Some Inclusion Relations Associated with Generalized Fractional Differintegral Operator

Abstract

A generalized fractional differintegral operator is used to define some new subclasses of analytic functions in the open unit disk 𝕌. For each of these new function classes, several inclusion relationships are established.

1. Introduction and Definitions

Let 𝔸 be the class of normalized functions 𝑓 of the form 𝑓(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑧^𝑛 (1) which are analytic in the open unit disk 𝕌={𝑧∈𝐷,|𝑧|<1}. If 𝑓∈𝔸 is given by (1) and 𝑔∈𝔸 given by 𝑔(𝑧)=𝑧+∑^∞_𝑛=2𝑏_𝑛𝑧^𝑛 in 𝑧∈𝕌, then the Hadamard product (or convolution) of 𝑓 and 𝑔 is defined by 𝑓∗𝑔(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑏_𝑛𝑧^𝑛. (2) Let 𝑃_𝑘(𝛼) denote the class of functions ℎ(𝑧) analytic in the unit disk 𝕌, satisfying the properties ℎ(0)=1 and ₀^2𝜋ℝℎ(𝑧)−𝛼1−𝛼𝑑𝜃≤𝑘𝜋𝑧=𝑟𝑒^𝑖𝜃;0≤𝛼<1;𝑘≥2. (3) This class 𝑃_𝑘(𝛼) has been introduced in [1]. Note that, for 𝛼=0, we obtain the class 𝑃_𝑘 defined and studied in [2], and for 𝑘=2, we have the class 𝑃(𝛼) of functions with positive real part greater than 𝛼. In particular,𝑃(0) is the class 𝑃 of functions with positive real part. From (3), we can easily deduce that ℎ∈𝑃_𝑘(𝛼) if and only if ℎ(𝑧)=𝑘4+12ℎ₁(𝑧)−𝑘4−12ℎ₂(𝑧),ℎ₁,ℎ₂∈𝑃(𝛼). (4) Following the recent investigations [3] (see also [4, 5]), we have the following subclasses: 𝑅_𝑘(𝛼)=𝑓∈𝐴∶𝑧𝑓^′(𝑧)𝑓(𝑧)∈𝑃_𝑘(𝛼),𝑉,𝑧∈𝕌_𝑘(𝛼)=𝑓∈𝐴∶𝑧𝑓^′(𝑧)^′𝑓^′(𝑧)∈𝑃_𝑘(𝛼),𝑃,𝑧∈𝕌_′𝑘(𝛼)=𝑓∈𝐴∶𝑓^′(𝑧)∈𝑃_𝑘(𝛼),𝑇,𝑧∈𝕌_𝑘𝛽,𝛼=𝑓∈𝐴∶𝑔∈𝑅₂(𝛼),𝑧𝑓^′(𝑧)𝑔(𝑧)∈𝑃_𝑘𝛽,𝑇,𝑧∈𝕌_∗𝑘=𝛽,𝛼𝑓∈𝐴∶𝑔∈𝑉₂(𝛼),𝑧𝑓^′(𝑧)^′𝑔^′(𝑧)∈𝑃_𝑘𝛽,𝑧∈𝕌. (5) We note that the class 𝑅₂(𝛼)=𝑆^∗(𝛼) and 𝑉₂(𝛼)=𝒦(𝛼) are, respectively, the subclasses of 𝔸, consisting of functions which are starlike of order 𝛼 and convex of order 𝛼 in 𝕌. The class 𝑇^∗₂(𝛽,𝛼)=𝐶^∗(𝛽,𝛼) was considered by Noor [6], and 𝑇^∗₂(0,0)=𝐶^∗ is the class of quasiconvex univalent functions which was first introduced and studied in [7]. It can be easily seen from the above definition that 𝑓(𝑧)∈𝑉_𝑘(𝛼)⟺𝑧𝑓^′(𝑧)∈𝑅_𝑘(𝛼),𝑓(𝑧)∈𝑇^∗_𝑘𝛽,𝛼⟺𝑧𝑓^′(𝑧)∈𝑇_𝑘𝛽,𝛼. (6)(7) We recall here the following family of generalized fractional integral operators due to Srivastava et al. [8] and generalized fractional derivative operators due to Raina and Nahar [9] (see also [10, 11]).

