Convexity and Spirallikeness Conditions for Two New General Integral Operators

Abstract

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

1. Introduction

Let 𝒜_𝑝(𝑛) denote the class of all functions of the form 𝑓(𝑧)=𝑧^𝑝+^∞_{𝑘=𝑝+𝑛}𝑎_𝑘𝑧^𝑘𝑝,𝑛∈ℕ={1,2,…} (1) which is analytic in the open unit disc 𝕌={𝑧∈ℂ∶|𝑧|<1}.

In particular, we set 𝒜_𝑝(1)∶=𝒜_𝑝,𝒜₁(1)=𝒜₁∶=𝒜. (2)

The fractional derivative of order 𝛼 is defined, for a function 𝑓, by 𝐷^𝛼_𝑧𝑓(𝑧)=1Γ()𝑑1−𝛼𝑑𝑧^𝑧₀𝑓(𝜁)(𝑧−𝜁)^𝛼𝑑𝜁(0≤𝛼<1), (3) 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 𝐷^𝛼_𝑧𝑧^𝑘=Γ(𝑘+1)Γ()𝑧𝑘−𝛼+1^𝑘−𝛼(0≤𝛼<1,𝑘∈ℕ). (4)

Using the operator Ω^𝛼_𝑝∶𝒜_𝑝(𝑛)→𝒜_𝑝(𝑛), Ω^𝛼_𝑝𝑓(𝑧)=Γ𝑝−𝛼+1Γ𝑧𝑝+1^𝛼𝐷^𝛼_𝑧𝑓(𝑧),=𝑧^𝑝+^∞_{𝑘=𝑝+𝑛}Γ(𝑘+1)Γ𝑝−𝛼+1ΓΓ()𝑎𝑝+1𝑘−𝛼+1^𝑘𝑧^𝑘,𝛼≠𝑝+1,𝑝+2,𝑝+3,…. (5)

Bulut [1] defined the general differential operator 𝐷^𝑚,𝛼_{𝜆,𝑙,𝑝}, namely, the generalization of the generalized Al-Oboudi differential operator as follows: 𝐷⁰𝑓(𝑧)(𝑧),𝐷=𝑓_{1,𝛼𝜆,𝑙,𝑝}𝑓(𝑧)=𝑝−𝜆𝑝+𝑙Ω𝑝+𝑙^𝛼𝑝𝑓(𝑧)+𝜆𝑧Ω𝑝+𝑙^𝛼𝑝𝑓(𝑧)^′=𝐷^𝛼_{𝜆,𝑙,𝑝}𝑓(𝑧),𝐷)(𝜆,𝑙≥0,0≤𝛼<1_{2,𝛼𝜆,𝑙,𝑝}𝑓(𝑧)=𝐷^𝛼_{𝜆,𝑙,𝑝}𝐷^1,𝛼_{𝜆,𝑙,𝑝}𝑓(𝑧),⋮𝐷^𝑚,𝛼_{𝜆,𝑙,𝑝}𝑓(𝑧)=𝐷^𝛼_{𝜆,𝑙,𝑝}𝐷^{𝑚−1,𝛼}_{𝜆,𝑙,𝑝}𝑓(𝑧)(𝑚∈ℕ). (6)

If 𝑓 is given by (1), then by (5) and (6), we see that 𝐷^𝑚,𝛼_{𝜆,𝑙,𝑝}𝑓(𝑧)=𝑧^𝑝+^∞_{𝑘=𝑝+𝑛}Ψ_𝑘,𝑚𝛼,𝜆,𝑙,𝑝𝑎_𝑘𝑧^𝑘𝑚∈ℕ₀=ℕ∪{0}, (7) where Ψ_𝑘,𝑚=𝛼,𝜆,𝑙,𝑝Γ(𝑘+1)Γ(𝑝−𝛼+1)Γ(𝑝+1)Γ(𝑘−𝛼+1)𝑝+𝜆𝑘−𝑝+𝑙𝑝+𝑙^𝑚. (8)

By using the operator 𝐷^𝑚,𝛼_{𝜆,𝑙,𝑝} defined by (7), we introduce the new classes 𝒰𝒮^{𝑚,𝑝,𝑛}_{𝛼,𝜆,𝑙}(𝛿,𝛽,𝑏) and 𝒰𝒦^{𝑚,𝑝,𝑛}_{𝛼,𝜆,𝑙}(𝛿,𝛽,𝑏) as follows.

Definition 1. A function 𝑓∈𝒜_𝑝(𝑛) is in the class 𝒰𝒮^{𝑚,𝑝,𝑛}_{𝛼,𝜆,𝑙}(𝛿,𝛽,𝑏) if and only if 𝑓 satisfies ℜ1𝑝+𝑏𝑧𝐷^𝑚,𝛼_{𝜆,𝑙,𝑝}𝑓(𝑧)^′𝐷^𝑚,𝛼_{𝜆,𝑙,𝑝}𝑓(𝑧)1−𝑝>𝛿𝑏𝑧𝐷^𝑚,𝛼_{𝜆,𝑙,𝑝}𝑓(𝑧)^′𝐷^𝑚,𝛼_{𝜆,𝑙,𝑝}𝑓(𝑧)−𝑝+𝛽, (9) where 𝑧∈𝕌, 𝑏∈ℂ−{0}, 𝛿≥0, 0≤𝛽<𝑝 and 𝐷^𝑚,𝛼_{𝜆,𝑙,𝑝} is defined by (7).

