# On Certain Classes ofp-Valent Functions by Using Complex-Order and Differential Subordination

Authors: Abdolreza Tehranchi; Adem Kılıçman
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

The aim of the present paper is to study thep-valent analytic functions in the unit disk and satisfy the differential subordinations z(ℐp(r,λ)f(z))(j+1)/(p-j)(ℐp(r,λ)f(z))(j)≺(a+(aB+(A-B)β)z)/a(1+Bz), where Ip(r,λ) is an operator defined by Sălăgean and β is a complex number. Further we define a new related integral operator and also study the Fekete-Szego problem by proving some interesting properties.


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## Body

## 1. Introduction

Let𝒜 be the class of  analytic functions in Δ={z∈ℂ:|z|<1}. Let 𝒜p denote the class of all analytic functions in the form of
(1.1)f(z)=ezp-∑n=p-12p-1tn-p+1zn-p+1+F21(a,b;c;z),|z|<1,
where F1(a,b;c;z) is Gaussian hypergeometric function defined by
(1.2)F21(a,b;c;z)=∑n=0∞(a,n)(b,n)(c,n)n!zn,(a,n)=Γ(a+n)Γ(a)=a(a+1,n-1),c>b>0,c>a+b,tn-p+1=(a,n-p+1)(b,n-p+1)(c,n-p+1)(n-p+1)!,e>0.
Note that it is easy to see that these functions are analytic in the unit disk Δ; for more details on hypergeometric functions 2F1(a,b;c.z), see [1, 2].Definition 1.1.
A functionf∈𝒜p is said to be in the class Sp*(α), p-valently starlike functions of order α, if it satisfies Re{zf′(z)/f(z)}>α,(0≤α<p,z∈Δ). We write Sp*(0)=Sp*, the class of p-valently starlike functions in Δ.
Similarly, a functionf∈𝒜p is said to be in the class 𝒞p(α), p-valently convex of order α, if it satisfies Re{1+zf′′(z)/f′(z)}>α,(0≤α<p,z∈Δ).
Leth(z) be analytic and h(0)=p. A function f∈𝒜p is in the class Sp*(h) if (1.3)zf′(z)f(z)≺h(z),z∈Δ.
The class Sp*(h) and a corresponding convex class 𝒞p(h) were defined by Ma and Minda in [3]. Similar results which are related to the convex class can also be obtained easily from the corresponding functions in Sp*(h). For example,  (i)
ifp=1 and
(1.4)h(z)=1+z1-z,
then the classes reduce to the usual classes of starlike and convex functions;(ii)
ifh(z)=(1+(1-2α)z)/(1-z) where 0≤α<1, then the classes are reduced to the usual classes of starlike and convex functions of order α;(iii)
ifh(z)=p((1+Az)/(1+Bz)), where -1≤B<A≤1, then the classes are reduced to the class of Janowski starlike functions Sp*[A,B] which is defined by
(1.5)Sp*[A,B]={f∈Ap:zf′f≺p1+Az1+Bz,-1≤B<A≤1,z∈Δ};(iv)
ifh(z)=((1+z)/(1-z))α where p=1 and 0<α≤1, then the classes reduce to the classes of strongly starlike and convex functions of order α that consists of univalent functions f∈𝒜 satisfing
(1.6)|arg(zf′(z)f(z))|<απ2,0<α≤1,z∈Δ
or equivalently we have
(1.7)SS*(α)={f∈Ap:zf′f≺(1+z1-z)α,0<α≤1,z∈Δ}.
