# Some Inclusion Relationships of Certain Subclasses ofp-Valent Functions Associated with a Family of Integral Operators

Authors: M. K. Aouf; R. M. El-Ashwah; Ahmed M. Abd-Eltawab
Journal: ISRN Mathematical Analysis
 (2013)

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## Abstract

By making use of the new integral operatorℜβ,pα,γ, we introduce and investigate several new subclasses of p-valent starlike, p-valent convex, p-valent close-to-convex, and p-valent quasi-convex functions. In particular, we establish some inclusion relationships associated with the aforementioned integral operators. Some of the results established in this paper would provide extensions of those given in earlier works.


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## Body

## 1. Introduction

LetA(p) denote the class of functions of the form
(1)f(z)=zp+∑k=1∞ak+pzk+p(p∈ℕ={1,2,3,…}),
which are analytic and p-valent in the unit disc U={z:z∈ℂ, and |z|<1} and let A(1)=A.A functionf∈A(p) is said to be in the class Sp*(λ) of p-valent starlike functions of order λ in U if and only if
(2)Re(zf′(z)f(z))>λ(z∈U;0≤λ<p).
The class Sp*(λ) was introduced by Patil and Thakare [1].Owa [2] introduced the class Kp(λ) of p-valent convex of order λ in U if and only if
(3)Re(1+zf′′(z)f′(z))>λ(z∈U;0≤λ<p).
It is easy to observe from (2) and (3) that
(4)f(z)∈Kp(λ)⟺zf′(z)p∈Sp*(λ).We denote bySp*=Sp*(0) and Kp=Kp(0) where Sp* and Kp are the classes of p-valently starlike functions and p-valently convex functions, respectively, (see Goodman [3]).For a functionf∈A(p), we say that f∈Cp(η,λ) if there exists a function g∈Sp*(λ) such that
(5)Re(zf′(z)g(z))>η(z∈U;0≤λ,η<p).
Functions in the class Cp(η,λ) are called p-valent close-to-convex functions of order η and type λ. The class Cp(η,λ) was studied by Aouf [4] and the class C1(η,λ) was studied by Libera [5].Noor [6, 7] introduced and studied the classes Cp*(η,λ) and C1*(η,λ) as follows.A functionf∈A(p) is said to be in the class Cp*(η,λ) of quasi-convex functions of order η and type λ if there exists a function g∈Kp(λ) such that
(6)Re{(zf′(z))′g′(z)}>η(z∈U;0≤λ,η<p).
It follows from (5) and (6) that
(7)f(z)∈Cp*(η,λ)⟺zf′(z)p∈Cp(η,λ).For functionsf∈A(p) given by (1) and g∈A(p) given by
(8)g(z)=zp+∑k=1∞bk+pzk+p(p∈ℕ),
the Hadamard product (or convolution) of f and g is given by
(9)(f*g)(z)=zp+∑k=1∞ak+pbk+pzk+p=(g*f)(z).For the functionf∈A(p), we introduced the operator ℜβ,pα,γ:A(p)→A(p) as follows:
(10)ℜβ,pα,γf(z)=(p+α+β-γp+β-1)(α-γ+1)zβ×∫0z(1-tz)α-γtβ-1f(t)dt=Γ(p+α+β-γ+1)Γ(p+β)Γ(α-γ+1)1zβ×∫0z(1-tz)α-γtβ-1f(t)dt=zp+Γ(p+α+β-γ+1)Γ(p+β)×∑k=1∞[Γ(β+p+k)Γ(α+β+p+k-γ+1)]ak+pzk+p00(β>-p;α≥γ-1;γ>0;p∈ℕ;z∈U).From (10), it is easy to verify that
(11)z(ℜβ,pα+1,γf(z))′=(α+β+p-γ+1)ℜβ,pα,γf(z)-(α+β-γ+1)ℜβ,pα+1,γf(z).Remark 1.
Consider (i) Forγ=1,
(12)ℜβ,pα,1f(z)=Qβ,pαf(z)=(p+α+β-1p+β-1)αzβ∫0z(1-tz)α-1tβ-1f(t)dt=Γ(p+α+β)Γ(p+β)Γ(α)1zβ∫0z(1-tz)α-1tβ-1f(t)dt=zp+Γ(p+α+β)Γ(p+β)×∑k=1∞[Γ(β+p+k)Γ(α+β+p+k)]ak+pzk+p00000(β>-p;α≥0;p∈ℕ;z∈U),