Definition 2. A function 𝑓∈𝒜_𝑝(𝑛) is in the class 𝒰𝒦^{𝑚,𝑝,𝑛}_{𝛼,𝜆,𝑙}(𝛿,𝛽,𝑏) if and only if 𝑓 satisfies ℜ1𝑝+𝑏𝑧𝐷1+^{𝑚,𝛼𝜆,𝑙,𝑝}𝑓(𝑧)^′′𝐷^𝑚,𝛼_{𝜆,𝑙,𝑝}𝑓(𝑧)^′1−𝑝>𝛿𝑏𝑧𝐷1+^{𝑚,𝛼𝜆,𝑙,𝑝}𝑓(𝑧)^′′𝐷^𝑚,𝛼_{𝜆,𝑙,𝑝}𝑓(𝑧)^′−𝑝+𝛽, (10) where 𝑧∈𝕌, 𝑏∈ℂ−{0}, 𝛿≥0, 0≤𝛽<𝑝, and 𝐷^𝑚,𝛼_{𝜆,𝑙,𝑝} is defined by (7).

Note that 𝑓∈𝒰𝒦^{𝑚,𝑝,𝑛}_{𝛼,𝜆,𝑙}⟺1𝛿,𝛽,𝑏𝑝𝑧𝑓^′∈𝒰𝒮^{𝑚,𝑝,𝑛}_{𝛼,𝜆,𝑙}𝛿,𝛽,𝑏. (11)

Remark 3. (i) For 𝛿=0, we have the classes 𝒰𝒮^{𝑚,𝑝,𝑛}_{𝛼,𝜆,𝑙}0,𝛽,𝑏≡𝒮^{𝑚,𝑝,𝑛}_{𝛼,𝜆,𝑙}𝛽,𝑏,𝒰𝒦^{𝑚,𝑝,𝑛}_{𝛼,𝜆,𝑙}0,𝛽,𝑏≡𝒦^{𝑚,𝑝,𝑛}_{𝛼,𝜆,𝑙}𝛽,𝑏 (12) introduced and studied by Bulut [1]. In particular, we get the classes 𝒰𝒮^𝑚,1,1_𝛼,𝜆,00,𝛽,𝑏≡𝒮^𝑚_𝛼,𝜆𝛽,𝑏,𝒰𝒦^𝑚,1,1_𝛼,𝜆,00,𝛽,𝑏≡𝒦^𝑚_𝛼,𝜆𝛽,𝑏 (13) introduced and studied by Bulut [2].(ii) For 𝛿=0 and 𝛽=0, we have the classes 𝒰𝒮^{𝑚,𝑝,𝑛}_{𝛼,𝜆,𝑙}(0,0,𝑏)≡𝒮^{𝑚,𝑝,𝑛}_{𝛼,𝜆,𝑙}(𝑏),𝒰𝒦^{𝑚,𝑝,𝑛}_{𝛼,𝜆,𝑙}(0,0,𝑏)≡𝒦^{𝑚,𝑝,𝑛}_{𝛼,𝜆,𝑙}(𝑏). (14)In particular, we set 𝒰𝒮^0,𝑝,𝑛_{𝛼,𝜆,𝑙}(0,0,𝑏)≡𝒮^𝑝,𝑛(𝑏),𝒰𝒦^0,𝑝,𝑛_{𝛼,𝜆,𝑙}(0,0,𝑏)≡𝒦^𝑝,𝑛(𝑏) (15) for the 𝑝-valently starlike and 𝑝-valently convex functions with complex order 𝑏(𝑏∈ℂ−{0}), respectively.(iii) For 𝑚=0, we have the classes 𝒰𝒮^0,𝑝,𝑛_{𝛼,𝜆,𝑙}𝛿,𝛽,𝑏≡𝒮^𝑝,𝑛=𝛿,𝛽,𝑏𝑓∈𝒜_𝑝(𝑛)1∶ℜ𝑝+𝑏𝑧𝑓^′(𝑧)𝑓(𝑧)1−𝑝>𝛿𝑏𝑧𝑓^′(𝑧)𝑓(𝑧)−𝑝+𝛽,𝒰𝒦^0,𝑝,𝑛_{𝛼,𝜆,𝑙}𝛿,𝛽,𝑏≡𝒦^𝑝,𝑛=𝛿,𝛽,𝑏𝑓∈𝒜_𝑝(𝑛)1∶ℜ𝑝+𝑏1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)1−𝑝>𝛿𝑏1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)−𝑝+𝛽. (16)In particular, we set 𝒰𝒮^0,1,1_{𝛼,𝜆,𝑙}𝛿,𝛽,𝑏≡𝒮𝛿,𝛽,𝑏,𝒰𝒦^0,1,1_{𝛼,𝜆,𝑙}𝛿,𝛽,𝑏≡𝒦𝛿,𝛽,𝑏. (17) Also, we have the classes 𝒰𝒮^0,1,1_{𝛼,𝜆,𝑙}𝛿,𝛽,1≡𝒮𝒟𝛿,𝛽,𝒰𝒦^0,1,1_{𝛼,𝜆,𝑙}𝛿,𝛽,1≡𝒦𝒟𝛿,𝛽 (18) introduced and studied by Shams et al. [3].(iv) For 𝛿=0 and 𝑚=0, we have the classes 𝒰𝒮^0,𝑝,𝑛_{𝛼,𝜆,𝑙}0,𝛽,𝑏≡𝒮^𝑝,𝑛=𝛽,𝑏𝑓∈𝒜_𝑝(𝑛)1∶ℜ𝑝+𝑏𝑧𝑓^′(𝑧)𝑓(𝑧)−𝑝>𝛽,𝒰𝒦^0,𝑝,𝑛_{𝛼,𝜆,𝑙}0,𝛽,𝑏≡𝒦^𝑝,𝑛=𝛽,𝑏𝑓∈𝒜_𝑝(𝑛)1∶ℜ𝑝+𝑏1+𝑧𝑓^′′(𝑧)𝑓^′(𝑧)−𝑝>𝛽 (19)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 𝒰𝒮^0,1,1_{𝛼,𝜆,𝑙}0,𝛽,𝑏≡𝒮𝛽,𝑏,𝒰𝒦^0,1,1_{𝛼,𝜆,𝑙}0,𝛽,𝑏≡𝒦𝛽,𝑏𝑏∈ℂ−{0},0≤𝛽<1. (20) Note that for 𝑏=1, we have 𝒰𝒮^0,1,1_{𝛼,𝜆,𝑙}0,𝛽,1≡𝒮^∗𝛽,𝒰𝒦^0,1,1_{𝛼,𝜆,𝑙}0,𝛽,1≡𝒦𝛽 (21) which are the classes of starlike and convex functions type 𝛽(0≤𝛽<1), respectively.(v) For 𝛿=1, 𝑚=0, 𝑝=𝑛=1, and 𝑏=1, we have the classes 𝒰𝒮^0,1,1_{𝛼,𝜆,𝑙}1,𝛽,1≡𝒰𝒮𝒯𝛽,𝒰𝒦^0,1,1_{𝛼,𝜆,𝑙}𝛿,𝛽,1≡𝒰𝒞𝒱𝛽 (22)of uniformly starlike and uniformly convex functions of order 𝛽(0≤𝛽<1), respectively, investigated by Rønning [4]. In particular, we get the classes 𝒰𝒮^0,1,1_{𝛼,𝜆,𝑙}(1,0,1)≡𝒰𝒮𝒯,𝒰𝒦^0,1,1_{𝛼,𝜆,𝑙}𝛿,𝛽,1≡𝒰𝒞𝒱 (23) of uniformly starlike and uniformly convex functions, respectively, first defined by Goodman [5].