In the literature, there are several works and many researchers have been studying the related problems. For example, Obradović and Owa [4], Silverman [5], Obradowič and Tuneski [6], and Tuneski [7] have studied the properties of classes of functions which are defined in terms of the ratio of 1+zf′′(z)/f′(z)andzf′(z)/f(z).Definition 1.2.
A functionf∈𝒜p is said to be p-valent Bazilevic of type η and order α if there exists a function g∈Sp* such that
(1.8)Re{zf′(z)f1-η(z)gη(z)}>α(z∈Δ)
for some η(η≥0) and α(0≤α<p). We denote by ℬp(η,α), the subclass of 𝒜p consisting of all such functions. In particular, a function in ℬp(1,α)=ℬp(α) is said to be p-valently close-to-convex of order α in Δ.Definition 1.3.
Letf and g be analytic functions in Δ, then we say f is subordinate to g and denoted by f≺g if there exists a Schwarz function w(z), analytic in Δ with w(0)=0 and |w(z)|<1, such that f(z)=g(w(z)),z∈Δ. In particular, if the function g is univalent in Δ, the above subordination is equivalent to f(0)=g(0) and f(Δ)⊂g(Δ). Also, we say that g is superordinate to f; see [8].Definition 1.4.
Motivated by the multiplier transformation on𝒜, we define the operator ℐp(r,λ); by the following infinite series when f(z)=zp+∑n=p+1∞anzn then
(1.9)Ip(r,λ)f(z)=zp+∑n=1+p∞(n+λp+λ)ranzn(λ≥0).
Sălăgean derivative operator is closely related to the operator ℐp(r,λ); see [9]. In [10], Uralegaddi and Somanatha also studied the case ℐ1(r,1)=ℐr. The operator ℐ1(r,λ)=ℐrλ was studied recently by Cho and Srivastava [11] and Cho and Kim [12].Definition 1.5.
Differential operator, for eachf(z)=zp+∑n=p+1∞anzn, we have
(1.10)f(j)(z)=p!(p-j)!zp-j+∑n=1+p∞n!(n-j)!anzn-j,
where n,p∈N,p>j, and j∈N0={0}∪N. In particular, if j=0 we have f(0)(z)=f(z).Definition 1.6.
A functionf∈𝒜p is said to be in the class 𝒜p(λ,r,j;h) if it satisfies the following subordination:
(1.11)z(Ip(r,λ)f(z))(j+1)(p-j)(Ip(r,λ)f(z))(j)≺h(z),
and in this study we consider
(1.12)h(z)=1+A-Baβz1+Bz,z∈Δ,
where -1≤B<A≤1,a>0 and β(≠0) is a complex number; so we denote 𝒜¯p(λ,r,j;h)=𝒜¯p(λ,r,j,β,a,A,B). Then we say that f(z) is superordinate to h(z) if f(z) satisfies the following:
(1.13)h(z)≺z(Ip(r,λ)f(z))(j+1)(p-j)(Ip(r,λ)f(z))(j),
where h(z) is analytic in Δ and h(0)=1.
Further we note that if(1.14)z(Ip(r,λ)f(z))(j+1)(Ip(r,λ)f(z))(j)≺(p-j)[a+(aB+(A-B)β)z]a(1+Bz)=h(z).
By choosing j=r=0,p=1, so h(0)=1, then f(z)∈S*(h). For a=A=β=1,B=-1, and p≥1, we have f(z)∈Sp*(1). But if a=β=1 and j=r=0 and -1≤B<A≤1, then f(z)∈S*[A,B], a class of Janowski starlike functions. If we put p=a=β=A=1,B=-1, then f(z)∈SS*(1) classes of strongly starlike. By Definition 1.2, if g(z)∈S*, univalent starlike, and j=r=0 and p=a=A=β=1,B=-1 and if Re{zf′(z)/f(z)g2(z)}>1, then f(z)∈ℬ(2,1) is a class Bazilevic functions of type η=2 and order α=1.