where the operator Qβ,pα was introduced and studied by Liu and Owa [8], and Qβ,1α=Qβα, where the operator Qβα was introduced and studied by Jung et al. [9];
(ii) Forα=γ=1 and β=c,
(13)ℜc,p1,1f(z)=Jc,p(z)=p+czc∫0ztc-1f(t)dt=zp+∑k=1∞(c+pc+k+p)ak+pzk+p000000(c>-p;p∈ℕ;z∈U),
where Jc,p(z) is the familiar integral operator, which was defined by Cho and Kim [10]. The operator Jc,1(z)=Jc(z) was introduced by Bernardi [11] and we note that J1,1(z)=J(z) was introduced and studied by Libera [12] and Livingston [13].
The main object of this paper is to investigate the various inclusion relationships for each of the following subclasses of the normalized analytic function classA(p) which are defined here by means of the operator ℜβ,pα,γ given by (10).Definition 2.
In conjunction with (2) and (10),
(14)Sα,β,γ,p*(λ)={f:f∈A(p),ℜβ,pα,γf∈Sp*(λ),0000≤λ<p,p∈ℕℜβ,pα,γ}.Definition 3.
In conjunction with (3) and (10),
(15)Kα,β,γ,p(λ)={f:f∈A(p),ℜβ,pα,γf∈Kp(λ),0000≤λ<p,p∈ℕℜβ,pα,γ}.Definition 4.
In conjunction with (5) and (10),
(16)Cα,β,γ,p(η,λ)={f:f∈A(p),ℜβ,pα,γf∈Cp(η,λ),0000≤λ,η<p,p∈ℕℜβ,pα,γ}.Definition 5.
In conjunction with (6) and (10),
(17)Cα,β,γ,p*(η,λ)={f:f∈A(p),ℜβ,pα,γf∈Cp*(η,λ),0000≤λ,η<p,p∈ℕℜβ,pα,γ}.Remark 6.
Consider (I) Forγ=1, in the above definitions, we have
(18)Sα,β,1,p*(λ)=Sα,β,p*(λ)={f:f∈A(p),Qβ,pαf∈Sp*(λ),0000≤λ<p,p∈ℕQβ,pα},Kα,β,1,p(λ)=Kα,β,p(λ)={f:f∈A(p),Qβ,pαf∈Kp(λ),0000≤λ<p,p∈ℕQβ,pα},Cα,β,1,p(η,λ)=Cα,β,p(η,λ)={f:f∈A(p),Qβ,pαf∈Cp(η,λ),0000≤λ,η<p,p∈ℕQβ,pα},Cα,β,1,p*(η,λ)=Cα,β,p*(η,λ)={f:f∈A(p),Qβ,pαf∈Cp*(η,λ),0000≤λ,η<p,p∈ℕQβ,pα}.
(II) Forα=γ=1 and β=c(c>-p), in the above definitions, we have
(19)S1,c,1,p*(λ)=Sc,p*(λ)={f:f∈A(p),Jc,pf(z)∈Sp*(λ),0000≤λ<p,p∈ℕJc,p},K1,c,1,p(λ)=Kc,p(λ)={f:f∈A(p),Jc,pf(z)∈Kp(λ),0000≤λ<p,p∈ℕJc,p},C1,c,1,p(η,λ)=Cc,p(η,λ)={f:f∈A(p),Jc,pf(z)∈Cp(η,λ),0000≤λ,η<p,p∈ℕJc,p},C1,c,1,p*(η,λ)=Cc,p*(η,λ)={f:f∈A(p),Jc,pf(z)∈Cp*(η,λ),0000≤λ,η<p,p∈ℕJc,p}.
(III) Forγ=p=1, in the above definitions, we have
(20)Sα,β,1,1*(λ)=Sα,β*(λ)={f:f∈A,Qβαf∈S*(λ),0≤λ<1},Kα,β,1,1(λ)=Kα,β(λ)={f:f∈A,Qβαf∈K(λ),0≤λ<1},Cα,β,1,1(η,λ)=Cα,β(η,λ)={f:f∈A,Qβαf∈C(η,λ),0≤λ,η<1},Cα,β,1,1*(η,λ)=Cα,β*(η,λ)={f:f∈A,Qβαf∈C*(η,λ),0000≤λ,η<1},
where the classes Sα,β*(λ), Kα,β(λ), Cα,β(η,λ), and Cα,β*(η,λ) were introduced and studied by Gao et al. [14].In order to establish our main results, we need the following lemma due to Miller and Mocanu [15].Lemma 7 (see [15]).
LetΘ be a complex-valued function such that
(21)Θ:D⟶ℂ,D⊂ℂ×ℂ(ℂisthecomplexplane),
and let u=u1+iu2, v=v1+iv2. Suppose that Θ(u,v) satisfies the following conditions:(i)
Θ(u,v) is continuous in D;(ii)
(1,0)∈D and Re{Θ(1,0)}>0;(iii)
Re{Θ(iu2,v1)}≤0 for all (iu2,v1)∈D such that v1≤-(1/2)(1+u22).Let(22)q(z)=1+q1z+q2z2+⋯
be analytic in U such that (q(z),zq′(z))∈D for all z∈U. If
(23)Re{Θ(q(z),zq′(z))}>0(z∈U),
then
(24)Re{q(z)}>0(z∈U).


