# Convexity and Spirallikeness Conditions for Two New General Integral Operators

Authors: H. Özlem Güney; Serap Bulut
Journal: Journal of Mathematics
 (2013)

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## Abstract

We define two new general integral operators for certain analytic functions in the unit disc𝕌 and give some sufficient conditions for these integral operators on some subclasses of analytic functions.


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## Body

## 1. Introduction

Let𝒜p(n) denote the class of all functions of the form
(1)f(z)=zp+∑k=p+n∞‍akzk(p,n∈ℕ={1,2,…})
which is analytic in the open unit disc 𝕌={z∈ℂ:|z|<1}.In particular, we set(2)𝒜p(1):=𝒜p,𝒜1(1)=𝒜1:=𝒜.The fractional derivative of orderα is defined, for a function f, by
(3)Dzαf(z)=1Γ(1-α)ddz∫0z‍f(ζ)(z-ζ)αdζ(0≤α<1),
where the function f is analytic in a simply connected region of the complex z-plane containing the origin, and the multiplicity of (z-ζ)-α is removed by requiring log(z-ζ) to be real when z-ζ>0.It readily follows from (3) that
(4)Dzαzk=Γ(k+1)Γ(k-α+1)zk-α(0≤α<1,k∈ℕ).Using the operatorΩpα:𝒜p(n)→𝒜p(n),
(5)Ωpαf(z)=Γ(p-α+1)Γ(p+1)zαDzαf(z),=zp+∑k=p+n∞Γ(k+1)Γ(p-α+1)Γ(p+1)Γ(k-α+1)akzk,α≠p+1,p+2,p+3,….Bulut [1] defined the general differential operator Dλ,l,pm,α, namely, the generalization of the generalized Al-Oboudi differential operator as follows:
(6)D0f(z)=f(z),Dλ,l,p1,αf(z)=p-λp+lp+lΩpαf(z)+λp+lz(Ωpαf(z))′=Dλ,l,pαf(z)(λ,l≥0,0≤α<1),Dλ,l,p2,αf(z)=Dλ,l,pα(Dλ,l,p1,αf(z)),⋮Dλ,l,pm,αf(z)=Dλ,l,pα(Dλ,l,pm-1,αf(z))(m∈ℕ).Iff is given by (1), then by (5) and (6), we see that
(7)Dλ,l,pm,αf(z)=zp+∑k=p+n∞‍Ψk,m(α,λ,l,p)akzk(m∈ℕ0=ℕ∪{0}),
where
(8)Ψk,m(α,λ,l,p)=[Γ(k+1)Γ(p-α+1)Γ(p+1)Γ(k-α+1)p+λ(k-p)+lp+l]m.By using the operatorDλ,l,pm,α defined by (7), we introduce the new classes 𝒰𝒮α,λ,lm,p,n(δ,β,b) and 𝒰𝒦α,λ,lm,p,n(δ,β,b) as follows.Definition 1.
A functionf∈𝒜p(n) is in the class 𝒰𝒮α,λ,lm,p,n(δ,β,b) if and only if f satisfies
(9)ℜ{p+1b(z(Dλ,l,pm,αf(z))′Dλ,l,pm,αf(z)-p)}>δ|1b(z(Dλ,l,pm,αf(z))′Dλ,l,pm,αf(z)-p)|+β,
where z∈𝕌, b∈ℂ-{0}, δ≥0, 0≤β<p and Dλ,l,pm,α is defined by (7).Definition 2.
A functionf∈𝒜p(n) is in the class 𝒰𝒦α,λ,lm,p,n(δ,β,b) if and only if f satisfies
(10)ℜ{p+1b(1+z(Dλ,l,pm,αf(z))′′(Dλ,l,pm,αf(z))′-p)}>δ|1b(1+z(Dλ,l,pm,αf(z))′′(Dλ,l,pm,αf(z))′-p)|+β,
where z∈𝕌, b∈ℂ-{0}, δ≥0, 0≤β<p, and Dλ,l,pm,α is defined by (7).Note that(11)f∈𝒰𝒦α,λ,lm,p,n(δ,β,b)⟺1pzf′∈𝒰𝒮α,λ,lm,p,n(δ,β,b).Remark 3.