Remark 4. If we set 𝛿=0 and 𝑏=𝑒^−𝑖𝜃cos𝜃(|𝜃|<𝜋/2) in Definitions 1 and 2, then we have the classes 𝒮^{𝑚,𝑝,𝑛}_{𝛼,𝜆,𝑙}=𝛽,𝜃𝑓∈𝒜_𝑝(𝑛)∶ℜ𝑒^𝑖𝜃𝑧𝐷^𝑚,𝛼_{𝜆,𝑙,𝑝}𝑓(𝑧)^′𝐷^𝑚,𝛼_{𝜆,𝑙,𝑝}𝑓(𝑧),𝒦>𝛽cos𝜃,0≤𝛽<𝑝_{𝑚,𝑝,𝑛𝛼,𝜆,𝑙}=𝛽,𝜃𝑓∈𝒜_𝑝(𝑛)∶ℜ𝑒^𝑖𝜃𝑧𝐷1+^{𝑚,𝛼𝜆,𝑙,𝑝}𝑓(𝑧)^′′𝐷^𝑚,𝛼_{𝜆,𝑙,𝑝}𝑓(𝑧)^′>𝛽cos𝜃,0≤𝛽<𝑝, (24)(25) respectively. Also, we put 𝒮^0,𝑝,𝑛_{𝛼,𝜆,𝑙}𝛽,𝜃≡𝒮^𝑝,𝑛𝛽,𝜃,𝒦^0,𝑝,𝑛_{𝛼,𝜆,𝑙}𝛽,𝜃≡𝒦^𝑝,𝑛𝛽,𝜃. (26) In particular, we set 𝒮^0,1,1_{𝛼,𝜆,𝑙}𝛽,𝜃≡𝒮^𝜃_𝛽|𝜃|<𝜋2,𝒦,0≤𝛽<1_{0,1,1𝛼,𝜆,𝑙}𝛽,𝜃≡𝒦^𝜃_𝛽|𝜃|<𝜋2,0≤𝛽<1 (27) 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. Let 𝜂∈ℕ, 𝑚=(𝑚₁,…,𝑚_𝜂)∈ℕ^𝜂₀, and 𝑘=(𝑘₁,…,𝑘_𝜂)∈ℝ^𝜂₊. One defines the following general integral operators: ℐ^𝑚,𝑘_𝑝,𝜂∶𝒜_𝑝(𝑛)^𝜂⟶𝒜_𝑝(𝑛),ℐ^𝑚,𝑘_𝑝,𝜂𝑓₁,…,𝑓_𝜂=ℱ_{𝑝,𝜂,𝑚,𝑘},ℱ_{𝑝,𝜂,𝑚,𝑘}(𝑧)=^𝑧₀𝑝𝑡^𝜂𝑝−1_𝑗=1𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)𝑡^𝑝𝑘𝑗𝒥𝑑𝑡,_{𝑚,𝑘𝑝,𝜂}∶𝒜_𝑝(𝑛)^𝜂⟶𝒜_𝑝(𝑛),𝒥^𝑚,𝑘_𝑝,𝜂𝑓₁,…,𝑓_𝜂=𝒢_{𝑝,𝜂,𝑚,𝑘},𝒢_{𝑝,𝜂,𝑚,𝑘}(𝑧)=^𝑧₀𝑝𝑡^𝜂𝑝−1_𝑗=1𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)^′𝑝𝑡^{𝑝−1𝑘𝑗}𝑑𝑡, (28)(29)(30)(31) where 𝑧∈𝕌, 𝑓_𝑗∈𝒜_𝑝(𝑛), and 𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝} is defined by (7) for 1≤𝑗≤𝜂.