## 2. The Main Results

In this section, we give several inclusion relationships for analytic function classes, which are associated with the integral operatorℜβ,pα,γ. Unless otherwise mentioned, we assume throughout this paper that β>-p, α≥γ-1,  γ>0, p∈ℕ, and z∈U.Theorem 8.
Let0≤λ<p. Then
(25)Sα,β,γ,p*(λ)⊂Sα+1,β,γ,p*(λ).Proof.
Letf∈Sα,β,γ,p*(λ) and set
(26)z(ℜβ,pα+1,γf(z))′ℜβ,pα+1,γf(z)-λ=(p-λ)q(z),
where q(z) is given by (22). By using identity (11), we obtain
(27)ℜβ,pα,γf(z)ℜβ,pα+1,γf(z)=λ+(p-λ)q(z)+(α+β-γ+1)α+β+p-γ+1.
Differentiating (27) logarithmically with respect to z, we obtain
(28)z(ℜβ,pα,γf(z))′ℜβ,pα,γf(z)=z(ℜβ,pα+1,γf(z))′ℜβ,pα+1,γf(z)+(p-λ)zq′(z)(p-λ)q(z)+λ+α+β-γ+1=λ+(p-λ)q(z)+(p-λ)zq′(z)(p-λ)q(z)+λ+α+β-γ+1.
We now choose u=q(z)=u1+iu2 and v=zq′(z)=v1+iv2, and define the function Θ by
(29)Θ(u,v)=(p-λ)u+(p-λ)v(p-λ)u+λ+α+β-γ+1.
Then, clearly Θ(u,v) satisfies the following conditions:(i)
Θ(u,v) is continuous in D=(ℂ∖{(λ+α+β-γ+1)/(λ-p)})×ℂ;(ii)
(1,0)∈D and Re{Θ(1,0)}=p-λ>0;(iii)
for all(iu2,v1)∈D such that v1≤-(1/2)(1+u22) we have
(30)Re{Θ(iu2,v1)}=Re{(p-λ)v1(p-λ)iu2+λ+α+β-γ+1}=(p-λ)(λ+α+β-γ+1)v1(p-λ)2u22+(λ+α+β-γ+1)2≤-(p-λ)(1+u22)(λ+α+β-γ+1)2[(p-λ)2u22+(λ+α+β-γ+1)2]<0,which shows that the function Θ satisfies the hypotheses of Lemma 7. Consequently, we easily obtain the inclusion relationship (25).Theorem 9.
Let0≤λ<p, p∈ℕ. Then
(31)Kα,β,γ,p(λ)⊂Kα+1,β,γ,p(λ).Proof.
Letf∈Kα,β,γ,p(λ). Then, from Definition 3, we have
(32)ℜβ,pα,γf∈Kp(λ),(0≤λ<p;p∈ℕ).
Furthermore, in view of the relationship (4), we find that
(33)zp(ℜβ,pα,γf(z))′∈Sp*(λ),
that is, that
(34)ℜβ,pα,γ(zf′(z)p)∈Sp*(λ).
Thus, by using Definition 2 and Theorem 8, we have
(35)zf′(z)p∈Sp*(α,β,γ,λ)⊂Sp*(α+1,β,γ,λ),
which implies that
(36)Kα,β,γ,p(λ)⊂Kα+1,β,γ,p(λ).
The proof of Theorem 9 is thus completed.Theorem 10.
Let0≤λ, η<p. Then
(37)Cα,β,γ,p(η,λ)⊂Cα+1,β,γ,p(η,λ).Proof.
Letf∈Cα,β,γ,p(η,λ). Then there exists a function Ψ∈Sp*(λ) such that
(38)Re{z(ℜβ,pα,γf(z))′Ψ(z)}>η(0≤λ,η<p,z∈U).
We put
(39)ℜβ,pα,γg(z)=Ψ(z),
so that we have
(40)g∈Sα,β,γ,p*(λ),Re{z(ℜβ,pα,γf(z))′ℜβ,pα,γg(z)}>η(z∈U).
We next put
(41)z(ℜβ,pα+1,γf(z))′ℜβ,pα+1,γg(z)=η+(p-η)q(z),
where q(z) is given by (22). Thus, by using identity (11), we obtain