(i) Forδ=0, we have the classes
(12)𝒰𝒮α,λ,lm,p,n(0,β,b)≡𝒮α,λ,lm,p,n(β,b),𝒰𝒦α,λ,lm,p,n(0,β,b)≡𝒦α,λ,lm,p,n(β,b)
introduced and studied by Bulut [1]. In particular, we get the classes
(13)𝒰𝒮α,λ,0m,1,1(0,β,b)≡𝒮α,λm(β,b),𝒰𝒦α,λ,0m,1,1(0,β,b)≡𝒦α,λm(β,b)
introduced and studied by Bulut [2].(ii)
Forδ=0 and β=0, we have the classes
(14)𝒰𝒮α,λ,lm,p,n(0,0,b)≡𝒮α,λ,lm,p,n(b),𝒰𝒦α,λ,lm,p,n(0,0,b)≡𝒦α,λ,lm,p,n(b).In particular, we set
(15)𝒰𝒮α,λ,l0,p,n(0,0,b)≡𝒮p,n(b),𝒰𝒦α,λ,l0,p,n(0,0,b)≡𝒦p,n(b)
for the p-valently starlike and p-valently convex functions with complex order b(b∈ℂ-{0}), respectively.(iii)
Form=0, we have the classes
(16)𝒰𝒮α,λ,l0,p,n(δ,β,b)≡𝒮p,n(δ,β,b)={f∈𝒜p(n):ℜ{p+1b(zf′(z)f(z)-p)}>δ|1b(zf′(z)f(z)-p)|+β},𝒰𝒦α,λ,l0,p,n(δ,β,b)≡𝒦p,n(δ,β,b)={f∈𝒜p(n):ℜ{p+1b(1+zf′′(z)f′(z)-p)}>δ|1b(1+zf′′(z)f′(z)-p)|+β}.In particular, we set
(17)𝒰𝒮α,λ,l0,1,1(δ,β,b)≡𝒮(δ,β,b),𝒰𝒦α,λ,l0,1,1(δ,β,b)≡𝒦(δ,β,b).
Also, we have the classes
(18)𝒰𝒮α,λ,l0,1,1(δ,β,1)≡𝒮𝒟(δ,β),𝒰𝒦α,λ,l0,1,1(δ,β,1)≡𝒦𝒟(δ,β)
introduced and studied by Shams et al. [3].(iv)
Forδ=0 and m=0, we have the classes
(19)𝒰𝒮α,λ,l0,p,n(0,β,b)≡𝒮p,n(β,b)={f∈𝒜p(n):ℜ{p+1b(zf′(z)f(z)-p)}>β},𝒰𝒦α,λ,l0,p,n(0,β,b)≡𝒦p,n(β,b)={f∈𝒜p(n):ℜ{p+1b(1+zf′′(z)f′(z)-p)}>β}of p-valently starlike and p-valently convex functions with complex order b(b∈ℂ-{0}) and type β(0≤β<p), respectively. In particular, we set
(20)𝒰𝒮α,λ,l0,1,1(0,β,b)≡𝒮(β,b),𝒰𝒦α,λ,l0,1,1(0,β,b)≡𝒦(β,b)(b∈ℂ-{0},0≤β<1).
Note that for b=1, we have
(21)𝒰𝒮α,λ,l0,1,1(0,β,1)≡𝒮*(β),𝒰𝒦α,λ,l0,1,1(0,β,1)≡𝒦(β)
which are the classes of starlike and convex functions type β(0≤β<1), respectively.(v)
Forδ=1, m=0, p=n=1, and b=1, we have the classes
(22)𝒰𝒮α,λ,l0,1,1(1,β,1)≡𝒰𝒮𝒯(β),𝒰𝒦α,λ,l0,1,1(δ,β,1)≡𝒰𝒞𝒱(β)of uniformly starlike and uniformly convex functions of order β(0≤β<1), respectively, investigated by Rønning [4]. In particular, we get the classes
(23)𝒰𝒮α,λ,l0,1,1(1,0,1)≡𝒰𝒮𝒯,𝒰𝒦α,λ,l0,1,1(δ,β,1)≡𝒰𝒞𝒱
of uniformly starlike and uniformly convex functions, respectively, first defined by Goodman [5].Remark 4.