Definition 1. Let 𝜆>0 and 𝜇,𝜂∈ℝ. Then the fractional integral operator ℐ^{𝜆,𝜇,𝜂}_0,𝑧 is defined by ℐ^{𝜆,𝜇,𝜂}_0,𝑧𝑓(𝑧)=𝑧^{−𝜆−𝜇}Γ(𝜆)^𝑧₀(𝑧−𝜁)₂^𝜆−1𝐹₁𝜁𝜆+𝜇,−𝜂;𝜆;1−𝑧𝑓(𝜁)𝑑𝜁, (8) where the function 𝑓(𝑧) is analytic in a simply connected region of the complex 𝑧-plane containing the origin, with the order 𝑓(𝑧)=𝑂|𝑧|^𝜀𝑧→0;𝜀>max0,𝜇−𝜂−1, (9) and the multiplicity (𝑧−𝜁)^𝜆−1 is removed by requiring log(𝑧−𝜁) to be real when 𝑧−𝜁>0.

Definition 2. For 𝑚−1≦𝜆<𝑚 and 𝜇,𝜂∈ℝ and 𝑚∈ℕ, the fractional derivative operator 𝒥^{𝜆,𝜇,𝜂}_0,𝑧 is defined by 𝒥^{𝜆,𝜇,𝜂}_0,𝑧𝑓(𝑧)=𝑑^𝑚𝑑𝑧^𝑚𝑧^𝜆−𝜇Γ()×𝑚−𝜆_𝑧0(𝑧−𝜁)₂^{𝑚−𝜆−1}𝐹₁𝜁𝜇−𝜆,𝑚−𝜂;𝑚−𝜆;1−𝑧𝑓(𝜁)𝑑𝜁, (10) where the multiplicity (𝑧−𝜁)^𝑚−𝜆 is removed as in the above definition.

The operators ℐ^{𝜆,𝜇,𝜂}_0,𝑧 and 𝒥^{𝜆,𝜇,𝜂}_0,𝑧 include the Riemann-Liouville and Erdelyi-Kober operators of fractional calculus (see, e.g., [10]). Using the hypotheses of Definitions 1 and 2 the generalized fractional differintegral operator 𝒮^{𝜆,𝜇,𝜂}_0,𝑧∶𝔸→𝔸 (11) is defined by (see also [12–15]) 𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑓(𝑧)=Γ2−𝜇Γ2+𝜂−𝜆Γ𝑧2+𝜂−𝜇^𝜇𝒥^{𝜆,𝜇,𝜂}_0,𝑧𝑓(𝑧),,ΓΓ0≤𝜆<𝜂+22−𝜇2+𝜂−𝜆Γ𝑧2+𝜂−𝜇^𝜇ℐ^{−𝜆,𝜇,𝜂}_0,𝑧𝑓(𝑧),(),.𝜆<0𝑧∈𝕌,𝜇,𝜂∈ℝ,min−𝜇,𝜂−𝜆,𝜂−𝜇>−2 (12) It is easily seen from (12) that for a function 𝑓 of the form (1), we have 𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑓(𝑧)ΓΓ=𝑧+2−𝜇2+𝜂−𝜆Γ×2+𝜂−𝜇_∞𝑛=2Γ(𝑛+1)Γ𝑛+𝜂−𝜇+1ΓΓ𝑎𝑛−𝜇+1𝑛+𝜂−𝜆+1^𝑛𝑧^𝑛=𝑧₃𝐹₂1,2,2+𝜂−𝜇;2−𝜇,2+𝜂−𝜆;𝑧∗𝑓(𝑧). (13)