Remark 6. (i) For 𝜂=1, 𝑚₁=𝑚, 𝑘₁=𝑘, and 𝑓₁=𝑓, we have the integral operators ℱ_{𝑝,𝑚,𝑘}(𝑧)=^𝑧₀𝑝𝑡^𝑝−1𝐷^𝑚,𝛼_{𝜆,𝑙,𝑝}𝑓(𝑡)𝑡^𝑝𝑘𝒢𝑑𝑡,_{𝑝,𝑚,𝑘}(𝑧)=^𝑧₀𝑝𝑡^𝑝−1𝐷^𝑚,𝛼_{𝜆,𝑙,𝑝}𝑓(𝑡)^′𝑝𝑡^𝑝−1𝑘𝑑𝑡. (32)(33)(ii) For 𝛼=0, 𝜆=1, and 𝑙=0, we have the integral operators ℱ_{𝑝,𝑛,𝑚,𝑘}(𝑧)=^𝑧₀𝑝𝑡^𝑝−1𝐷^𝑚1𝑓₁(𝑡)𝑡^𝑝𝑘1⋯𝐷^𝑚𝑛𝑓_𝑛(𝑡)𝑡^𝑝𝑘𝑛𝒢𝑑𝑡,_{𝑝,𝑛,𝑚,𝑘}(𝑧)=^𝑧₀𝑝𝑡^𝑝−1𝐷^𝑚1𝑓₁(𝑡)^′𝑝𝑡^{𝑝−1𝑘1}⋯𝐷^𝑚𝑛𝑓_𝑛(𝑡)^′𝑝𝑡^{𝑝−1𝑘𝑛}𝑑𝑡 (34)introduced by Saltık et al. [8].(iii) For 𝑚₁=⋯=𝑚_𝜂=0, we have the integral operators ℱ_𝑝(𝑧)=^𝑧₀𝑝𝑡^𝑝−1𝑓₁(𝑡)𝑡^𝑝𝑘1⋯𝑓_𝑛(𝑡)𝑡^𝑝𝑘𝑛𝒢𝑑𝑡,_𝑝(𝑧)=^𝑧₀𝑝𝑡^𝑝−1𝑓^′₁(𝑡)𝑝𝑡^{𝑝−1𝑘1}⋯𝑓^′_𝑛(𝑡)𝑝𝑡^{𝑝−1𝑘𝑛}𝑑𝑡 (35)introduced by Frasin [9].

2. Sufficient Conditions for the Operator ℱ_{𝑝,𝜂,𝑚,𝑘}

Theorem 7. Let 𝜂∈ℕ, 𝑚=(𝑚₁,…,𝑚_𝜂)∈ℕ^𝜂₀, and 𝑘=(𝑘₁,…,𝑘_𝜂)∈ℝ^𝜂₊. Also let 𝑏∈ℂ−{0}, 𝛿_𝑗≥0, 0≤𝛽_𝑗<𝑝, and 𝑓_𝑗∈𝒰𝒮^{𝑚𝑗,𝑝,𝑛}_{𝛼,𝜆,𝑙}(𝛿_𝑗,𝛽_𝑗,𝑏) for 1≤𝑗≤𝜂. If 0≤𝑝+^𝜂_𝑗=1𝑘_𝑗𝛽_𝑗−𝑝<𝑝, (36) then the integral operator ℱ_{𝑝,𝜂,𝑚,𝑘}, defined by (29), is in the class 𝒦^𝑝,𝑛(𝜏,𝑏), where 𝜏=𝑝+^𝜂_𝑗=1𝑘_𝑗𝛽_𝑗−𝑝. (37)