If we setδ=0 and b=e-iθcosθ(|θ|<π/2) in Definitions 1 and 2, then we have the classes
(24)𝒮α,λ,lm,p,n(β,θ)={f∈𝒜p(n):ℜ{eiθz(Dλ,l,pm,αf(z))′Dλ,l,pm,αf(z)}>βcosθ,0≤β<p{{eiθz(Dλ,l,pm,αf(z))′Dλ,l,pm,αf(z)}}},(25)𝒦α,λ,lm,p,n(β,θ)={f∈𝒜p(n):ℜ{eiθ(1+z(Dλ,l,pm,αf(z))′′(Dλ,l,pm,αf(z))′)}>βcosθ,0≤β<p{(1+z(Dλ,l,pm,αf(z))′′(Dλ,l,pm,αf(z))′)}},
respectively. Also, we put
(26)𝒮α,λ,l0,p,n(β,θ)≡𝒮p,n(β,θ),𝒦α,λ,l0,p,n(β,θ)≡𝒦p,n(β,θ).
In particular, we set
(27)𝒮α,λ,l0,1,1(β,θ)≡𝒮βθ(|θ|<π2,0≤β<1),𝒦α,λ,l0,1,1(β,θ)≡𝒦βθ(|θ|<π2,0≤β<1)
for the the class of θ-spirallike functions of order β [6], and the class of θ-Robertson functions of order β [7], respectively.
We now define the following general integral operators.Definition 5.
Letη∈ℕ, m=(m1,…,mη)∈ℕ0η, and k=(k1,…,kη)∈ℝ+η. One defines the following general integral operators:
(28)ℐp,ηm,k:𝒜p(n)η⟶𝒜p(n),ℐp,ηm,k(f1,…,fη)=ℱp,η,m,k,(29)ℱp,η,m,k(z)=∫0z‍ptp-1∏j=1η‍(Dλ,l,pmj,αfj(t)tp)kjdt,(30)𝒥p,ηm,k:𝒜p(n)η⟶𝒜p(n),𝒥p,ηm,k(f1,…,fη)=𝒢p,η,m,k,(31)𝒢p,η,m,k(z)=∫0z‍ptp-1∏j=1η‍((Dλ,l,pmj,αfj(t))′ptp-1)kjdt,
where z∈𝕌, fj∈𝒜p(n), and Dλ,l,pmj,α is defined by (7) for 1≤j≤η.Remark 6.
(i) Forη=1, m1=m, k1=k, and f1=f, we have the integral operators
(32)ℱp,m,k(z)=∫0z‍ptp-1(Dλ,l,pm,αf(t)tp)kdt,(33)𝒢p,m,k(z)=∫0z‍ptp-1((Dλ,l,pm,αf(t))′ptp-1)kdt.(ii)
Forα=0, λ=1, and l=0, we have the integral operators
(34)ℱp,n,m,k(z)=∫0z‍ptp-1(Dm1f1(t)tp)k1⋯(Dmnfn(t)tp)kndt,𝒢p,n,m,k(z)=∫0z‍ptp-1((Dm1f1(t))′ptp-1)k1⋯((Dmnfn(t))′ptp-1)kndtintroduced by Saltık et al. [8].(iii)
Form1=⋯=mη=0, we have the integral operators
(35)ℱp(z)=∫0z‍ptp-1(f1(t)tp)k1⋯(fn(t)tp)kndt,𝒢p(z)=∫0z‍ptp-1(f1′(t)ptp-1)k1⋯(fn′(t)ptp-1)kndtintroduced by Frasin [9].