(42)z(ℜβ,pα,γf(z))′ℜβ,pα,γg(z)=ℜβ,pα,γ(zf′(z))ℜβ,pα,γg(z)=(z[ℜβ,pα+1,γ(zf′(z))]′ℜβ,pα+1,γ000+(α+β-γ+1)ℜβ,pα+1,γ(zf′(z))[ℜβ,pα+1,γ(zf′(z))]′ℜβ,pα+1,γ)×(ℜβ,pα+1,γz[ℜβ,pα+1,γg(z)]′00000+(α+β-γ+1)ℜβ,pα+1,γg(z)[ℜβ,pα+1,γg(z)]′)-1=(z[ℜβ,pα+1,γ(zf′(z))]′ℜβ,pα+1,γg(z)0000+(α+β-γ+1)ℜβ,pα+1,γ(zf′(z))ℜβ,pα+1,γg(z)z[ℜβ,pα+1,γ(zf′(z))]′ℜβ,pα+1,γg(z))×(z[ℜβ,pα+1,γg(z)]′ℜβ,pα+1,γg(z)000000+(α+β-γ+1)z[ℜβ,pα+1,γg(z)]′ℜβ,pα+1,γg(z))-1.
Since g∈Sα,β,γ,p*(λ), then from Theorem 8 we have g∈Sα+1,β,γ,p*(λ), so that we can put
(43)z[ℜβ,pα+1,γg(z)]′ℜβ,pα+1,γg(z)=λ+(p-λ)G(z),
where
(44)G(z)=g1(x,y)+ig2(x,y),Re(G(z))=g1(x,y)>0(z∈U).
Then
(45)z(ℜβ,pα,γf(z))′ℜβ,pα,γg(z)=(z[ℜβ,pα+1,γ(zf′(z))]′ℜβ,pα+1,γg(z)0000+(α+β-γ+1)(η+(p-η)q(z))z[ℜβ,pα+1,γ(zf′(z))]′ℜβ,pα+1,γg(z))×((p-λ)G(z)+λ+α+β-γ+1)-1.
We thus find from (41) that
(46)z(ℜβ,pα+1,γf(z))′=ℜβ,pα+1,γg(z)[η+(p-η)q(z)].
Differentiating both sides of (46) with respect to z, we obtain
(47)z[z(ℜβ,pα+1,γf(z))′]′ℜβ,pα+1,γg(z)=(p-η)zq′(z)+z[ℜβ,pα+1,γg(z)]′ℜβ,pα+1,γg(z)[η+(p-η)q(z)]=(p-η)zq′(z)+[λ+(p-λ)G(z)][η+(p-η)q(z)].
By substituting (47) into (45), we have
(48)z(ℜβ,pα,γf(z))′ℜβ,pα,γg(z)-η=(p-η)q(z)+(p-η)zq′(z)(p-λ)G(z)+λ+α+β-γ+1.
The remainder of our proof of Theorem10 is much akin to that of Theorem 8. We, therefore, choose to omit the details involved.Theorem 11.
Let0≤λ, η<p. Then
(49)Cα,β,γ,p*(η,λ)⊂Cα+1,β,γ,p*(η,λ).Proof.
Just as we derived Theorem9 as a consequence of Theorem 8 by using the equivalence (4), we can also prove Theorem 11 by using Theorem 10 in conjunction with the equivalence (7).Our main results in Theorems8–11, can thus be applied with a view to deducing the following corollaries.Takingγ=1 in Theorems 8–11 above, we obtain the following corollary.Corollary 12.
Let0≤λ, η<p. Then
(50)Sα,β,p*(λ)⊂Sα+1,β,p*(λ),Kα,β,p(λ)⊂Kα+1,β,p(λ),Cα,β,p(η,λ)⊂Cα+1,β,p(η,λ),Cα,β,p*(η,λ)⊂Cα+1,β,p*(η,λ).Remark 13.
Takingp=1 in Corollary 12, we obtain the results obtained by Gao et al. [14, Theorems 1-4].
Takingα=γ=1 and β=c(c>-p) in Theorems 8–11, we obtain the following corollary.Corollary 14.
Let0≤λ, η<p. Then
(51)S1,c,p*(λ)⊂S2,c,p*(λ),K1,c,p(λ)⊂K2,c,p(λ),C1,c,p(η,λ)⊂C2,c,p(η,λ),C1,c,p*(η,λ)⊂C2,c,p*(η,λ).Remark 15.
Takingp=1 in Corollary 14, we obtain the results obtained by Gao et al. [14, Corollary  1–4].