Proof. From the definition (29), we observe that ℱ_{𝑝,𝜂,𝑚,𝑘}(𝑧)∈𝒜_𝑝(𝑛). On the other hand, it is easy to see that ℱ_{𝑝,𝜂,𝑚,𝑘}(𝑧)^′=𝑝𝑧^𝜂𝑝−1_𝑗=1𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑧)𝑧^𝑝𝑘𝑗. (38) Differentiating (38) logarithmically and multiplying by 𝑧, we obtain 𝑧ℱ_{𝑝,𝜂,𝑚,𝑘}(𝑧)^′′ℱ_{𝑝,𝜂,𝑚,𝑘}(𝑧)^′=𝑝−1+^𝜂_𝑗=1𝑘_𝑗𝑧𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)^′𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)−𝑝 (39) or equivalently 𝑧ℱ1+^{𝑝,𝜂,𝑚,𝑘}(𝑧)^′′ℱ_{𝑝,𝜂,𝑚,𝑘}(𝑧)^′−𝑝=^𝜂_𝑗=1𝑘_𝑗𝑧𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)^′𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)−𝑝. (40) Then, by multiplying (40) with 1/𝑏, we have 1𝑏𝑧ℱ1+^{𝑝,𝜂,𝑚,𝑘}(𝑧)^′′ℱ_{𝑝,𝜂,𝑚,𝑘}(𝑧)^′=−𝑝_𝜂𝑗=1𝑘_𝑗1𝑏𝑧𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)^′𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)−𝑝 (41) or equivalently 1𝑝+𝑏𝑧ℱ1+^{𝑝,𝜂,𝑚,𝑘}(𝑧)^′′ℱ_{𝑝,𝜂,𝑚,𝑘}(𝑧)^′=−𝑝_𝜂𝑗=1𝑘_𝑗1𝑝+𝑏𝑧𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)^′𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)−𝑝+𝑝−𝑝^𝜂_𝑗=1𝑘_𝑗. (42) Since 𝑓_𝑗∈𝒰𝒮^{𝑚𝑗,𝑝,𝑛}_{𝛼,𝜆,𝑙}(𝛿_𝑗,𝛽_𝑗,𝑏)(1≤𝑗≤𝜂), we get ℜ1𝑝+𝑏𝑧ℱ1+^{𝑝,𝜂,𝑚,𝑘}(𝑧)^′′ℱ_{𝑝,𝜂,𝑚,𝑘}(𝑧)^′=−𝑝_𝜂𝑗=1𝑘_𝑗ℜ1𝑝+𝑏𝑧𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)^′𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)−𝑝+𝑝−𝑝^𝜂_𝑗=1𝑘_𝑗>^𝜂_𝑗=1𝑘_𝑗𝛿_𝑗1𝑏𝑧𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)^′𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)−𝑝+𝑝+^𝜂_𝑗=1𝑘_𝑗𝛽_𝑗−𝑝. (43) Since ^𝜂_𝑗=1𝑘_𝑗𝛿_𝑗1𝑏𝑧𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)^′𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)−𝑝>0, (44) the integral operator ℱ_{𝑝,𝜂,𝑚,𝑘}∈𝒦^𝑝,𝑛(𝜏,𝑏) with 𝜏=𝑝+^𝜂_𝑗=1𝑘_𝑗𝛽_𝑗−𝑝. (45)

Corollary 8. In Theorem 7, letting(i)𝑝=𝑛=1, 𝛿_𝑗=0, and 𝑚_𝑗=0(1≤𝑗≤𝜂), one obtains [10, Theorem 1].(ii)𝑝=𝑛=1, 𝛿_𝑗=0, and 𝑚_𝑗=0, 𝛽_𝑗=𝛽∈[0,1)(1≤𝑗≤𝜂), one obtains [11, Theorem 1].(iii)𝑝=𝑛=1, 𝛿_𝑗=0, and 𝑚_𝑗=0, 𝛽_𝑗=0(1≤𝑗≤𝜂), one obtains [12, Theorem 1].

Putting 𝜂=1, 𝑚₁=𝑚, 𝑘₁=𝑘, 𝛽₁=𝛽, 𝛿₁=𝛿, and 𝑓₁=𝑓 in Theorem 7, we have the following.

Corollary 9. Let 𝑚∈ℕ₀ and 𝑘>0. Also let 𝑏∈ℂ−{0}, 𝛿≥0, 0≤𝛽<𝑝, and 𝑓∈𝒰𝒮^{𝑚,𝑝,𝑛}_{𝛼,𝜆,𝑙}(𝛿,𝛽,𝑏). If 0≤𝑝+𝑘𝛽−𝑝<𝑝, (46) then the integral operator ℱ_{𝑝,𝑚,𝑘}, defined by (32), is in the class 𝒦^𝑝,𝑛(𝜇,𝑏), where 𝜇=𝑝+𝑘𝛽−𝑝. (47)

Putting 𝑝=𝑛=1 and 𝑚=0 in Corollary 9, we have the following.

Corollary 10. Let 𝑘>0, 𝑏∈ℂ−{0}, 𝛿≥0, 0≤𝛽<1, and 𝑓∈𝒮(𝛿,𝛽,𝑏). If 0≤1+𝑘𝛽−1<1, (48) then the integral operator ℱ_𝑘(𝑧)=^𝑧₀𝑓(𝑡)𝑡^𝑘𝑑𝑡 (49) is convex of complex order 𝑏(𝑏∈ℂ−{0}) and type 𝜌=1+𝑘(𝛽−1); that is, ℱ_𝑘∈𝒦(𝜌,𝑏).

Putting 𝑝=𝑛=1, 𝑚_𝑗=0(1≤𝑗≤𝜂), and 𝑏=1 in Theorem 7, we have the following.

Corollary 11. Let 𝜂∈ℕ, 𝑘=(𝑘₁,…,𝑘_𝜂)∈ℝ^𝜂₊, 𝛿_𝑗≥0, 0≤𝛽_𝑗<1, and 𝑓_𝑗∈𝒮𝒟(𝛿_𝑗,𝛽_𝑗) for 1≤𝑗≤𝜂. If 0≤1+^𝜂_𝑗=1𝑘_𝑗𝛽_𝑗−1<1, (50) then the integral operator ℱ(𝑧)=^𝑧₀𝑓₁(𝑡)𝑡^𝑘1⋯𝑓_𝑛(𝑡)𝑡^𝑘𝑛𝑑𝑡 (51) is in the class 𝒦(𝜎), where 𝜎=1+^𝜂_𝑗=1𝑘_𝑗𝛽_𝑗−1. (52)

Theorem 12. Let 𝜂∈ℕ, 𝑚=(𝑚₁,…,𝑚_𝜂)∈ℕ^𝜂₀, and 𝑘=(𝑘₁,…,𝑘_𝜂)∈ℝ^𝜂₊. Also let |𝜃|<𝜋/2, 0≤𝛽_𝑗<𝑝, and 𝑓_𝑗∈𝒮^{𝑚𝑗,𝑝,𝑛}_{𝛼,𝜆,𝑙}(𝛽_𝑗,𝜃) for 1≤𝑗≤𝜂. If 0≤𝑝+^𝜂_𝑗=1𝑘_𝑗𝛽_𝑗−𝑝<𝑝, (53) then the integral operator ℱ_{𝑝,𝜂,𝑚,𝑘}, defined by (29), is in the class 𝒦^𝑝,𝑛(𝜏,𝜃), where 𝜏=𝑝+^𝜂_𝑗=1𝑘_𝑗𝛽_𝑗−𝑝. (54)

Proof. From (24), (25), and (40), we get the desired result.