We note that the operator 𝒮^{𝜆,𝜇,𝜂}_0,𝑧 is a generalization of many other operators, for example, (i)𝒮^0,0,0_0,𝑧𝑓(𝑧)(𝑧),()𝒮=𝑓ii^1,1,1_0,𝑧𝑓(𝑧)=𝒮^1,0,0_0,𝑧𝑓(𝑧)=𝑧𝑓^′(𝑧),(iii)𝒮^2,1,1_0,𝑧𝑓(𝑧)=𝑧𝑓^′(𝑧)+𝑧²𝑓^′′(𝑧),(iv)𝒮^{𝜆,𝜆,𝜂}_0,𝑧𝑓(𝑧)=𝒮^𝜆,𝜇,0_0,𝑧𝑓(𝑧),(v)𝒮^{−𝜆,0,𝛼−1}_0,𝑧𝑓(𝑧)=Ω^𝜆_𝛼𝑓(𝑧)=𝛼𝜆+𝛼𝜆𝑧^𝛼×^𝑧₀𝜁^𝛼−1𝜁1−𝑧^𝜆−1𝑓(𝜁)𝑑𝜁, (14) which is a multiplier transformation operator Ω^𝜆_𝛼 studied by Jung et al. [16], (vi)𝒮^{−1,0,𝛼−1}_0,𝑧𝑓(𝑧)=ℐ_𝛼𝑓(𝑧)=𝛼+1𝑧^𝛼𝑧₀𝜁^𝛼−1𝑓(𝜁)𝑑𝜁(𝛼>−1) (15) which is the generalized Bernardi-Libra-Livingston integral operator. It can easily be verified from (12) that 𝑧𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑓(𝑧)^′=𝜂−𝜆+1𝒮^{𝜆+1,𝜇,𝜂}_0,𝑧𝑓(𝑧)−𝜂−𝜆𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑓(𝑧). (16) For the generalized fractional differintegral operator 𝒮^{𝜆,𝜇,𝜂}_0,𝑧, we now define the following subclasses of 𝔸.

Definition 3. Let 𝑓(𝑧)∈𝔸. Then 𝑓(𝑧)∈𝑅^{𝜆,𝜇,𝜂}(𝑘,𝛼) if and only if 𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑓(𝑧)∈𝑅_𝑘(𝛼), for 𝑧∈𝕌.

Definition 4. Let 𝑓(𝑧)∈𝔸. Then 𝑓(𝑧)∈𝑉^{𝜆,𝜇,𝜂}(𝑘,𝛼) if and only if 𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑓(𝑧)∈𝑉_𝑘(𝛼), for 𝑧∈𝕌.

Definition 5. Let 𝑓(𝑧)∈𝔸. Then 𝑓(𝑧)∈𝑇^{𝜆,𝜇,𝜂}(𝑘,𝛽,𝛼) if and only if 𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑓(𝑧)∈𝑇_𝑘(𝛽,𝛼), for 𝑧∈𝕌.

Definition 6. Let 𝑓(𝑧)∈𝔸. Then 𝑓(𝑧)∈𝑇_∗^{𝜆,𝜇,𝜂}(𝑘,𝛽,𝛼) if and only if 𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑓(𝑧)∈𝑇^∗_𝑘(𝛽,𝛼), for 𝑧∈𝕌.

Definition 7. Let 𝑓(𝑧)∈𝔸. Then 𝑓(𝑧)∈𝑃^′_𝑘(𝜆,𝜇,𝜂,𝛼) if and only if 𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑓(𝑧)∈𝑃^′_𝑘(𝛼), for 𝑧∈𝕌.

In this paper we establish some inclusion relationships and some other interesting properties for these subclasses.

2. Main Inclusion Relationships

We recall first the following necessary lemma.

Lemma 8 (see [17]). Let 𝑢=𝑢₁+𝑖𝑢₂ and 𝑣=𝑣₁+𝑖𝑣₂, and let 𝜙(𝑢,𝑣) be a complex-valued function satisfying the conditions:(i)𝜙(𝑢,𝑣) is continuous in 𝐷⊂ℂ²,(ii)(1,0)∈𝐷 and ℝ𝜙(1,0)>0,(iii)ℝ𝜙(𝑖𝑢₂,𝑣₁)≤0 whenever (𝑖𝑢₂,𝑣₁)∈𝐷 and 𝑣₁≤−(1/2)(1+𝑢²₂).