## 2. Main Results

Theorem 2.1.
Let the functionf(z) be of the form (1.1). If some A,B,(-1<B<A≤1), and β(≠0) are complex numbers and
(2.1)∑m=p+1∞γλr(m,p)[a(1+B)(δ(m,j+1)-δ(m,j)(p-j))-δ(m,j)(A-B)(p-j)|β|]km<|β|e(A-B)δ(p,j+1),
then f(z)∈𝒜¯p(λ,r,j,β,a,A,B), where γλr(m,p)=((m+λ)/(p+λ))r,δ(m,j)=m!/(m-j)! and for r,j∈ℕ0,λ>0,p∈ℕ,j<p. The result is sharp.Proof.
Since the functionf(z) in the theorem can be expressed in the form
(2.2)f(z)=ezp+∑n=2p∞kn-p+1zn-p+1orf(z)=ezp+∑m=p+1∞kmzm,
where m=n-p+1 and km=(a,m)(b,m)/(c,m)m!, and also we have for all r,j∈𝒩0,
(2.3)(Ip(r,λ)f(z))(j)=ep!(p-j)!zp-j+∑m=p+1∞(m+λp+λ)rm!(m-j)!kmzm-j=eδ(p,j)zp-j+∑m=p+1∞γλr(m,p)δ(m,j)kmzm-j,
now, assume that the condition (2.1) holds true. We show that f∈𝒜¯p(λ,r,j,β,a,A,B). Equivalently, we prove that
(2.4)|az(Ip(r,λ)f(z))(j+1)-a(p-j)(Ip(r,λ)f(z))(j)(p-j)R(Ip(r,λ)f(z))(j)-Baz(Ip(r,λ)f(z))(j+1)|<1,
where R=aB+(A-B)β. But we have
(2.5)|az(Ip(r,λ)f(z))(j+1)-a(p-j)(Ip(r,λ)f(z))(j)(p-j)R(Ip(r,λ)f(z))(j)-Baz(Ip(r,λ)f(z))(j+1)|=|[a∑m=p+1∞γλr(m,p)(δ(m,j+1)-δ(m,j)(p-j))kmzm-j][β(A-B)δ(p,j+1)ezp-j-∑m=p+1∞γλr(m,p)(aBC-δ(m,j)(A-B)β(p-j))kmzm-j]|<{[a∑m=p+1∞γλr(m,p)(δ(m,j+1)-δ(m,j)(p-j))km][∑m=p+1∞|β|e(A-B)δ(p,j+1)-∑m=p+1∞γλr(m,p)(aBC-δ(m,j)(A-B)|β|(p-j))km]}<1.
where 𝒞 denotes (δ(m,j+1)-δ(m,j)(p-j)).
The last inequality is true by (2.1) and this completes the proof. The result is sharp for the functions fm(z) defined in Δ by
(2.6)fm(z)=ezp+|β|e(A-B)δ(p,j+1)γλr(m,p)[a(1+B)(δ(m,j+1)-δ(m,j)(p-j))-δ(m,j)(A-B)(p-j)|β|]zm
form≥p+1.Remark 2.2.
We observe that ifB≠0, the converse of the above theorem needs not be true. For instance, consider the function f(z) defined by
(2.7)z(Ip(r,λ)f(z))(j+1)((p-j)Ip(r,λ)f(z))(j)≺a-Sgn(B)(pB+(A-B)β)za(1-Sgn(B)Bz),
where Sgn(B)=1,0,-1 thus accordingly B>0,B=0, and B<0. It is easily seen that f(z)∈𝒜¯p(λ,r,j,β,a,A,B) and
(2.8)km=-(1+B)βe(A-B)δ(p,j+1)Sgn(B)m-p-1Bm-p-2[a(1+B)(δ(m,j+1)+δ(m,j)(p-j))+δ(m,j)(A-B)(p-j)β]γλr(m,p)(m≥p+1)
so that
(2.9)∑m=p+1∞γλr(m,p)[a(1+B)(δ(m,j+1)-δ(m,j)(p-j))-β(A-B)δ(m,j)(p-j)βe(A-B)δ(p,j+1)]|km|=(1+B)∑m=p+1∞(B)m-p-2=1+B1-B>1,
where A,B are satisfying the conditions -1≤B<A≤1,0<B<1. This establishes our claim.Theorem 2.3.
If the functionf(z)∈𝒜¯p(λ,r,j,β,a,A,B), then
(2.10)|km|≤|β|(A-B)δ(p,j+1)eaγλr(m,p)(δ(m,j+1)-δ(m,j)(p-j)),m≥p-1,