Putting 𝜂=1, 𝑚₁=𝑚, 𝑘₁=𝑘, 𝛽₁=𝛽 and 𝑓₁=𝑓 in Theorem 12, we have the following.

Corollary 13. Let 𝑚∈ℕ₀ and 𝑘>0. Also let |𝜃|<𝜋/2, 0≤𝛽<𝑝, and 𝑓∈𝒮^{𝑚,𝑝,𝑛}_{𝛼,𝜆,𝑙}(𝛽,𝜃). If 0≤𝑝+𝑘𝛽−𝑝<𝑝, (55) then the integral operator ℱ_{𝑝,𝑚,𝑘}, defined by (32), is in the class 𝒦^𝑝,𝑛(𝜇,𝜃), where 𝜇=𝑝+𝑘𝛽−𝑝. (56)

Putting 𝑝=𝑛=1 and 𝑚=0 in Corollary 13, we have the following.

Corollary 14. Let 𝑘>0, |𝜃|<𝜋/2, 0≤𝛽<1, and 𝑓∈𝒮^𝜃_𝛽. If 0≤1+𝑘𝛽−1<1, (57) then the integral operator ℱ_𝑘, defined by (49), is in the class 𝒦^𝜃_𝜌, where 𝜌=1+𝑘𝛽−1. (58)

Putting 𝑝=𝑛=1 and 𝑚_𝑗=0(1≤𝑗≤𝜂) in Theorem 12, we have the following.

Corollary 15. Let 𝜂∈ℕ, 𝑘=(𝑘₁,…,𝑘_𝜂)∈ℝ^𝜂₊, 0≤𝛽_𝑗<1, and 𝑓_𝑗∈𝒮^𝜃_𝛽𝑗(|𝜃|<𝜋/2) for 1≤𝑗≤𝜂. If 0≤1+^𝜂_𝑗=1𝑘_𝑗𝛽_𝑗−1<1, (59) then the integral operator ℱ, defined by (51), is in the class 𝒦^𝜃_𝜎, where 𝜎=1+^𝜂_𝑗=1𝑘_𝑗𝛽_𝑗−1. (60)

Theorem 16. Let 𝜂∈ℕ, 𝑚=(𝑚₁,…,𝑚_𝜂)∈ℕ^𝜂₀, and 𝑘=(𝑘₁,…,𝑘_𝜂)∈ℝ^𝜂₊. Also let 𝑏∈ℂ−{0}, 𝛿_𝑗≥0, and 0≤𝛽_𝑗<𝑝 for 1≤𝑗≤𝜂. If 𝑧𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)^′𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)−𝑝>−𝑝+∑^𝜂_𝑗=1𝑘_𝑗𝛽_𝑗−𝑝(1/|𝑏|)∑^𝜂_𝑗=1𝑘_𝑗𝛿_𝑗 (61) for all 1≤𝑗≤𝜂, then the integral operator ℱ_{𝑝,𝜂,𝑚,𝑘}, defined by (29), is 𝑝-valently convex of complex order 𝑏(𝑏∈ℂ−{0}); that is, ℱ_{𝑝,𝜂,𝑚,𝑘}∈𝒦^𝑝,𝑛(𝑏).

Proof. From (43) and (61), we easily get that the integral operator ℱ_{𝑝,𝜂,𝑚,𝑘} is 𝑝-valently convex of complex order 𝑏.

3. Sufficient Conditions for the Operator 𝒢_{𝑝,𝜂,𝑚,𝑘}

Theorem 17. Let 𝜂∈ℕ, 𝑚=(𝑚₁,…,𝑚_𝜂)∈ℕ^𝜂₀, and 𝑘=(𝑘₁,…,𝑘_𝜂)∈ℝ^𝜂₊. Also let 𝑏∈ℂ−{0}, 𝛿_𝑗≥0, 0≤𝛽_𝑗<𝑝, and 𝑓_𝑗∈𝒰𝒦^{𝑚𝑗,𝑝,𝑛}_{𝛼,𝜆,𝑙}(𝛿_𝑗,𝛽_𝑗,𝑏) for 1≤𝑗≤𝜂. If 0≤𝑝+^𝜂_𝑗=1𝑘_𝑗𝛽_𝑗−𝑝<𝑝, (62) then the integral operator 𝒢_{𝑝,𝜂,𝑚,𝑘}, defined by (31), is in the class 𝒦^𝑝,𝑛(𝜏,𝑏), where 𝜏=𝑝+^𝜂_𝑗=1𝑘_𝑗𝛽_𝑗−𝑝. (63)