If ℎ(𝑧)=1+∑^∞_𝑛=2𝑐_𝑛𝑧^𝑛 is a function analytic in 𝕌 such that (ℎ(𝑧),𝑧ℎ^′(𝑧))∈𝐷 and ℝe(𝜙(ℎ(𝑧),𝑧ℎ^′(𝑧)))>0, for 𝑧∈𝕌, then ℝe(ℎ(𝑧))>0 for 𝑧∈𝕌.

Our first main inclusion relationship is given by the theorem below.

Theorem 9. Let 𝑓∈𝔸,  0≤𝛼<1,  𝜂−𝜆>1/8, and  min{−𝜇,𝜂−𝜇}>−2; then 𝑅^{𝜆+1,𝜇,𝜂}(𝑘,𝛼)⊂𝑅^{𝜆,𝜇,𝜂}𝑘,𝛼₁, (17) where 𝛼₁×2+=22𝛼𝜂−2𝛼𝜆+1𝜂−𝜆−𝛼+14𝜂−𝜆+𝛼²+4𝜂−𝜆−𝛼+9⁻¹ (18) and 0≤𝛼<𝛼₁<1.

Proof. Let 𝑓∈𝑅^{𝜆+1,𝜇,𝜂}(𝑘,𝛼). Then upon setting 𝑧𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑓(𝑧)^′𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑓(𝑧)(𝑧)=𝑘=𝑝4+12𝑝₁(𝑧)−𝑘4−12𝑝₂(𝑧)(𝑧∈𝕌), (19) we see that the function 𝑝(𝑧) is analytic in 𝕌, with 𝑝(0)=1 in 𝑧∈𝕌. Using the identity (16) in (19) and differentiating with respect to 𝑧, we get 𝑧𝒮^{𝜆+1,𝜇,𝜂}_0,𝑧𝑓(𝑧)^′𝒮^{𝜆+1,𝜇,𝜂}_0,𝑧𝑓(𝑧)=𝑝(𝑧)+𝑧𝑝^′(𝑧)𝑝(𝑧)+𝜂−𝜆∈𝑃_𝑘(𝛼)(𝑧∈𝕌). (20) Let 𝜙(𝑧)=^∞_𝑗=1𝜂−𝜆+𝑗𝑧𝜂−𝜆+1^𝑗. (21) Then, by convolution technique, (see [18]), we have 𝑝(𝑧)∗𝜙(𝑧)𝑧=𝑝(𝑧)+𝑧𝑝^′(𝑧)𝑝(𝑧)=𝑘+𝜂−𝜆4+12𝑝₁(𝑧)∗𝜙(𝑧)𝑧−𝑘4−12𝑝₂(𝑧)∗𝜙(𝑧)𝑧 (22) and this implies that 𝑝_𝑖(𝑧)+𝑧𝑝^′_𝑖(𝑧)𝑝_𝑖(𝑧)(𝛼).+𝜂−𝜆∈𝑃)(𝑧∈𝕌,𝑖=1,2 (23) We want to show that 𝑝_𝑖(𝑧)∈𝑃(𝛼₁), where 𝛼₁ is given by (18), and this will show that 𝑝(𝑧)∈𝑃_𝑘(𝛼₁) for 𝑧∈𝕌. Let 𝑝_𝑖(𝑧)=1−𝛼₁ℎ_𝑖(𝑧)+𝛼₁(𝑧∈𝕌,𝑖=1,2). (24) Then in view of (23) and (24), we obtain, for 𝑧∈𝕌,  𝑖=1,2ℝ1−𝛼₁ℎ_𝑖(𝑧)+𝛼₁−𝛼+1−𝛼₁𝑧ℎ^′_𝑖(𝑧)1−𝛼₁ℎ_𝑖(𝑧)+𝛼₁+𝜂−𝜆>0. (25) We now form a function 𝜙(𝑢,𝑣) by choosing 𝑢=ℎ_𝑖(𝑧) and 𝑣=𝑧ℎ^′_𝑖(𝑧) in (25). Thus 𝜙(𝑢,𝑣)=1−𝛼₁𝑢+𝛼₁−𝛼+1−𝛼₁𝑣1−𝛼₁𝑢+𝛼₁+𝜂−𝜆. (26) We can easily see that the first two conditions of Lemma 8 are easily satisfied as 𝜙(𝑢,𝑣) is continuous in 𝐷=ℂ−(−(𝛼₁+𝜂−𝜆)/(1−𝛼₁))×ℂ,  (1,0)∈𝐷, and ℝ{𝜙(1,0)}>0. Now for 𝑣₁≤−(1/2)(1+𝑢²₂) we obtain ℝ𝜙𝑖𝑢₂,𝑣₁=ℝ1−𝛼₁𝑖𝑢₂+𝛼₁−𝛼+1−𝛼₁𝑣₁1−𝛼₁𝑖𝑢₂+𝛼₁+𝜂−𝜆=𝛼₁−𝛼+1−𝛼₁𝑣₁𝛼₁+𝜂−𝜆𝛼₁+𝜂−𝜆²+1−𝛼₁²𝑢²₂≤𝛼₁1−𝛼−21−𝛼₁𝛼₁+𝜂−𝜆1+𝑢²₂𝛼₁+𝜂−𝜆²+1−𝛼₁²𝑢²₂=𝐴+𝐵𝑢²₂,2𝐶 (27) where 𝐴=(𝛼₁+𝜂−𝜆){2(𝛼₁−𝛼)(𝛼₁+𝜂−𝜆)−(1−𝛼₁)},  𝐵=(1−𝛼₁){2(𝛼₁−𝛼)(1−𝛼₁)−(𝛼₁+𝜂−𝜆)}, and 𝐶=(𝛼₁+𝜂−𝜆)²+(1−𝛼₁)²𝑢²₂>0.
We note that ℝ{𝜙(𝑖𝑢₂,𝑣₁}≤0 if and only if 𝐴≤0 and 𝐵≤0. From 𝐴≤0, we obtain 𝛼₁ as given by (18), and 𝐵≤0 gives us 0≤𝛼₁<1. This completes the proof.