## 2. Sufficient Conditions for the Operatorℱp,η,m,k

Theorem 7.
Letη∈ℕ, m=(m1,…,mη)∈ℕ0η, and k=(k1,…,kη)∈ℝ+η. Also let b∈ℂ-{0}, δj≥0, 0≤βj<p, and fj∈𝒰𝒮α,λ,lmj,p,n(δj,βj,b) for 1≤j≤η. If
(36)0≤p+∑j=1η‍kj(βj-p)<p,
then the integral operator ℱp,η,m,k, defined by (29), is in the class 𝒦p,n(τ,b), where
(37)τ=p+∑j=1η‍kj(βj-p).Proof.
From the definition (29), we observe that ℱp,η,m,k(z)∈𝒜p(n). On the other hand, it is easy to see that
(38)(ℱp,η,m,k(z))′=pzp-1∏j=1η‍(Dλ,l,pmj,αfj(z)zp)kj.
Differentiating (38) logarithmically and multiplying by z, we obtain
(39)z(ℱp,η,m,k(z))′′(ℱp,η,m,k(z))′=(p-1)+∑j=1η‍kj(z(Dλ,l,pmj,αfj(t))′Dλ,l,pmj,αfj(t)-p)
or equivalently
(40)1+z(ℱp,η,m,k(z))′′(ℱp,η,m,k(z))′-p=∑j=1η‍kj(z(Dλ,l,pmj,αfj(t))′Dλ,l,pmj,αfj(t)-p).
Then, by multiplying (40) with 1/b, we have
(41)1b(1+z(ℱp,η,m,k(z))′′(ℱp,η,m,k(z))′-p)=∑j=1η‍kj1b(z(Dλ,l,pmj,αfj(t))′Dλ,l,pmj,αfj(t)-p)
or equivalently
(42)p+1b(1+z(ℱp,η,m,k(z))′′(ℱp,η,m,k(z))′-p)=∑j=1η‍kj(p+1b(z(Dλ,l,pmj,αfj(t))′Dλ,l,pmj,αfj(t)-p))+p-p∑j=1η‍kj.
Since fj∈𝒰𝒮α,λ,lmj,p,n(δj,βj,b)(1≤j≤η), we get
(43)ℜ{p+1b(1+z(ℱp,η,m,k(z))′′(ℱp,η,m,k(z))′-p)}=∑j=1η‍kjℜ{p+1b(z(Dλ,l,pmj,αfj(t))′Dλ,l,pmj,αfj(t)-p)}+p-p∑j=1η‍kj>∑j=1η‍kjδj|1b(z(Dλ,l,pmj,αfj(t))′Dλ,l,pmj,αfj(t)-p)|+p+∑j=1η‍kj(βj-p).
Since
(44)∑j=1η‍kjδj|1b(z(Dλ,l,pmj,αfj(t))′Dλ,l,pmj,αfj(t)-p)|>0,
the integral operator ℱp,η,m,k∈𝒦p,n(τ,b) with
(45)τ=p+∑j=1η‍kj(βj-p).Corollary 8.
In Theorem7, letting(i)
p=n=1, δj=0, and mj=0(1≤j≤η), one obtains [10, Theorem 1].(ii)
p=n=1, δj=0, and  mj=0, βj=β∈[0,1)(1≤j≤η), one obtains [11, Theorem 1].(iii)
p=n=1, δj=0, and mj=0, βj=0(1≤j≤η), one obtains [12, Theorem 1].Puttingη=1, m1=m, k1=k, β1=β, δ1=δ, and f1=f in Theorem 7, we have the following.Corollary 9.
Letm∈ℕ0 and k>0. Also let b∈ℂ-{0}, δ≥0, 0≤β<p, and f∈𝒰𝒮α,λ,lm,p,n(δ,β,b). If
(46)0≤p+k(β-p)<p,
then the integral operator ℱp,m,k, defined by (32), is in the class 𝒦p,n(μ,b), where
(47)μ=p+k(β-p).Puttingp=n=1 and m=0 in Corollary 9, we have the following.Corollary 10.