Proof. From the definition (31), we observe that 𝒢_{𝑝,𝜂,𝑚,𝑘}(𝑧)∈𝒜_𝑝(𝑛). On the other hand, it is easy to see that 𝒢_{𝑝,𝜂,𝑚,𝑘}(𝑧)^′=𝑝𝑧^𝜂𝑝−1_𝑗=1𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑧)^′𝑝𝑧^{𝑝−1𝑘𝑗}. (64) Differentiating (64) logarithmically and multiplying by 𝑧, we obtain 𝑧𝒢_{𝑝,𝜂,𝑚,𝑘}(𝑧)^′′𝒢_{𝑝,𝜂,𝑚,𝑘}(𝑧)^′=𝑝−1+^𝜂_𝑗=1𝑘_𝑗𝑧𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)^′′𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)^′−𝑝−1 (65) or equivalently 𝑧𝒢1+^{𝑝,𝜂,𝑚,𝑘}(𝑧)^′′𝒢_{𝑝,𝜂,𝑚,𝑘}(𝑧)^′=−𝑝_𝜂𝑗=1𝑘_𝑗𝑧𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)^′′𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)^′−𝑝−1. (66) Then, by multiplying (66) with 1/𝑏, we have 1𝑏𝑧𝒢1+^{𝑝,𝜂,𝑚,𝑘}(𝑧)^′′𝒢_{𝑝,𝜂,𝑚,𝑘}(𝑧)^′=−𝑝_𝜂𝑗=1𝑘_𝑗1𝑏𝑧𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)^′′𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)^′−𝑝−1 (67) or equivalently 1𝑝+𝑏𝑧𝒢1+^{𝑝,𝜂,𝑚,𝑘}(𝑧)^′′𝒢_{𝑝,𝜂,𝑚,𝑘}(𝑧)^′=−𝑝_𝜂𝑗=1𝑘_𝑗1𝑝+𝑏𝑧𝐷1+^{𝑚𝑗,𝛼𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)^′′𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)^′−𝑝+𝑝−𝑝^𝜂_𝑗=1𝑘_𝑗. (68) Since 𝑓_𝑗∈𝒰𝒦^{𝑚𝑗,𝑝,𝑛}_{𝛼,𝜆,𝑙}(𝛿_𝑗,𝛽_𝑗,𝑏)(1≤𝑗≤𝜂), we get ℜ1𝑝+𝑏𝑧𝒢1+^{𝑝,𝜂,𝑚,𝑘}(𝑧)^′′𝒢_{𝑝,𝜂,𝑚,𝑘}(𝑧)^′=−𝑝_𝜂𝑗=1𝑘_𝑗ℜ1𝑝+𝑏𝑧𝐷1+^{𝑚𝑗,𝛼𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)^′′𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)^′−𝑝+𝑝−𝑝^𝜂_𝑗=1𝑘_𝑗>^𝜂_𝑗=1𝑘_𝑗𝛿_𝑗1𝑏𝑧𝐷1+^{𝑚𝑗,𝛼𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)^′′𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)^′−𝑝+𝑝+^𝜂_𝑗=1𝑘_𝑗𝛽_𝑗−𝑝. (69) Since ^𝜂_𝑗=1𝑘_𝑗𝛿_𝑗1𝑏𝑧𝐷1+^{𝑚𝑗,𝛼𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)^′′𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)^′−𝑝>0, (70) the integral operator 𝒢_{𝑝,𝜂,𝑚,𝑘}∈𝒦^𝑝,𝑛(𝜏,𝑏) with 𝜏=𝑝+^𝜂_𝑗=1𝑘_𝑗𝛽_𝑗−𝑝. (71)

Corollary 18. In Theorem 17, letting(i)𝑝=𝑛=1, 𝛿_𝑗=0, and 𝑚_𝑗=0(1≤𝑗≤𝜂), one obtains [10, Theorem 3].(ii)𝑝=𝑛=1, 𝛿_𝑗=0, and 𝑚_𝑗=0, 𝛽_𝑗=𝛽∈[0,1)(1≤𝑗≤𝜂), one obtains [11, Theorem 3].(iii)𝑝=𝑛=1, 𝛿_𝑗=0, and 𝑚_𝑗=0, 𝛽_𝑗=0(1≤𝑗≤𝜂), one obtains [12, Theorem 2].

Putting 𝜂=1, 𝑚₁=𝑚, 𝑘₁=𝑘, 𝛽₁=𝛽, 𝛿₁=𝛿, and 𝑓₁=𝑓 in Theorem 17, we have the following.

Corollary 19. Let 𝑚∈ℕ₀ and 𝑘>0. Also let 𝑏∈ℂ−{0}, 𝛿≥0, 0≤𝛽<𝑝 and 𝑓∈𝒰𝒦^{𝑚,𝑝,𝑛}_{𝛼,𝜆,𝑙}(𝛿,𝛽,𝑏). If 0≤𝑝+𝑘𝛽−𝑝<𝑝, (72) then the integral operator 𝒢_{𝑝,𝑚,𝑘}, defined by (33), is in the class 𝒦^𝑝,𝑛(𝜇,𝑏), where 𝜇=𝑝+𝑘𝛽−𝑝. (73)

Putting 𝑝=𝑛=1 and 𝑚=0 in Corollary 19, we have the following.

Corollary 20. Let 𝑘>0, 𝑏∈ℂ−{0}, 𝛿≥0, 0≤𝛽<1, and 𝑓∈𝒦(𝛿,𝛽,𝑏). If 0≤1+𝑘𝛽−1<1, (74) then the integral operator 𝒢_𝑘(𝑧)=^𝑧₀𝑓^′(𝑡)^𝑘𝑑𝑡 (75) is convex of complex order 𝑏(𝑏∈ℂ−{0}) and type 𝜌=1+𝑘(𝛽−1); that is, 𝒢_𝑘∈𝒦(𝜌,𝑏).

Putting 𝑝=𝑛=1, 𝑚_𝑗=0(1≤𝑗≤𝜂), and 𝑏=1 in Theorem 17, we have the following.