Theorem 10. Let 𝑓∈𝔸,  0≤𝛼<1,  𝜂−𝜆>1/8, and min{−𝜇,𝜂−𝜇}>−2; then 𝑉^{𝜆+1,𝜇,𝜂}(𝑘,𝛼)⊂𝑉^{𝜆,𝜇,𝜂}𝑘,𝛼₁, (28) where 𝛼₁ is given by (18).

Proof. To prove the inclusion relationship, we observe (in view of Theorem 9) that 𝑓(𝑧)∈𝑉^{𝜆+1,𝜇,𝜂}(𝑘,𝛼)⟺𝑧𝑓^′(𝑧)∈𝑅^{𝜆+1,𝜇,𝜂}(𝑘,𝛼)⟹𝑧𝑓^′(𝑧)∈𝑅^{𝜆,𝜇,𝜂}𝑘,𝛼₁⟺𝑓(𝑧)∈𝑉^{𝜆,𝜇,𝜂}𝑘,𝛼₁, (29) which establishes Theorem 10.

Theorem 11. Let 𝑓∈𝔸,   0≤𝛼,  𝛽<1,  𝜂−𝜆>1/8, and min{−𝜇,𝜂−𝜇}>−2; then 𝑇^{𝜆+1,𝜇,𝜂}𝑘,𝛽,𝛼⊂𝑇^{𝜆,𝜇,𝜂}𝑘,𝛽₁,𝛼₁, (30) where 𝛼₁ is given by (18) and 𝛽<𝛽₁<1 is defined in the proof.