where -1≤B<A≤1 and 0<a<A-B≤1, and the estimate is sharp.Proof.
We have(2.11)z(Ip(r,λ)f(z))(j+1)(p-j)(Ip(r,λ)f(z))(j)=a+(aB+(A-B)β)w(z)a(1+Bw(z)),
where w(z)=∑i=p+1∞wi-pzi-p is defined as in the Definition 1.3. Now we can write
(2.12)a∑i=p+1∞γλr(i,p)(δ(i,j+1)-δ(i,j)(p-j))kizi-j={β(A-B)δ(p,j+1)ezp-j-∑i=p+1∞γλr(i,p)(δ(i,j+1)Ba-δ(i,j)(p-j)R)kizi-j∑i=p+1∞γλr(i,p)}∑i=p+1∞wi-pzi-p,
where R=aB+(A-B)β. Now if we equalize the coefficients of the same power of z in both sides, then we have
(2.13)a∑i=p-1mγλr(i,p)(δ(i,j+1)-δ(i,j)(p-j))kizi-j+∑i=m+1∞cizi-j={β(A-B)δ(p,j+1)ezp-j-∑i=p+1m-1γλr(i,p)(δ(i,j+1)Ba-δ(i,j)(p-j)R)kizi-j}w(z),
where ci’s are suitable constants. By multiplying each side of the above equation by its conjugate and letting |z|=1, r→1-, we get
(2.14)a2∑i=p-1mγλ2r(i,p)(δ(i,j+1)-δ(i,j)(p-j))2|ki|2≤[|β|(A-B)δ(p,j+1)e]2+∑i=p+1m-1γλ2r(i,p)(δ(i,j+1)Ba-δ(i,j)(p-j)R)2|ki|2
so that
(2.15)a2γλ2r(m,p)(δ(m,j+1)-δ(m,j)(p-j))2|km|2≤[|β|(A-B)δ(p,j+1)e]2-(1-a2)∑i=p+1m-1γλ2r(i,p)(δ(i,j+1)Ba-δ(i,j)(p-j)R)2|ki|2.
Since -1<B<A≤1 and 0<a<A-B<1, we have
(2.16)|km|≤|β|(A-B)δ(p,j+1)eaγλr(m,p)(δ(m,j+1)-δ(m,j)(p-j)),m≥p-1
and this completes the proof. Note that the estimate in (2.10) is sharp for the functions fm(z) defined in Δ; when j=r=0 in (1.3), then
(2.17)fm(z)=exp⁡[∫0zp(ψ(t)R+a)t(1+Bψ(t))dt],m≥1+p,
where |ψ(t)|<1, z∈Δ and R=aB+(A-B)β. We can choose ψ(t)=tm.Theorem 2.4 ([Fekete-Szego Problem]).
Let the functionf(z), given by (2.2), be in the class 𝒜¯p(λ,j,β,a,A,B) and μ any complex number. Then
(2.18)|kp+2-μkp+12|≤(A-B)δ(p,j+1)e|β|2aγλr(p+2,p)×max⁡{1,|2γλr(p+2,p)δ(p+2,j)(aγλr(p+1,p)δ(p+1,j)+μ(A-B)δ(p,j+1)eβ)a(γλr(p+1,p))2|}.Proof.
On using the coefficients ofzp+1 and zp+2, we get
(2.19)kp+1=(A-B)βeδ(p,j+1)aγλr(p+1,p)δ(p+1,j)w1,kp+2=(A-B)eβδ(p,j+1)aγλr(p+2,p)2δ(p+2,j)w2-(A-B)eβδ(p,j+1)aγλr(p+1,p)δ(p+1,j)w12.
By using [13] that
(2.20)|w2-ρw12|≤max⁡{1,|ρ|}
for every complex number ρ, then we can write
(2.21)|kp+2-μkp+12|=|1a(A-B)δ(p,j+1)eβ(w22γλr(p+2,p)δ(p+2,j)-w12γλr(p+1,p)δ(p+1,j))-μ((A-B)δ(p,j+1)eβaγλr(p+1,p)δ(p+1,j))2w12|=|(A-B)δ(p,j+1)eβ2aγλr(p+2,p)δ(p+2,j)w2-(A-B)δ(p,j+1)eβaγλr(p+1,p)δ(p+1,j)-μ((A-B)δ(p,j+1)eβ)2(aγλr(p+1,p)δ(p+1,j))2w12|=(A-B)δ(p,j+1)e|β|2aγλr(p+2,p)|w2-hw12|,
where
(2.22)h=2γλr(p+2,p)δ(p+2,j)(aγλr(p+1,p)δ(p+1,j)-μ(A-B)eβδ(p,j+1))a(γλr(p+1,p)δ(p+1,j))2.