Corollary 21. Let 𝜂∈ℕ, 𝑘=(𝑘₁,…,𝑘_𝜂)∈ℝ^𝜂₊, 𝛿_𝑗≥0, 0≤𝛽_𝑗<1, and 𝑓_𝑗∈𝒦𝒟(𝛿_𝑗,𝛽_𝑗) for 1≤𝑗≤𝜂. If 0≤1+^𝜂_𝑗=1𝑘_𝑗𝛽_𝑗−1<1, (76) then the integral operator 𝒢(𝑧)=^𝑧₀𝑓^′₁(𝑡)^𝑘1⋯𝑓^′_𝑛(𝑡)^𝑘𝑛𝑑𝑡 (77) is in the class 𝒦(𝜎), where 𝜎=1+^𝜂_𝑗=1𝑘_𝑗𝛽_𝑗−1. (78)

Theorem 22. Let 𝜂∈ℕ, 𝑚=(𝑚₁,…,𝑚_𝜂)∈ℕ^𝜂₀, and 𝑘=(𝑘₁,…,𝑘_𝜂)∈ℝ^𝜂₊. Also let |𝜃|<𝜋/2, 0≤𝛽_𝑗<𝑝, and 𝑓_𝑗∈𝒦^{𝑚𝑗,𝑝,𝑛}_{𝛼,𝜆,𝑙}(𝛽_𝑗,𝜃) for 1≤𝑗≤𝜂. If 0≤𝑝+^𝜂_𝑗=1𝑘_𝑗𝛽_𝑗−𝑝<𝑝, (79) then the integral operator 𝒢_{𝑝,𝜂,𝑚,𝑘}, defined by (31), is in the class 𝒦^𝑝,𝑛(𝜏,𝜃), where 𝜏=𝑝+^𝜂_𝑗=1𝑘_𝑗𝛽_𝑗−𝑝. (80)

Proof. From (25) and (66), we get the desired result.

Putting 𝜂=1, 𝑚₁=𝑚, 𝑘₁=𝑘, 𝛽₁=𝛽, and 𝑓₁=𝑓 in Theorem 22, we have the following.

Corollary 23. Let 𝑚∈ℕ₀ and 𝑘>0. Also let |𝜃|<𝜋/2, 0≤𝛽<𝑝, and 𝑓∈𝒦^{𝑚,𝑝,𝑛}_{𝛼,𝜆,𝑙}(𝛽,𝜃). If 0≤𝑝+𝑘𝛽−𝑝<𝑝, (81) then the integral operator 𝒢_{𝑝,𝑚,𝑘}, defined by (33), is in the class 𝒦^𝑝,𝑛(𝜇,𝜃), where 𝜇=𝑝+𝑘𝛽−𝑝. (82)

Putting 𝑝=𝑛=1 and 𝑚=0 in Corollary 23, we have the following.

Corollary 24. Let 𝑘>0, |𝜃|<𝜋/2, 0≤𝛽<1, and 𝑓∈𝒦^𝜃_𝛽. If 0≤1+𝑘𝛽−1<1, (83) then the integral operator 𝒢_𝑘, defined by (75), is in the class 𝒦^𝜃_𝜌, where 𝜌=1+𝑘𝛽−1. (84)

Putting 𝑝=𝑛=1 and 𝑚_𝑗=0(1≤𝑗≤𝜂) in Theorem 22, we have the following.

Corollary 25. Let 𝜂∈ℕ, 𝑘=(𝑘₁,…,𝑘_𝜂)∈ℝ^𝜂₊, 0≤𝛽_𝑗<1, and 𝑓_𝑗∈𝒦^𝜃_𝛽𝑗(|𝜃|<𝜋/2) for 1≤𝑗≤𝜂. If 0≤1+^𝜂_𝑗=1𝑘_𝑗𝛽_𝑗−1<1, (85) then the integral operator 𝒢, defined by (77), is in the class 𝒦^𝜃_𝜎, where 𝜎=1+^𝜂_𝑗=1𝑘_𝑗𝛽_𝑗−1. (86)

Theorem 26. Let 𝜂∈ℕ, 𝑚=(𝑚₁,…,𝑚_𝜂)∈ℕ^𝜂₀, and 𝑘=(𝑘₁,…,𝑘_𝜂)∈ℝ^𝜂₊. Also let 𝑏∈ℂ−{0}, 𝛿_𝑗≥0, and 0≤𝛽_𝑗<𝑝 for 1≤𝑗≤𝜂. If 𝑧𝐷1+^{𝑚𝑗,𝛼𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)^′′𝐷^{𝑚𝑗,𝛼}_{𝜆,𝑙,𝑝}𝑓_𝑗(𝑡)^′−𝑝>−𝑝+∑^𝜂_𝑗=1𝑘_𝑗𝛽_𝑗−𝑝(1/|𝑏|)∑^𝜂_𝑗=1𝑘_𝑗𝛿_𝑗 (87) for all 1≤𝑗≤𝜂, then the integral operator 𝒢_{𝑝,𝜂,𝑚,𝑘}, defined by (31), is 𝑝-valently convex of complex order 𝑏(𝑏∈ℂ−{0}); that is, 𝒢_{𝑝,𝜂,𝑚,𝑘}∈𝒦^𝑝,𝑛(𝑏).

Proof. From (69) and (87), we easily get that the integral operator 𝒢_{𝑝,𝜂,𝑚,𝑘} is 𝑝-valently convex of complex order 𝑏.