Proof. Let 𝑓(𝑧)∈𝑇^{𝜆+1,𝜇,𝜂}(𝑘,𝛽,𝛼). Then there exists 𝑔(𝑧)∈𝑅^{𝜆+1,𝜇,𝜂}(2,𝛼) such that 𝑧𝒮^{𝜆+1,𝜇,𝜂}_0,𝑧𝑓(𝑧)^′𝒮^{𝜆+1,𝜇,𝜂}_0,𝑧𝑔(𝑧)∈𝑃_𝑘𝛽𝑧∈𝕌,0≤𝛽<1. (31) Since 𝑅^{𝜆+1,𝜇,𝜂}(2,𝛼)⊂𝑅^{𝜆,𝜇,𝜂}(2,𝛼₁), therefore we have 𝑧𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑔(𝑧)^′𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑔(𝑧)=1−𝛼₁𝑞(𝑧)+𝛼₁=1−𝛼₁𝑞₁+𝜄𝑞₂+𝛼₁, (32) where 𝑞(𝑧)∈𝑃. Next consider, 𝑧𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑓(𝑧)^′𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑔(𝑧)=1−𝛽₁ℎ(𝑧)+𝛽₁ (33) and proceed on the same technique of Theorem 9 (see also [5]); we obtain 𝑓(𝑧)∈𝑇^{𝜆,𝜇,𝜂}(𝑘,𝛽₁,𝛼₁), where 𝛽₁=2𝛽1−𝛼₁𝑞₁+𝛼₁+𝜂−𝜆²+1−𝛼₁²𝑞²₂+1−𝛼₁𝑞₁+𝛼₁×2+𝜂−𝜆1−𝛼₁𝑞₁+𝛼₁+𝜂−𝜆²+1−𝛼₁²𝑞²₂+1−𝛼₁𝑞₁+𝛼₁+𝜂−𝜆⁻¹. (34)

In view of relation (7), an application of Theorem 11 is obtained as follows.

Theorem 12. Let 𝑓∈𝔸,   0≤𝛼,  𝛽<1,  𝜂−𝜆>1/8, and min{−𝜇,𝜂−𝜇}>−2; then 𝑇_∗^{𝜆+1,𝜇,𝜂}𝑘,𝛽,𝛼⊂𝑇_∗^{𝜆,𝜇,𝜂}𝑘,𝛽₁,𝛼₁, (35) where 𝛽₁ and 𝛼₁ are as in Theorem 11.

Lemma 13 (see [19]). Let 𝑝(𝑧) be analytic in 𝕌 with 𝑝(0)=1 and ℝ𝑝(𝑧)>0,  𝑧∈𝕌. Then, for 𝑠>0 and 𝜂₁≠−1 (complex), ℝ𝑝(𝑧)+𝑠𝑧𝑝^′(𝑧)𝑝(𝑧)+𝜂₁>0,𝑓𝑜𝑟|𝑧|<𝑟₀, (36) where 𝑟₀ is given by 𝑟₀=|1+𝜂₁|/𝑚+(𝑚²−|𝜂²₁−1|)^1/2 and 𝑚=2(𝑠+1)²+|𝜂₁|²−1, and this result is the best possible.

Theorem 14. Let 𝑧∈𝕌,  𝑓∈𝑅^{𝜆,𝜇,𝜂}(𝑘,0). Then 𝑓∈𝑅^{𝜆+1,𝜇,𝜂}(𝑘,0) for |𝑧|<𝑟₀, where 𝑟₀=𝜂−𝜆+1𝑚+𝑚²−𝜂−𝜆²−1,𝑚=7+𝜂−𝜆². (37) This radius is exact.