## 3. A Set of Integral-Preserving Properties

In this section, we present several integral-preserving properties of the analytic function classes introduced here. In order to obtain the integral-preserving properties involving the integral operatorJc,p defined by (13).Theorem 16.
Letc be any real number and c>-p. If f(z)∈Sα,β,γ,p*(λ), then Jc,p(z)∈Sα,β,γ,p*(λ), where Jc,pf(z) is defined by (13).Proof.
From (13), we have
(52)z(ℜβ,pα,γJc,p(z))′=(c+p)ℜβ,pα,γ-cℜβ,pα,γJc,p(z).
Letf∈Sα,β,γ,p*(λ) and set
(53)z(ℜβ,pα,γJc,pf(z))′ℜβ,pα,γJc,pf(z)-λ=(p-λ)q(z),
where q(z) is given by (22). By using identity (52), we obtain
(54)ℜβ,pα,γf(z)ℜβ,pα,γJc,pf(z)=λ+(p-λ)q(z)+cc+p.
Differentiating (54) logarithmically with respect to z, we obtain
(55)z(ℜβ,pα,γf(z))′ℜβ,pα,γf(z)=z(ℜβ,pα,γJc,pf(z))′ℜβ,pα,γJc,pf(z)+(p-λ)zq′(z)(p-λ)q(z)+λ+c=λ+(p-λ)q(z)+(p-λ)zq′(z)(p-λ)q(z)+λ+c.
We now choose u=q(z)=u1+iu2, and v=zq′(z)=v1+iv2, and define the function Θ by
(56)Θ(u,v)=(p-λ)u+(p-λ)v(p-λ)u+λ+c.
It is easy to see that the function Θ(u,v) satisfies the conditions of Lemma 7, and the remaining part of the proof of Theorem 16 is similar to that of Theorem 8.Takingγ=1 in Theorem 16, we obtain the following corollary.Corollary 17.
Letc be any real number and c>-p. If f(z)∈Sα,β,p*(λ), then Jc,p(z)∈Sα,β,p*(λ), where Jc,pf(z) is defined by (13).Theorem 18.
Letc be any real number and c>-p. If f(z)∈Kα,β,γ,p(λ), then Jc,p(z)∈Kα,β,γ,p(λ), where Jc,pf(z) is defined by (13).Proof.
By applying Theorem16 in conjunction with the equivalence (4), it follows that
(57)f(z)∈Kα,β,γ,p(λ)⟺zpf′(z)∈Sα,β,γ,p*(λ)⟹Jc,p(zpf′(z))∈Sα,β,γ,p*(λ)⟺zp(Jc,pf(z))′∈Sα,β,γ,p*(λ)⟹Jc,pf(z)∈Kα,β,γ,p(λ),
which proves Theorem 18.Takingγ=1 in Theorem 18, we obtain the following corollary.Corollary 19.
Letc be any real number and c>-p. If f(z)∈Kα,β,p(λ), then Jc,p(z)∈Kα,β,p(λ), where Jc,pf(z) is defined by (13).Theorem 20.
Letc be any real number and c>-p. If f(z)∈Cα,β,γ,p(η,λ), then Jc,p(z)∈Cα,β,γ,p(η,λ), where Jc,pf(z) is defined by (13).Proof.
Letf∈Cα,β,γ,p(η,λ). Then there exists a function Ψ∈Sp*(λ) such that
(58)Re{z(ℜβ,pα,γf(z))′Ψ(z)}>η(0≤λ,η<p,z∈U).
We put
(59)ℜβ,pα,γg(z)=Ψ(z),
so that we have
(60)g∈Sα,β,γ,p*(λ),Re{z(ℜβ,pα,γf(z))′ℜβ,pα,γg(z)}>η(z∈U).