Letk>0, b∈ℂ-{0}, δ≥0, 0≤β<1, and f∈𝒮(δ,β,b). If
(48)0≤1+k(β-1)<1,
then the integral operator
(49)ℱk(z)=∫0z‍(f(t)t)kdt
is convex of complex order b(b∈ℂ-{0}) and type ρ=1+k(β-1); that is, ℱk∈𝒦(ρ,b).Puttingp=n=1, mj=0(1≤j≤η), and b=1 in Theorem 7, we have the following.Corollary 11.
Letη∈ℕ, k=(k1,…,kη)∈ℝ+η, δj≥0, 0≤βj<1, and fj∈𝒮𝒟(δj,βj) for 1≤j≤η. If
(50)0≤1+∑j=1η‍kj(βj-1)<1,
then the integral operator
(51)ℱ(z)=∫0z‍(f1(t)t)k1⋯(fn(t)t)kndt
is in the class 𝒦(σ), where
(52)σ=1+∑j=1η‍kj(βj-1).Theorem 12.
Letη∈ℕ, m=(m1,…,mη)∈ℕ0η, and k=(k1,…,kη)∈ℝ+η. Also let |θ|<π/2, 0≤βj<p, and fj∈𝒮α,λ,lmj,p,n(βj,θ) for 1≤j≤η. If
(53)0≤p+∑j=1η‍kj(βj-p)<p,
then the integral operator ℱp,η,m,k, defined by (29), is in the class 𝒦p,n(τ,θ), where
(54)τ=p+∑j=1η‍kj(βj-p).Proof.
From (24), (25), and (40), we get the desired result.Puttingη=1, m1=m, k1=k, β1=β and f1=f in Theorem 12, we have the following.Corollary 13.
Letm∈ℕ0 and k>0. Also let |θ|<π/2, 0≤β<p, and f∈𝒮α,λ,lm,p,n(β,θ). If
(55)0≤p+k(β-p)<p,
then the integral operator ℱp,m,k, defined by (32), is in the class 𝒦p,n(μ,θ), where
(56)μ=p+k(β-p).Puttingp=n=1 and m=0 in Corollary 13, we have the following.Corollary 14.
Letk>0, |θ|<π/2, 0≤β<1, and f∈𝒮βθ. If
(57)0≤1+k(β-1)<1,
then the integral operator ℱk, defined by (49), is in the class 𝒦ρθ, where
(58)ρ=1+k(β-1).Puttingp=n=1 and mj=0(1≤j≤η) in Theorem 12, we have the following.Corollary 15.
Letη∈ℕ, k=(k1,…,kη)∈ℝ+η, 0≤βj<1, and fj∈𝒮βjθ(|θ|<π/2) for 1≤j≤η. If
(59)0≤1+∑j=1η‍kj(βj-1)<1,
then the integral operator ℱ, defined by (51), is in the class 𝒦σθ, where
(60)σ=1+∑j=1η‍kj(βj-1).Theorem 16.
Letη∈ℕ, m=(m1,…,mη)∈ℕ0η, and k=(k1,…,kη)∈ℝ+η. Also let b∈ℂ-{0}, δj≥0, and 0≤βj<p for 1≤j≤η. If
(61)|z(Dλ,l,pmj,αfj(t))′Dλ,l,pmj,αfj(t)-p|>-p+∑j=1ηkj(βj-p)(1/|b|)∑j=1ηkjδj
for all 1≤j≤η, then the integral operator ℱp,η,m,k, defined by (29), is p-valently convex of complex order b(b∈ℂ-{0}); that is, ℱp,η,m,k∈𝒦p,n(b).Proof.
From (43) and (61), we easily get that the integral operator ℱp,η,m,k is p-valently convex of complex order b.


## 3. Sufficient Conditions for the Operator𝒢p,η,m,k

Theorem 17.
