝒜 satisfies
(23)|zf′′(z)f′(z)+f′(z)-1|<32(z∈𝒰),
then f(z) is close-to-convex in 𝒰.Puttingμ=2, λ=1, and l=0 in Theorem 5, one  obtains the following.Corollary 9.
Iff(z)∈𝒜p(n) satisfies
(24)|1+zf′′(z)f′(z)-zf′(z)f(z)+γ((zpf(z))z1-p(f′(z)-p))|<(p-α)(γ(2p-α)+1)2p-α,
for some α(0≤α<p), then  f(z)∈𝒮p*(n,α).Puttingp=γ=n=1 in Corollary 9, one easily obtains the following result due to Owa [21].Corollary 10.
Iff(z)∈𝒜 satisfies
(25)|zf′′(z)f′(z)|<(1-α)(3-α)2-α,(z∈𝒰),
for some α(0≤α<1), then f(z)∈𝒮*(α). In particular, if  f(z)∈𝒜 satisfies
(26)|zf′′(z)f′(z)|<32(z∈𝒰),
then f(z) is starlike in 𝒰.Remark 11.
We note that the results obtained by the author  [14, Theorem 2.1, Corollaries 2.2–2.5] are not corrects. The correct results are given by Corollaries 6, 7, and 9.Next we prove the following.Theorem 12.
Iff(z)∈𝒜p(n)  satisfies
(27)Re{[(zpJpm(λ,l)f(z))μ-1z1-p(Jpm(λ,l)f(z))′]((zpJpm(λ,l)f(z))μ-1z1-p(Jpm(λ,l)f(z))′+l+pλl+pλ×(Jpm+1(λ,l)f′(z)Jpm(λ,l)f′(z)-(μ-1)Jpm+1(λ,l)f(z)Jpm(λ,l)f(z)))}>δ(l(2-μ)λ+δ+n2)+p(δ(2-μ)λ-n2).
then f(z)∈𝔹pm(n,λ,l,μ,α), where 0≤δ<p.Proof .
Define the functionq(z) by
(28)(zpJpm(λ,l)f(z))μ-1z1-p(Jpm(λ,l)f(z))′=δ+(p-δ)q(z).
Then, we see that q(z)=1+qnzn+qn+1zn+1+⋯ is analytic in  𝒰. A computation shows that
(29)[(zpJpm(λ,l)f(z))μ-1z1-p(Jpm(λ,l)f(z))′]{(zpJpm(λ,l)f(z))μ-1z1-p(Jpm(λ,l)f(z))′+l+pλ(Jpm+1(λ,l)f′(z)Jpm(λ,l)f′(z)-(μ-1)Jpm+1(λ,l)f(z)Jpm(λ,l)f(z))}=(p-δ)zq′(z)+(p-δ)2q2(z)+(p-δ)q(z)×((l+p)(2-μ)λ+2δ)+δ(l+p)(2-μ)λ+δ2=Φ(q(z),zq′(z);z),
where
(30)Φ(r,s;t)=(p-δ)s+(p-δ)2r2+(p-δ)r((l+p)(2-μ)λ+2δ)+δ(l+p)(2-μ)λ+δ2.
For all real x,y satisfying y≤-n(1+x2)/2, we have
(31)ReΦ(ix,y;z)=(p-δ)y-(p-δ)2x2+δ(l+p)(2-μ)λ+δ2≤-n2(p-δ)-(p-δ)[n2+p-δ]x2+δ(l+p)(2-μ)λ+δ2≤δ(l+p)(2-μ)λ+δ2-n2(p-δ)=δ(l(2-μ)λ+δ+n2)+p(δ(2-μ)λ-n2).
Let Ω={w:Rew>δ(l(2-μ)/λ+δ+n/2)+p(δ(2-μ)/λ-n/2)}. Then Φ(q(z),zq′(z);z)∈Ω and Φ(ix,y;z)∉Ω for all real x and y≤-n(1+x22)/2, z∈𝒰. By using Lemma 3, we have Req(z)>0, that is, f(z)∈𝔹pm(n,λ,l,μ,α).Puttingμ=λ=1 and m=l=0 in Theorem 12, we have the following.Corollary 13 (see [14]).
Iff(z)∈𝒜p(n) satisfies
(32)Re{(z1-pf′(z))2+z1-pf′(z)+z2-pf′′(z)}>δ(δ+n2)+p(δ-n2).
then f(z)∈𝒞p(n,δ),  where 0≤δ<p.  In particular, if  f(z)∈𝒜 satisfies
(33)Re{(f′(z))2+f′(z)+zf′′(z)}>-12
then f(z) is close-to-convex in 𝒰.Puttingm=l=0,  λ=1, and μ=2 in Theorem 12, one has the following.Corollary 14 (see [14]).
Iff(z)∈𝒜p(n) satisfies
(34)Re(zf′(z)f(z)+z2f′′(z)f(z))>δ(δ+n2)-n2p,
then f(z)∈𝒮p*(n,δ), where 0≤δ<p. In particular, if f(z)∈𝒜 satisfies
(35)Re(zf′(z)f(z)+z2f′′(z)f(z))>-12.
thenf(z) is starlike in 𝒰.