We next put
(61)z(ℜβ,pα,γJc,pf(z))′ℜβ,pα,γJc,pg(z)=η+(p-η)q(z),
where q(z) is given by (22). Thus, by using identity (52), we obtain
(62)z(ℜβ,pα,γJc,pf(z))′ℜβ,pα,γJc,pg(z)=ℜβ,pα,γ(z(Jc,pf(z))′)ℜβ,pα,γJc,pg(z)=(ℜβ,pα,γ(()(z(Jc,pf(z))′)z[ℜβ,pα,γ(z(Jc,pf(z))′)]′000+cℜβ,pα,γ(z(Jc,pf(z))′))×(z[ℜβ,pα,γJc,pg(z)]′000000+cℜβ,pα,γJc,pg(z)z[ℜβ,pα,γJc,pg(z)]′)-1=(z[ℜβ,pα,γ(z(Jc,pf(z))′)]′ℜβ,pα,γJc,pg(z)00000+cℜβ,pα,γ(z(Jc,pf(z))′)ℜβ,pα,γJc,pg(z))×(z[ℜβ,pα,γJc,pg(z)]′ℜβ,pα,γJc,pg(z)+c)-1.
Since g∈Sα,β,γ,p*(λ), then from Theorem 16 we have Jc,pg(z)∈Sα,β,γ,p*(λ), so that we can put
(63)z[ℜβ,pα,γJc,pg(z)]′ℜβ,pα,γJc,pg(z)=λ+(p-λ)G(z),
where
(64)G(z)=g1(x,y)+ig2(x,y),Re(G(z))=g1(x,y)>0000000000000000000000000000000000000000(z∈U).
Then
(65)z(ℜβ,pα,γJc,pf(z))′ℜβ,pα,γJc,pg(z)=(z[ℜβ,pα,γ(z(Jc,pf(z))′)]′ℜβ,pα,γJc,pg(z)0000+c(η+(p-η)q(z))z[ℜβ,pα,γ(z(Jc,pf(z))′)]′ℜβ,pα,γJc,pg(z))×((p-λ)G(z)+λ+c)-1.
We thus find from (61) that
(66)z(ℜβ,pα,γJc,pf(z))′=ℜβ,pα,γJc,pg(z)[η+(p-η)q(z)].
Differentiating both sides of (66) with respect to z, we obtain
(67)z[z(ℜβ,pα,γJc,pf(z))′]′ℜβ,pα,γJc,pg(z)=(p-η)zq′(z)+z[ℜβ,pα,γJc,pg(z)]′ℜβ,pα,γJc,pg(z)[η+(p-η)q(z)]=(p-η)zq′(z)+[λ+(p-λ)G(z)][η+(p-η)q(z)].
By substituting (67) into (65), we have
(68)z(ℜβ,pα,γf(z))′ℜβ,pα,γg(z)-η=(p-η)q(z)+(p-η)zq′(z)(p-λ)G(z)+λ+c.
The remainder of our proof of Theorem20 is much akin to that of Theorem 8. We, therefore, choose to omit the details involved.Takingγ=1 in Theorem 20, we obtain the following corollary.Corollary 21.
Letc be any real number and c>-p. If f(z)∈Cα,β,p(η,λ), then Jc,p(z)∈Cα,β,p(η,λ), where Jc,pf(z) is defined by (13).Theorem 22.
Letc be any real number and c>-p. If f(z)∈Cα,β,γ,p*(η,λ), then Jc,p(z)∈Cα,β,γ,p*(η,λ), where Jc,pf(z) is defined by (13).Proof.
Just as we derived Theorem18 as a consequence of Theorem 16 by using the equivalence (4), we can also prove Theorem 22 by using Theorem 20 in conjunction with the equivalence (7).Takingγ=1 in Theorem 22, we obtain the following corollary.Corollary 23.
Letc be any real number and c>-p. If f(z)∈Cα,β,γ,p*(η,λ), then Jc,p(z)∈Cα,β,γ,p*(η,λ), where Jc,pf(z) is defined by (13).



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*Source: 384170-2013-09-28.xml*