Letη∈ℕ, m=(m1,…,mη)∈ℕ0η, and k=(k1,…,kη)∈ℝ+η. Also let b∈ℂ-{0}, δj≥0, 0≤βj<p, and fj∈𝒰𝒦α,λ,lmj,p,n(δj,βj,b) for 1≤j≤η. If
(62)0≤p+∑j=1η‍kj(βj-p)<p,
then the integral operator 𝒢p,η,m,k, defined by (31), is in the class 𝒦p,n(τ,b), where
(63)τ=p+∑j=1η‍kj(βj-p).Proof.
From the definition (31), we observe that 𝒢p,η,m,k(z)∈𝒜p(n). On the other hand, it is easy to see that
(64)(𝒢p,η,m,k(z))′=pzp-1∏j=1η‍((Dλ,l,pmj,αfj(z))′pzp-1)kj.
Differentiating (64) logarithmically and multiplying by z, we obtain
(65)z(𝒢p,η,m,k(z))′′(𝒢p,η,m,k(z))′=(p-1)+∑j=1η‍kj(z(Dλ,l,pmj,αfj(t))′′(Dλ,l,pmj,αfj(t))′-(p-1))
or equivalently
(66)1+z(𝒢p,η,m,k(z))′′(𝒢p,η,m,k(z))′-p=∑j=1η‍kj(z(Dλ,l,pmj,αfj(t))′′(Dλ,l,pmj,αfj(t))′-(p-1)).
Then, by multiplying (66) with 1/b, we have
(67)1b(1+z(𝒢p,η,m,k(z))′′(𝒢p,η,m,k(z))′-p)=∑j=1η‍kj1b(z(Dλ,l,pmj,αfj(t))′′(Dλ,l,pmj,αfj(t))′-(p-1))
or equivalently
(68)p+1b(1+z(𝒢p,η,m,k(z))′′(𝒢p,η,m,k(z))′-p)=∑j=1η‍kj(p+1b(1+z(Dλ,l,pmj,αfj(t))′′(Dλ,l,pmj,αfj(t))′-p))+p-p∑j=1η‍kj.
Since fj∈𝒰𝒦α,λ,lmj,p,n(δj,βj,b)(1≤j≤η), we get
(69)ℜ{p+1b(1+z(𝒢p,η,m,k(z))′′(𝒢p,η,m,k(z))′-p)}=∑j=1η‍kjℜ{p+1b(1+z(Dλ,l,pmj,αfj(t))′′(Dλ,l,pmj,αfj(t))′-p)}+p-p∑j=1η‍kj>∑j=1η‍kjδj|1b(1+z(Dλ,l,pmj,αfj(t))′′(Dλ,l,pmj,αfj(t))′-p)|+p+∑j=1η‍kj(βj-p).
Since
(70)∑j=1η‍kjδj|1b(1+z(Dλ,l,pmj,αfj(t))′′(Dλ,l,pmj,αfj(t))′-p)|>0,
the integral operator 𝒢p,η,m,k∈𝒦p,n(τ,b) with
(71)τ=p+∑j=1η‍kj(βj-p).Corollary 18.
In Theorem17, letting(i)
p=n=1, δj=0, and mj=0(1≤j≤η), one obtains [10, Theorem 3].(ii)
p=n=1, δj=0, and mj=0, βj=β∈[0,1)(1≤j≤η), one obtains [11, Theorem 3].(iii)
p=n=1, δj=0, and mj=0, βj=0(1≤j≤η), one obtains [12, Theorem 2].Puttingη=1, m1=m, k1=k, β1=β, δ1=δ, and f1=f in Theorem 17, we have the following.Corollary 19.
Letm∈ℕ0 and k>0. Also let b∈ℂ-{0}, δ≥0, 0≤β<p and f∈𝒰𝒦α,λ,lm,p,n(δ,β,b). If
(72)0≤p+k(β-p)<p,
then the integral operator 𝒢p,m,k, defined by (33), is in the class 𝒦p,n(μ,b), where
(73)μ=p+k(β-p).Puttingp=n=1 and m=0 in Corollary 19, we have the following.Corollary 20.