## 3. Argument Properties

Finally, we prove the following.Theorem 15.
Suppose that(36)(zpJpm(λ,l)f(z))μ-1z1-p(Jpm(λ,l)f(z))′≠δ
for  z∈𝒰  and  0≤δ<p. If f(z)∈𝒜p(n) satisfies
(37)-π2α1-tan-1(1-|a|1+|a|(α1+α2)(p-δ)2γ)<arg{((zpJpm(λ,l)f(z))μ-1z1-p(Jpm(λ,l)f(z))′)·([l+pλ(Jpm+1(λ,l)f′(z)Jpm(λ,l)f′(z)-(μ-1)Jpm+1(λ,l)f(z)Jpm(λ,l)f(z)+μ-2)]+γp-δ)-γδp-δ}<π2α2+tan-1(1-|a|1+|a|(α1+α2)(p-δ)2γ).
for  α1,α2,γ>0, then
(38)-π2α1<arg((zpJpm(λ,l)f(z))μ-1z1-p(Jpm(λ,l)f(z))′-δ)<π2α2.Proof.
Define the functionq(z) by
(39)q(z)=1p-δ((zpJpm(λ,l)f(z))μ-1z1-p(Jpm(λ,l)f(z))′-δ).
Then we see that  q(z)  analytic in 𝒰, q(0)=1, and q(z)≠0 for all  z∈𝒰. It follows from (39) that
(40)((zpJpm(λ,l)f(z))μ-1z1-p(Jpm(λ,l)(f(z))′)·([l+pλ(Jpm+1(λ,l)f′(z)Jpm(λ,l)f′(z)-(μ-1)Jpm+1(λ,l)f(z)Jpm(λ,l)f(z)+μ-2Jpm+1(λ,l)f′(z)Jpm(λ,l)f′(z))l+pλ]+γp-δ)-γδp-δ=(p-δ)zq′(z)+γq(z).
Suppose that there exists two pointsz1,z2∈𝒰 such that the condition (10) is satisfied, then by Lemma 4, we obtain (11) under the constraint (12). Therefore, we have
(41)arg(γq(z1)+(p-δ)zq′(z1))=argq(z1)+arg(γ+(p-δ)z1q′(z1)q(z1))=-π2α1+arg(γ-i(α1+α2)(p-δ)2β)=-π2α1-tan-1((α1+α2)(p-δ)2γβ)≤π2α1-tan-1(1-|a|1+|a|(α1+α2)(p-δ)2γ),arg(γq(z2)+(p-δ)zq′(z2))≥π2α2+tan-1(1-|a|1+|a|(α1+α2)(p-δ)2γ),
which contradict the assumption of the theorem. This completes the proof.Puttingμ=λ=1 and m=l=0 in Theorem 15, one has the following.Corollary 16 (see [14]).
Suppose thatz1-pf′(z)≠δ for  z∈𝒰  and  0≤δ<p. If  f(z)∈𝒜p(n) satisfies
(42)-π2α1-tan-1(1-|a|1+|a|(α1+α2)(p-δ)2γ)<arg{z1-pf′(z)(1+zf′′(z)f′(z)-p+γp-δ)-γδp-δ}<π2α2+tan-1(1-|a|1+|a|(α1+α2)(p-δ)2γ),
for α1,α2,γ>0, then
(43)-π2α1<arg(z1-pf′(z)-δ)<π2α2.
In particular, if f(z)∈𝒜 satisfies
(44)|arg(zf′′(z)+f′(z)(γ+1)-z(f′(z))2f(z))|<π2α+tan-1αγ,
for  γ>0, then
(45)|argf′(z)|<π2α,(0<α≤1).
Putting μ=2,  λ=1, and l=0 in Theorem 15, one has the following.Corollary 17 (see [14]).
Suppose thatzf′(z)/f(z)≠δ for z∈𝒰 and 0≤δ<p. If f(z)∈𝒜p(n) satisfies
(46)-π2α1-tan-1(1-|a|1+|a|(α1+α2)(p-δ)2γ)<arg{zf′(z)f(z)(1+zf′′(z)f′(z)-zf′(z)f(z)+γp-δ)-γδp-δ}<π2α2+tan-1(1-|a|1+|a|(α1+α2)(p-δ)2γ),
for α1,α2,γ>0, then
(47)-π2α1<arg(zf′(z)f(z)-δ)<π2α2.
In particular, if f(z)∈𝒜 satisfies
(48)|arg(z2f′′(z)f(z)-(zf′(z)f(z))2+zf′(z)f(z)(γ+1))|<π2α+tan-1αγ,
for γ>0, then
(49)|argzf′(z)f(z)|<π2α,(0<α≤1).Remark 18.
Taking different choices ofm,p,λ, and l in the above theorems, we obtain some sufficient conditions for starlikeness and close-to-convexity and some angular properties for functions belonging to new classes defined by the previously operators mentioned in Section 1.



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