## 3. Integral Operator

Now, we introduce a new integral operator which is denoted by𝒢η,p(z) on functions belonging to 𝒜¯p as follows: (3.1)Gη,p(z)=e(1-η)zp+ηp∫ϵzf(t)tdt(0<η<1,ϵ⟶0+),
and we verify the effect of this operator on (1.11); with a simple calculation, we have  (3.2)Gη,p(z)=e(1-η)zp+ηp[∫ϵz(etp-1+∑m=p+1∞kmtm-1)dt](0<η<1,ϵ⟶0+)=e(1-η)zp+ηezp+∑m=p+1∞ηpmkmzm=ezp+∑m=p+1∞dmzm,
where dm=(ηp/m)km. If we put r=j=0 in (1.11), then we obtain (3.3)zf′(z)pf(z)≺a+(aB+(A-B)β)za(1+Bz)
that is denoted by 𝒜¯p(λ,0,0,β,a,A,B)=𝒜¯p(λ,β,a,A,B).Now if we let𝒜¯p,η(λ,β,a,A,B) be a class of functions 𝒢η,p(z) analytic in Δ and defined by (3.1) where f(z)∈𝒜¯p(λ,β,a,A,B). Then on using (3.1) and definition of subordination, we have the following theorem.Theorem 3.1.
𝒢η,p(z)∈𝒜¯p,η(λ,β,a,A,B) if and only if
(3.4)zcgη,p′(z)+z2Gη,p′′(z)-ep2(1-η)zppzGη,p′-ep2(1-η)zp≺a+(aB+(A-B)β)za(1+Bz).Proof.
The conditions (3.3) and (3.1) give
(3.5)Gη,p′(z)=ep(1-η)zp-1+ηpf(z)zorf(z)=zGη,p′(z)ηp-e(1-η)zpη,f′(z)=1η,p(Gη,p′(z)+zGη,p′′(z))-epη(1-η)zp-1=1ηp(Hη,p(z))′-epη(1-η)zp-1,
where Hη,p(z)=z𝒢η,p′(z). By putting f′(z) and f(z) in (3.3), we obtain
(3.6)zf′(z)pf(z)=zHη,p′(z)-ep2(1-η)zpHη,p(z)-ep2(1-η)zp=zGη,p′(z)+z2Gη,p′′(z)-ep2(1-η)zppzGη,p′(z)-ep2(1-η)zp≺a+(aB+(A-B)β)za(1+Bz).
With a simple calculation on Fξ(z), we have
(3.7)Fξ(z)=p+ξzξ[∫0zsξ-1(esp+∑m=p+1∞kmsm)ds]=ezp+∑m=p+1∞p+ξm+ξkmzm=ezp+∑p+1∞bmzmwherebm=p+ξm+ξkm.
Let 𝒜¯p,ξ(λ,β,a,A,B) be the class of functions Fξ(z) analytic in Δ defined by f∈𝒜¯p(λ,β,a,A,B). We can write next theorem on using (3.3) and definition of subordination.Theorem 3.2.
TheFξ(z)∈𝒜¯p,ξ(λ,β,a,A,B) if and only if
(3.8)z((ξ+1)Fξ′(z)+zFξ′′(z))p(ξFξ(z)+zFξ′(z))≺a+(aB+(A-B)β)za(1+Bz).Proof.
Since(3.9)Fξ(z)=p+ξzξ∫0zsξ-1f(s)ds,
then we have
(3.10)f(z)=1p+ξ(ξFξ(z)+zFξ′(z)),f′(z)=1p+ξ((ξ+1)Fξ′(z)+zFξ′′(z)).
Now by making substitution f′(z) and f(z) in (3.3), we obtain
(3.11)zf′(z)pf(z)=(z/(p+ξ))((ξ+1)Fξ′(z)+zFξ′′(z))(p/(p+ξ))(ξFξ(z)+zFξ′(z))=z((ξ+1)Fξ′(z)+zFξ′′(z))p(ξFξ(z)+zFξ′(z))≺a+(aB+(A-B)β)za(1+Bz).



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