Letk>0, b∈ℂ-{0}, δ≥0, 0≤β<1, and f∈𝒦(δ,β,b). If
(74)0≤1+k(β-1)<1,
then the integral operator
(75)𝒢k(z)=∫0z‍(f′(t))kdt
is convex of complex order b(b∈ℂ-{0}) and type ρ=1+k(β-1); that is, 𝒢k∈𝒦(ρ,b).Puttingp=n=1, mj=0(1≤j≤η), and b=1 in Theorem 17, we have the following.Corollary 21.
Letη∈ℕ, k=(k1,…,kη)∈ℝ+η, δj≥0, 0≤βj<1, and fj∈𝒦𝒟(δj,βj) for 1≤j≤η. If
(76)0≤1+∑j=1η‍kj(βj-1)<1,
then the integral operator
(77)𝒢(z)=∫0z‍(f1′(t))k1⋯(fn′(t))kndt
is in the class 𝒦(σ), where
(78)σ=1+∑j=1η‍kj(βj-1).Theorem 22.
Letη∈ℕ, m=(m1,…,mη)∈ℕ0η, and k=(k1,…,kη)∈ℝ+η. Also let |θ|<π/2, 0≤βj<p, and fj∈𝒦α,λ,lmj,p,n(βj,θ) for 1≤j≤η. If
(79)0≤p+∑j=1η‍kj(βj-p)<p,
then the integral operator 𝒢p,η,m,k, defined by (31), is in the class 𝒦p,n(τ,θ), where
(80)τ=p+∑j=1η‍kj(βj-p).Proof.
From (25) and (66), we get the desired result.Puttingη=1, m1=m, k1=k, β1=β, and f1=f in Theorem 22, we have the following.Corollary 23.
Letm∈ℕ0 and k>0. Also let |θ|<π/2, 0≤β<p, and f∈𝒦α,λ,lm,p,n(β,θ). If
(81)0≤p+k(β-p)<p,
then the integral operator 𝒢p,m,k, defined by (33), is in the class 𝒦p,n(μ,θ), where
(82)μ=p+k(β-p).Puttingp=n=1 and m=0 in Corollary 23, we have the following.Corollary 24.
Letk>0, |θ|<π/2, 0≤β<1, and f∈𝒦βθ. If
(83)0≤1+k(β-1)<1,
then the integral operator 𝒢k, defined by (75), is in the class 𝒦ρθ, where
(84)ρ=1+k(β-1).Puttingp=n=1 and mj=0(1≤j≤η) in Theorem 22, we have the following.Corollary 25.
Letη∈ℕ, k=(k1,…,kη)∈ℝ+η, 0≤βj<1, and fj∈𝒦βjθ(|θ|<π/2) for 1≤j≤η. If
(85)0≤1+∑j=1η‍kj(βj-1)<1,
then the integral operator 𝒢, defined by (77), is in the class 𝒦σθ, where
(86)σ=1+∑j=1η‍kj(βj-1).Theorem 26.
Letη∈ℕ, m=(m1,…,mη)∈ℕ0η, and k=(k1,…,kη)∈ℝ+η. Also let b∈ℂ-{0}, δj≥0, and 0≤βj<p for 1≤j≤η. If
(87)|1+z(Dλ,l,pmj,αfj(t))′′(Dλ,l,pmj,αfj(t))′-p|>-p+∑j=1ηkj(βj-p)(1/|b|)∑j=1ηkjδj
for all 1≤j≤η, then the integral operator 𝒢p,η,m,k, defined by (31), is p-valently convex of complex order b(b∈ℂ-{0}); that is, 𝒢p,η,m,k∈𝒦p,n(b).Proof.
From (69) and (87), we easily get that the integral operator 𝒢p,η,m,k is p-valently convex of complex order b.



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*Source: 841837-2013-05-09.xml*