Proof. Let 𝑧𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑓(𝑧)^′𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑓(𝑧)(𝑧)=𝑘=𝑝4+12𝑝₁(𝑧)−𝑘4−12𝑝₂(𝑧), (38) where 𝑝∈𝑃_𝑘 and 𝑝₁,𝑝₂∈𝑃 in 𝕌. Using a similar argument as in Theorem 9, we obtain 𝑧𝒮^{𝜆+1,𝜇,𝜂}_0,𝑧𝑓(𝑧)^′𝒮^{𝜆+1,𝜇,𝜂}_0,𝑧𝑓(𝑧)=𝑝(𝑧)+𝑧𝑝^′(𝑧)(𝑧)=𝑘𝜂−𝜆+𝑝4+12𝑝₁(𝑧)+𝑧𝑝^′₁(𝑧)𝑝₁(𝑧)−𝑘+𝜂−𝜆4−12𝑝₂(𝑧)+𝑧𝑝^′₂(𝑧)𝑝₂(𝑧).+𝜂−𝜆 (39) Applying Lemma 13, we get ℝ𝑝_𝑖(𝑧)+𝑧𝑝^′_𝑖(𝑧)𝑝_𝑖(𝑧)|𝑧|+𝜂−𝜆>0for<𝑟₀, (40) where 𝑟₀ is given by (37). This completes our proof.

Lemma 15 (see [20]). Let 𝑝 be analytic in 𝔼 with 𝑝(0)=1, and 𝛿 is a complex number satisfying ℝ𝛿≥0,  (𝛿≠0); then ℝ[𝑝(𝑧)+𝛿𝑧𝑝^′(𝑧)]>𝛽,  (0≤𝛽<1) implies ℝ𝑝(𝑧)>𝛽+(1−𝛽)(2𝛾−1), where 𝛾 is given by 𝛾=¹₀1+𝑡^ℝ𝛿−1𝑑𝑡, (41) which is an increasing function of ℝ𝛿 and 1/2≤𝛾≤1. The estimate is sharp in the sense that bound cannot be improved.

Theorem 16. Let min{𝜂−𝜆−1,−𝜇,𝜂−𝜇}>−2. Then 𝑃^′_𝑘𝜆+1,𝜇,𝜂,𝛼⊂𝑃^′_𝑘𝜆,𝜇,𝜂,𝛼+(1−𝛼)2𝛾−1, (42) where 𝛾=¹₀1+𝑡^{1/(𝜂−𝜆+1)−1}𝑑𝑡, (43) which is an increasing function of 1/(𝜂−𝜆+1) and 1/2≤𝛾<1.

Proof. Let 𝑓(𝑧)∈𝑃^′_𝑘(𝜆+1,𝜇,𝜂,𝛼)𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑓(𝑧)^′(𝑧)=𝑘=𝐻4+12ℎ₁(𝑧)−𝑘4−12ℎ₂(𝑧), (44) where 𝐻(𝑧) is analytic and 𝐻(0)=1 in 𝕌. Identity (16) gives us 𝒮^{𝜆+1,𝜇,𝜂}_0,𝑧𝑓(𝑧)^′=𝐻(𝑧)+𝑧𝐻^′(𝑧)=𝑘𝜂−𝜆+14+12ℎ₁(𝑧)+𝑧ℎ^′₁(𝑧)−𝑘𝜂−𝜆+14−12ℎ₂(𝑧)+𝑧ℎ^′₂(𝑧).𝜂−𝜆+1 (45) This implies that ℝℎ_𝑖(𝑧)+𝑧ℎ^′_𝑖(𝑧)𝜂−𝜆+1>𝛼,𝑖=1,2. (46) Now using Lemma 15, we get the desired result.

Lemma 17 (see [21]). Let 𝜙 be convex and 𝑔 starlike in 𝕌. Then, for 𝐹 analytic in 𝕌 with 𝐹(0)=1,  (Φ∗𝐹𝑔)/(Φ∗𝑔) is contained in the convex hull of 𝐹(𝕌). By convex hull of a set 𝑋, one means the intersection of all convex sets that contain 𝑋.

Theorem 18. Let 𝜙 be a convex function and 𝑓∈𝑅^{𝜆,𝜇,𝜂}(2,𝛼). Then 𝜙∗𝑓∈𝑅^{𝜆,𝜇,𝜂}(2,𝛼).

Proof. Let 𝐺=𝜙∗𝑓, and let 𝜙(𝑧)=𝑧+^∞_𝑛=2𝑏_𝑛𝑧^𝑛,𝑓(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑛𝑧^𝑛. (47) Then 𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝐺(𝑧)=𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑧+^∞_𝑛=2𝑎_𝑛𝑏_𝑛𝑧^𝑛ΓΓ=𝑧+2−𝜇2+𝜂−𝜆Γ×2−𝜇+𝜂_∞𝑛=2Γ(𝑛+1)Γ𝑛−𝜇+𝜂+1ΓΓ𝑎𝑛−𝜇+1𝑛+𝜂−𝜆+1^𝑛𝑏_𝑛𝑧^𝑛=𝜙∗𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑓(𝑧). (48) Also, 𝑓(𝑧)∈𝑅^{𝜆,𝜇,𝜂}(2,𝛼). Therefore, 𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑓(𝑧)∈𝑅₂(𝛼)=𝑆^∗(𝛼). By logarithmic differentiation of (48), we have 𝑧𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝐺(𝑧)^′𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝐺(𝑧)=𝜙(𝑧)∗𝐹𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑓(𝑧)𝜙(𝑧)∗𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑓(𝑧), (49) where 𝐹(𝑧)=𝑧𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑓(𝑧)^′𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑓(𝑧) (50) is analytic in 𝕌, and 𝐹(0)=1. From Lemma 17, we see that 𝑧(𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝐺(𝑧))^′/𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝐺(𝑧) is contained in the convex hull of 𝐹(𝕌). Since 𝑧(𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝐺(𝑧))^′/𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝐺(𝑧) is analytic in 𝕌 and 𝐹(𝕌)⊂Ω={𝑊∶(𝑧(𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑊(𝑧))^′)/(𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝑊(𝑧))∈𝑃₂(𝛼)}, then 𝑧(𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝐺(𝑧))^′/𝒮^{𝜆,𝜇,𝜂}_0,𝑧𝐺(𝑧) lies in Ω. This implies that 𝐺=𝜙∗𝑓∈𝑅^{𝜆,𝜇,𝜂}(2,𝛼).

This is application of Theorem 18.

Theorem 19. The class 𝑅^{𝜆,𝜇,𝜂}(2,𝛼) is invariant under the following differintegral operators. That is if 𝑓∈𝑅^{𝜆,𝜇,𝜂}(2,𝛼), then so does 𝑓_𝑖, where 𝑓_𝑖 are given to us:(i)𝑓₁(𝑧)=∫^𝑧₀(𝑓(𝑡)/𝑡)𝑑𝑡, (ii)𝑓₂(𝑧)=(2/𝑧)∫^𝑧₀𝑓(𝑡)𝑑𝑡, (iii)𝑓₃(𝑧)=∫^𝑧₀((𝑓(𝑡)−𝑓(𝑥𝑡))/(𝑡−𝑥𝑡))𝑑𝑡,  |𝑥|≤1,  𝑥≠1,(iv)𝑓₄(𝑧)=((1+𝑐)/𝑧^𝑐)∫^𝑧₀𝑡^𝑐−1𝑓(𝑡)𝑑𝑡,  ℝe(𝑐)>0.

The proof immediately follows from Theorem 18. Since we can write 𝑓_𝑖=𝑓∗𝜙_𝑖 with 𝜙₁(𝑧)(),𝜙=−log1−𝑧₂(𝑧)()=−2𝑧+log1−𝑧𝑧,𝜙₃(𝑧)=11−𝑥log1−𝑥𝑧,𝜙1−𝑧₄(𝑧)=^∞_𝑚=11+𝑐𝑧𝑚+𝑐^𝑚ℝe𝑐>0, (51)each 𝜙_𝑖 is convex for 𝑖=1,2,3, and 4.

Acknowledgment

The authors would like to express their thanks to the referee for the careful reading and suggestions made for the improvement of the paper.